American Mathematical Society

Sharp global well-posedness for KdV and modified KdV on double-struck upper R and double-struck upper T

By J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao

Abstract

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all upper L squared -based Sobolev spaces upper H Superscript s where local well-posedness is presently known, apart from the upper H Superscript one-fourth Baseline left-parenthesis double-struck upper R right-parenthesis endpoint for mKdV and the upper H Superscript negative three-fourths endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura’s transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

1. Introduction

The initial value problem for the Korteweg-de Vries (KdV) equation,

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column partial-differential Subscript t Baseline u plus partial-differential Subscript x Superscript 3 Baseline u plus one-half partial-differential Subscript x Baseline u squared equals 0 comma 2nd Column u colon double-struck upper R times left-bracket 0 comma upper T right-bracket long right-arrow from bar double-struck upper R comma 2nd Row 1st Column u left-parenthesis 0 right-parenthesis equals phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis comma EndLayout EndLayout

has been shown to be locally well-posed (LWP) for s greater-than negative three-fourths period Kenig, Ponce and Vega Reference32 extended the local-in-time analysis of Bourgain Reference5, valid for s greater-than-or-equal-to 0 , to the range s greater-than negative three-fourths by constructing the solution of Equation1.1 on a time interval left-bracket 0 comma delta right-bracket with delta depending upon double-vertical-bar phi double-vertical-bar Subscript upper H Sub Superscript s Subscript left-parenthesis double-struck upper R right-parenthesis . Earlier results can be found in Reference4, Reference28, Reference23, Reference31, Reference12. We prove here that these solutions exist for t in an arbitrary time interval left-bracket 0 comma upper T right-bracket thereby establishing global well-posedness (GWP) of Equation1.1 in the full range s greater-than negative three-fourths period The corresponding periodic double-struck upper R -valued initial value problem for KdV

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column partial-differential Subscript t Baseline u plus partial-differential Subscript x Superscript 3 Baseline u plus one-half partial-differential Subscript x Baseline u squared equals 0 comma 2nd Column u colon double-struck upper T times left-bracket 0 comma upper T right-bracket long right-arrow from bar double-struck upper R comma 2nd Row 1st Column u left-parenthesis 0 right-parenthesis equals phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper T right-parenthesis EndLayout EndLayout

is known Reference32 to be locally well-posed for s greater-than-or-equal-to negative one-half . These local-in-time solutions are also shown to exist on an arbitrary time interval. Bourgain established Reference9 global well-posedness of Equation1.2 for initial data having (small) bounded Fourier transform. The argument in Reference9 uses the complete integrability of KdV. Analogous globalizations of the best known local-in-time theory for the focussing and defocussing modified KdV (mKdV) equations ( u squared in Equation1.1, Equation1.2 replaced by minus u cubed and u cubed , respectively) are also obtained in the periodic left-parenthesis s greater-than-or-equal-to one-half right-parenthesis and real line ( s greater-than one-fourth right-parenthesis settings.

The local-in-time theory globalized here is sharp (at least up to certain endpoints) in the scale of upper L squared -based Sobolev spaces upper H Superscript s . Indeed, recent examples Reference33 of Kenig, Ponce and Vega (see also Reference2, Reference3) reveal that focussing mKdV is ill-posed for s less-than one-fourth and that double-struck upper C -valued KdV ( u colon double-struck upper R times left-bracket 0 comma upper T right-bracket long right-arrow from bar double-struck upper C ) is ill-posed for s less-than negative three-fourths . (The local theory in Reference32 adapts easily to the double-struck upper C -valued situation.) A similar failure of local well-posedness below the endpoint regularities for the defocusing modified KdV and the double-struck upper R -valued KdV has been established Reference13 by Christ, Colliander and Tao. The fundamental bilinear estimate used to prove the local well-posedness result on the line was shown to fail for s less-than-or-equal-to negative three-fourths by Nakanishi, Takaoka and Tsutsumi Reference45. Nevertheless, a conjugation of the upper H Superscript one-fourth local well-posedness theory for defocusing mKdV using the Miura transform established Reference13 a local well-posedness result for KdV at the endpoint upper H Superscript negative three-fourths Baseline left-parenthesis double-struck upper R right-parenthesis . Global well-posedness of KdV at the negative three-fourths endpoint and for mKdV in upper H Superscript one-fourth remain open problems.

1.1. GWP below the conservation law

double-struck upper R -valued solutions of KdV satisfy upper L squared conservation: double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Baseline equals double-vertical-bar phi double-vertical-bar Subscript upper L squared . Consequently, a local well-posedness result with the existence lifetime determined by the size of the initial data in upper L squared may be iterated to prove global well-posedness of KdV for upper L squared data Reference5. What happens to solutions of KdV which evolve from initial data which are less regular than upper L squared ? Bourgain observed, in a context Reference8 concerning very smooth solutions, that the nonlinear Duhamel term may be smoother than the initial data. This observation was exploited Reference8, using a decomposition of the evolution of the high and low frequency parts of the initial data, to prove polynomial-in-time bounds for global solutions of certain nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations. In Reference10, Bourgain introduced a general high/low frequency decomposition argument to prove that certain NLS and NLW equations were globally well-posed below upper H Superscript 1 , the natural regularity associated with the conserved Hamiltonian. Subsequently, Bourgain’s high/low method has been applied to prove global well-posedness below the natural regularity of the conserved quantity in various settings Reference20, Reference50, Reference48, Reference34, including KdV Reference18 on the line. A related argument—directly motivated by Bourgain’s work—appeared in Reference29, Reference30 where the presence of derivatives in the nonlinearities leaves a Duhamel term which cannot be shown to be smoother than the initial data. Global rough solutions for these equations are constructed with a slightly different use of the original conservation law (see below).

We summarize the adaptation Reference18 of the high/low method to construct a solution of Equation1.1 for rough initial data. The task is to construct the global solution of Equation1.1 evolving from initial data phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis for s 0 less-than s less-than 0 with negative 3 slash 4 much-less-than s 0 less-than-or-equivalent-to 0 . The argument Reference18 accomplishes this task for initial data in a subset of upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis consisting of functions with relatively small low frequency components. Split the data phi equals phi 0 plus psi 0 with ModifyingAbove phi 0 With caret left-parenthesis k right-parenthesis equals chi Subscript left-bracket negative upper N comma upper N right-bracket Baseline left-parenthesis k right-parenthesis ModifyingAbove phi With caret left-parenthesis k right-parenthesis , where upper N equals upper N left-parenthesis upper T right-parenthesis is a parameter to be determined. The low frequency part phi 0 of phi is in upper L squared left-parenthesis double-struck upper R right-parenthesis (in fact phi 0 element-of upper H Superscript s for all s ) with a big norm while the high frequency part psi 0 is the tail of an upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis function and is therefore small (with large upper N ) in upper H Superscript sigma Baseline left-parenthesis double-struck upper R right-parenthesis for any sigma less-than s . The low frequencies are evolved according to KdV: phi 0 long right-arrow from bar u 0 left-parenthesis t right-parenthesis period The high frequencies evolve according to a “difference equation” which is selected so that the sum of the resulting high frequency evolution, psi 0 long right-arrow from bar v 0 left-parenthesis t right-parenthesis and the low frequency evolution solves Equation1.1. The key step is to decompose v 0 left-parenthesis t right-parenthesis equals upper S left-parenthesis t right-parenthesis psi 0 plus w 0 left-parenthesis t right-parenthesis , where upper S left-parenthesis t right-parenthesis is the solution operator of the Airy equation. For the selected class of rough initial data mentioned above, one can then prove that w 0 element-of upper L squared left-parenthesis double-struck upper R right-parenthesis and has a small (depending upon upper N ) upper L squared norm. Then an iteration of the local-in-time theory advances the solution to a long (depending on upper N ) time interval. An appropriate choice of upper N completes the construction.

The nonlinear Duhamel term for the “difference equation” mentioned above is

w 0 left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript t Baseline upper S left-parenthesis t minus t Superscript prime Baseline right-parenthesis left-parenthesis left-bracket v 0 squared left-parenthesis t Superscript prime Baseline right-parenthesis plus 2 u 0 left-parenthesis t Superscript prime Baseline right-parenthesis v 0 left-parenthesis t Superscript prime Baseline right-parenthesis right-bracket right-parenthesis d t Superscript prime Baseline period

The local well-posedness machinery Reference5, Reference32 allows us to prove that w 0 left-parenthesis t right-parenthesis element-of upper L squared left-parenthesis double-struck upper R right-parenthesis if we have the extra smoothing bilinear estimate

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel double-vertical-bar partial-differential Subscript x Baseline left-parenthesis u v right-parenthesis double-vertical-bar Subscript upper X Sub Subscript 0 comma b minus 1 Subscript Baseline less-than-or-equivalent-to double-vertical-bar u double-vertical-bar Subscript upper X Sub Subscript s comma b Subscript Baseline double-vertical-bar v double-vertical-bar Subscript upper X Sub Subscript s comma b Subscript Baseline comma s less-than 0 comma b equals one-half plus comma EndLayout

with the space upper X Subscript s comma b defined below (see Equation1.11). The estimate Equation1.3 is valid for functions u comma v such that ModifyingAbove u With caret comma ModifyingAbove v With caret are supported outside StartSet StartAbsoluteValue k EndAbsoluteValue less-than-or-equal-to 1 EndSet , in the range negative three-eighths less-than s Reference18, Reference15. The estimate Equation1.3 fails for s less-than negative three-eighths and this places an intrinsic limitation on how far the high/low frequency decomposition technique may be used to extend GWP for rough initial data. Also, Equation1.3 fails without some assumptions on the low frequencies of u and v , hence the initial data considered in the high/low argument of Reference18. We showed that the low frequency issue may indeed be circumvented in Reference15 by proving Equation1.1 is GWP in upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis comma s greater-than negative three-tenths . The approach in Reference15 does not rely on showing the nonlinear Duhamel term has regularity at the level of the conservation law. We review this approach now and motivate the nontrivial improvements of that argument leading to sharp global regularity results for Equation1.1 and Equation1.2.

1.2. The operator upper I and almost conserved quantities

Global well-posedness follows from (an iteration of) local well-posedness (results) provided the successive local-in-time existence intervals cover an arbitrary time interval left-bracket 0 comma upper T right-bracket . The length of the local-in-time existence interval is controlled from below by the size of the initial data in an appropriate norm. A natural approach to global well-posedness in upper H Superscript s is to establish upper bounds on double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper H Sub Superscript s for solutions u left-parenthesis t right-parenthesis which are strong enough to prove that left-bracket 0 comma upper T right-bracket may be covered by iterated local existence intervals. We establish appropriate upper bounds to carry out this general strategy by constructing almost conserved quantities and rescaling. The rescaling exploits the subcritical nature of the KdV initial value problem (but introduces technical issues in the treatment of the periodic problem). The almost conserved quantities are motivated by the following discussion of the upper L squared conservation property of solutions of KdV.

Consider the following Fourier proofFootnote1 This argument was known previously; see a similar argument in Reference27.âś– that double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Baseline equals double-vertical-bar phi double-vertical-bar Subscript upper L squared Baseline for-all t element-of double-struck upper R . By Plancherel,

double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Superscript 2 Baseline equals integral ModifyingAbove u With caret left-parenthesis xi right-parenthesis ModifyingAbove Above ModifyingAbove u With caret With bar left-parenthesis xi right-parenthesis d xi comma

where

ModifyingAbove u With caret left-parenthesis xi right-parenthesis equals integral e Superscript minus i x xi Baseline u left-parenthesis x right-parenthesis d x

is the (spatial) Fourier transform. Fourier transform properties imply

integral ModifyingAbove u With caret left-parenthesis xi right-parenthesis ModifyingAbove Above ModifyingAbove u With caret With bar left-parenthesis xi right-parenthesis d xi equals integral ModifyingAbove u With caret left-parenthesis xi right-parenthesis ModifyingAbove Above ModifyingAbove u With bar With caret left-parenthesis negative xi right-parenthesis d xi equals integral Underscript xi 1 plus xi 2 equals 0 Endscripts ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove Above ModifyingAbove u With bar With caret left-parenthesis xi 2 right-parenthesis period

Since we are assuming u is double-struck upper R -valued, we may replace ModifyingAbove Above ModifyingAbove u With bar With caret left-parenthesis xi 2 right-parenthesis by ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis . Hence,

double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Superscript 2 Baseline equals integral Underscript xi 1 plus xi 2 equals 0 Endscripts ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis period

We apply partial-differential Subscript t , and we use symmetry and the equation to find

partial-differential Subscript t Baseline left-parenthesis double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Superscript 2 Baseline right-parenthesis equals 2 i integral Underscript xi 1 plus xi 2 equals 0 Endscripts xi 1 cubed ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis minus i integral Underscript xi 1 plus xi 2 equals 0 Endscripts xi 1 ModifyingAbove u squared With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis period

The first expression is symmetric under the interchange of xi 1 and xi 2 so xi 1 cubed may be replaced by one-half left-parenthesis xi 1 cubed plus xi 2 cubed right-parenthesis . Since we are integrating on the set where xi 1 plus xi 2 equals 0 , the integrand is zero and this term vanishes. Calculating ModifyingAbove u squared With caret left-parenthesis xi right-parenthesis equals integral Underscript xi equals xi 1 plus xi 2 Endscripts ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis , the remaining term may be rewritten

StartLayout 1st Row with Label left-parenthesis 1.4 right-parenthesis EndLabel minus i integral Underscript xi 1 plus xi 2 plus xi 3 equals 0 Endscripts left-bracket xi 1 plus xi 2 right-bracket ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 3 right-parenthesis period EndLayout

On the set where xi 1 plus xi 2 plus xi 3 equals 0 , xi 1 plus xi 2 equals minus xi 3 which we symmetrize to replace xi 1 plus xi 2 in Equation1.4 by minus one-third left-parenthesis xi 1 plus xi 2 plus xi 3 right-parenthesis and this term vanishes as well. Summarizing, we have found that double-struck upper R -valued solutions u left-parenthesis t right-parenthesis of KdV satisfy

StartLayout 1st Row with Label left-parenthesis 1.5 right-parenthesis EndLabel StartLayout 1st Row 1st Column partial-differential Subscript t Baseline left-parenthesis double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Superscript 2 Baseline right-parenthesis 2nd Column equals minus i integral Underscript xi 1 plus xi 2 equals 0 Endscripts left-parenthesis xi 1 cubed plus xi 2 cubed right-parenthesis ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis 2nd Row 1st Column Blank 2nd Column plus StartFraction i Over 3 EndFraction integral Underscript xi 1 plus xi 2 plus xi 3 equals 0 Endscripts left-parenthesis xi 1 plus xi 2 plus xi 3 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 3 right-parenthesis EndLayout EndLayout

and both integrands on the right side vanish.

We introduce the (spatial) Fourier multiplier operator upper I u defined via

ModifyingAbove upper I u With caret left-parenthesis xi right-parenthesis equals m left-parenthesis xi right-parenthesis ModifyingAbove u With caret left-parenthesis xi right-parenthesis

with an arbitrary double-struck upper C -valued multiplier m . A formal imitation of the Fourier proof of upper L squared -mass conservation above reveals that for double-struck upper R -valued solutions of KdV we have

StartLayout 1st Row with Label left-parenthesis 1.6 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column partial-differential Subscript t Baseline left-parenthesis double-vertical-bar upper I u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Superscript 2 Baseline right-parenthesis equals minus StartFraction i Over 2 EndFraction integral Underscript xi 1 plus xi 2 equals 0 Endscripts left-bracket m left-parenthesis xi 1 right-parenthesis ModifyingAbove m With bar left-parenthesis xi 1 right-parenthesis plus m left-parenthesis xi 2 right-parenthesis ModifyingAbove m With bar left-parenthesis xi 2 right-parenthesis right-bracket StartSet xi 1 cubed plus xi 2 cubed EndSet ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis 2nd Row 1st Column Blank 2nd Column plus StartFraction i Over 6 EndFraction integral Underscript xi 1 plus xi 2 plus xi 3 equals 0 Endscripts sigma-summation Underscript j equals 1 Overscript 3 Endscripts left-bracket m left-parenthesis minus xi Subscript j Baseline right-parenthesis ModifyingAbove m With bar left-parenthesis minus xi Subscript j Baseline right-parenthesis plus m left-parenthesis xi Subscript j Baseline right-parenthesis ModifyingAbove m With bar left-parenthesis xi Subscript j Baseline right-parenthesis right-bracket xi Subscript j Baseline ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 3 right-parenthesis period EndLayout EndLayout

The term arising from the dispersion cancels since xi 1 cubed plus xi 2 cubed equals 0 on the set where xi 1 plus xi 2 equals 0 . The remaining trilinear term can be analyzed under various assumptions on the multiplier m giving insight into the time behavior of double-vertical-bar upper I u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared . Moreover, the flexibility in our choice of m may allow us to observe how the conserved upper L squared mass is moved around in frequency space during the KdV evolution.

Remark 1.1

Our use of the multiplier m to localize the upper L squared mass in frequency space is analogous to the use of cutoff functions to spatially localize the conserved density in physical space. In that setting, the underlying conservation law partial-differential Subscript t Baseline left-parenthesis conserved density right-parenthesis plus partial-differential Subscript x Baseline left-parenthesis flux right-parenthesis equals 0 is multiplied by a cutoff function. The localized flux term is no longer a perfect derivative and is then estimated, sometimes under an appropriate choice of the cutoff, to obtain bounds on the spatially localized energy.

Consider now the problem of proving well-posedness of Equation1.1 or Equation1.2, with s less-than 0 , on an arbitrary time interval left-bracket 0 comma upper T right-bracket . We define a spatial Fourier multiplier operator upper I which acts like the identity on low frequencies and like a smoothing operator of order StartAbsoluteValue s EndAbsoluteValue on high frequencies by choosing a smooth monotone multiplier satisfying

m left-parenthesis xi right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 comma 2nd Column StartAbsoluteValue xi EndAbsoluteValue less-than upper N comma 2nd Row 1st Column upper N Superscript negative s Baseline StartAbsoluteValue xi EndAbsoluteValue Superscript s Baseline comma 2nd Column StartAbsoluteValue xi EndAbsoluteValue greater-than 2 upper N period EndLayout

The parameter upper N marks the transition from low to high frequencies. When upper N equals 1 , the operator upper I is essentially the integration (since s less-than 0 ) operator upper D Superscript s . When upper N equals normal infinity , upper I acts like the identity operator. Note that double-vertical-bar upper I phi double-vertical-bar Subscript upper L squared is bounded if phi element-of upper H Superscript s . We prove a variant local well-posedness result which shows the length of the local existence interval left-bracket 0 comma delta right-bracket for Equation1.1 or Equation1.2 may be bounded from below by double-vertical-bar upper I phi double-vertical-bar Subscript upper L squared Superscript negative alpha Baseline comma alpha greater-than 0 , for an appropriate range of the parameter s . The basic idea is then to bound the trilinear term in Equation1.6 to prove, for a particular small beta greater-than 0 , that

StartLayout 1st Row with Label left-parenthesis 1.7 right-parenthesis EndLabel sup Underscript t element-of left-bracket 0 comma delta right-bracket Endscripts double-vertical-bar upper I u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Baseline less-than-or-equal-to double-vertical-bar upper I u left-parenthesis 0 right-parenthesis double-vertical-bar Subscript upper L squared Baseline plus c upper N Superscript negative beta Baseline double-vertical-bar upper I u left-parenthesis 0 right-parenthesis double-vertical-bar Subscript upper L squared Superscript 3 Baseline period EndLayout

If upper N is huge, Equation1.7 shows there is at most a tiny increment in double-vertical-bar upper I u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared as t evolves from 0 to delta . An iteration of the local theory under appropriate parameter choices gives global well-posedness in upper H Superscript s for certain s less-than 0 .

The strategy just described is enhanced with two extra ingredients: a multilinear correction technique and rescaling. The correction technique shows that, up to errors of smaller order in upper N , the trilinear term in Equation1.6 may be replaced by a quintilinear term improving Equation1.7 to

StartLayout 1st Row with Label left-parenthesis 1.8 right-parenthesis EndLabel sup Underscript t element-of left-bracket 0 comma delta right-bracket Endscripts double-vertical-bar upper I u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Baseline less-than-or-equal-to double-vertical-bar upper I u left-parenthesis 0 right-parenthesis double-vertical-bar Subscript upper L squared Baseline plus c upper N Superscript negative 3 minus three-fourths plus epsilon Baseline double-vertical-bar upper I u left-parenthesis 0 right-parenthesis double-vertical-bar Subscript upper L squared Superscript 5 Baseline comma EndLayout

where epsilon is tiny. The rescaling argument reduces matters to initial data phi of fixed size: double-vertical-bar upper I phi double-vertical-bar Subscript upper L squared Baseline tilde epsilon 0 much-less-than 1 . In the periodic setting, the rescaling we use forces us to track the dependence upon the spatial period in the local well-posedness theory Reference5, Reference32.

The main results obtained here are:

Theorem 1

The double-struck upper R -valued initial value problem Equation1.1 is globally well-posed for initial data phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis comma s greater-than negative three-fourths period

Theorem 2

The double-struck upper R -valued periodic initial value problem Equation1.2 is globally well-posed for initial data phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper T right-parenthesis comma s greater-than-or-equal-to negative one-half .

Theorem 3

The double-struck upper R -valued initial value problem for modified KdV Equation9.1 (focussing or defocussing) is globally well-posed for initial data phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis comma s greater-than one-fourth .

Theorem 4

The double-struck upper R -valued periodic initial value problem for modified KdV (focussing or defocussing) is globally well-posed for initial data phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper T right-parenthesis comma s greater-than-or-equal-to one-half .

The infinite-dimensional symplectic nonsqueezing machinery developed by S. Kuksin Reference36 identifies upper H Superscript negative one-half Baseline left-parenthesis double-struck upper T right-parenthesis as the Hilbert Darboux (symplectic) phase space for KdV. We anticipate that Theorem 3 will be useful in adapting these ideas to the KdV context. The main remaining issue is an approximation of the KdV flow using finite-dimensional Hamiltonian flows analogous to that obtained by Bourgain Reference6 in the NLS setting. We plan to address this topic in a forthcoming paper.

We conclude this subsection with a discussion culminating in a table which summarizes the well-posedness theory in SobolevFootnote2 There are results, e.g. Reference9, in function spaces outside the upper L squared -based Sobolev scale.âś– spaces upper H Superscript s for the polynomial generalized KdV equations. The initial value problem

StartLayout 1st Row with Label left-parenthesis 1.9 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column partial-differential Subscript t Baseline u plus partial-differential Subscript x Superscript 3 Baseline u plus-or-minus StartFraction 1 Over k EndFraction partial-differential Subscript x Baseline u Superscript k Baseline equals 0 comma 2nd Column u colon double-struck upper R times left-bracket 0 comma upper T right-bracket long right-arrow from bar double-struck upper R comma 2nd Row 1st Column u left-parenthesis 0 right-parenthesis equals phi element-of upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis comma EndLayout EndLayout

has the associated Hamiltonian

StartLayout 1st Row with Label left-parenthesis 1.10 right-parenthesis EndLabel upper H left-bracket u right-bracket equals integral one-half u Subscript x Superscript 2 Baseline minus-or-plus StartFraction 1 Over k left-parenthesis k plus 1 right-parenthesis EndFraction u Superscript k plus 1 Baseline d x period EndLayout

The replacement u long right-arrow from bar negative u shows that the plus-or-minus choice is irrelevant when k is even, but, when k is odd there are two distinct cases in Equation1.9: plus is called focussing and minus is called defocussing. The usefulness of the Hamiltonian in controlling the upper H Superscript 1 norm can depend upon the minus-or-plus choice in Equation1.10.

We now summarize the well-posedness theory for the generalized KdV equations. The notation D and F in Table 1 refers to the defocussing and focussing cases. We highlight with the notation ?? some issues which are not yet resolved (as far as we are aware).

Table 1.

double-struck upper R -Valued Generalized KdV on double-struck upper R Well-posedness Summary Table

Our results here and elsewhere Reference16, Reference14, Reference17 suggest that local well-posedness implies global well-posedness in subcritical dispersive initial value problems. In particular, we believe our methods will extend to prove GWP of mKdV in upper H Superscript one-fourth Baseline left-parenthesis double-struck upper R right-parenthesis and KdV in upper H Superscript negative three-fourths Baseline left-parenthesis double-struck upper R right-parenthesis and also extend the GWP intervals in the cases k greater-than-or-equal-to 4 . However, our results rely on the fact that we are considering the double-struck upper R -valued KdV equation and, due to a lack of conservation laws, we do not know if the local results for the double-struck upper C -valued KdV equation may be similarly globalized. An adaptation of techniques from Reference13 may provide ill-posedness results in the higher power defocussing cases. Blow up in the focussing supercritical ( k greater-than-or-equal-to 6 or, more generally, k element-of double-struck upper R with k greater-than 5 ) is expected to occur but no rigorous results in this direction have been so far obtained Reference39.

1.3. Outline

Sections 2 and 3 describe the multilinear correction technique which generates modified energies. Section 4 establishes useful pointwise upper bounds on certain multipliers arising in the multilinear correction procedure. These upper bounds are combined with a quintilinear estimate, in the double-struck upper R setting, to prove the bulk of Equation1.8 in Section 5. Section 6 contains the variant local well-posedness result and the proof of global well-posedness for Equation1.1 in upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis comma s greater-than negative three-fourths period We next consider the periodic initial value problem Equation1.2 with period lamda . Section 7 extends the local well-posedness theory for Equation1.2 to the lamda -periodic setting. Section 8 proves global well-posedness of Equation1.2 in upper H Superscript s Baseline left-parenthesis double-struck upper T right-parenthesis comma s greater-than-or-equal-to negative one-half . The last section exploits Miura’s transform to prove the corresponding global well-posedness results for the focussing and defocussing modified KdV equations.

1.4. Notation

We will use c comma upper C to denote various time independent constants, usually depending only upon s . In case a constant depends upon other quantities, we will try to make that explicit. We use upper A less-than-or-equivalent-to upper B to denote an estimate of the form upper A less-than-or-equal-to upper C upper B . Similarly, we will write upper A tilde upper B to mean upper A less-than-or-equivalent-to upper B and upper B less-than-or-equivalent-to upper A . To avoid an issue involving a logarithm, we depart from standard practice and write mathematical left-angle k mathematical right-angle equals 2 plus StartAbsoluteValue k EndAbsoluteValue period The notation a plus denotes a plus epsilon for an arbitrarily small epsilon . Similarly, a minus denotes a minus epsilon . We will make frequent use of the two-parameter spaces upper X Subscript s comma b Baseline left-parenthesis double-struck upper R times double-struck upper R right-parenthesis with norm

StartLayout 1st Row with Label left-parenthesis 1.11 right-parenthesis EndLabel double-vertical-bar u double-vertical-bar Subscript upper X Sub Subscript s comma b Subscript Baseline equals left-parenthesis integral integral mathematical left-angle xi mathematical right-angle Superscript 2 s Baseline mathematical left-angle tau minus xi cubed mathematical right-angle Superscript 2 b Baseline StartAbsoluteValue ModifyingAbove u With caret left-parenthesis xi comma tau right-parenthesis EndAbsoluteValue squared d xi d tau right-parenthesis Superscript one-half Baseline period EndLayout

For any time interval upper I , we define the restricted spaces upper X Subscript s comma b Baseline left-parenthesis upper R times upper I right-parenthesis by the norm

double-vertical-bar u double-vertical-bar Subscript upper X Sub Subscript s comma b Subscript left-parenthesis double-struck upper R times upper I right-parenthesis Baseline equals inf left-brace double-vertical-bar upper U double-vertical-bar Subscript upper X Sub Subscript s comma b Subscript Baseline colon upper U vertical-bar Subscript double-struck upper R times upper I Baseline equals u right-brace period

These spaces were first used to systematically study nonlinear dispersive wave problems by Bourgain Reference5. Klainerman and Machedon Reference37 used similar ideas in their study of the nonlinear wave equation. The spaces appeared earlier in a different setting in the works Reference46, Reference1 of Rauch, Reed, and M. Beals. We will systematically ignore constants involving pi in the Fourier transform, except in Section 7. Other notation is introduced during the developments that follow.

2. Multilinear forms

In this section, we introduce notation for describing certain multilinear operators; see for example Reference41, Reference40. Bilinear versions of these operators will generate a sequence of almost conserved quantities involving higher order multilinear corrections.

Definition 1

A k-multiplier is a function m colon double-struck upper R Superscript k Baseline long right-arrow from bar double-struck upper C . A k -multiplier is symmetric if m left-parenthesis xi 1 comma xi 2 comma ellipsis comma xi Subscript k Baseline right-parenthesis equals m left-parenthesis sigma left-parenthesis xi 1 comma xi 2 comma ellipsis comma xi Subscript k Baseline right-parenthesis right-parenthesis for all sigma element-of upper S Subscript k , the group of all permutations on k objects. The symmetrization of a k -multiplier m is the multiplier

StartLayout 1st Row with Label left-parenthesis 2.1 right-parenthesis EndLabel left-bracket m right-bracket Subscript s y m Baseline left-parenthesis xi 1 comma xi 2 comma ellipsis comma xi Subscript k Baseline right-parenthesis equals StartFraction 1 Over k factorial EndFraction sigma-summation Underscript sigma element-of upper S Subscript k Baseline Endscripts m left-parenthesis sigma left-parenthesis xi 1 comma xi 2 comma ellipsis comma xi Subscript k Baseline right-parenthesis right-parenthesis period EndLayout

The domain of m is double-struck upper R Superscript k ; however, we will only be interested in m on the hyperplane xi 1 plus dot dot dot plus xi Subscript k Baseline equals 0 .

Definition 2

A k -multiplier generates a k-linear functional or k-form acting on k functions u 1 comma ellipsis comma u Subscript k Baseline ,

StartLayout 1st Row with Label left-parenthesis 2.2 right-parenthesis EndLabel normal upper Lamda Subscript k Baseline left-parenthesis m semicolon u 1 comma ellipsis comma u Subscript k Baseline right-parenthesis equals integral Underscript xi 1 plus dot dot dot plus xi Subscript k Baseline equals 0 Endscripts m left-parenthesis xi 1 comma ellipsis comma xi Subscript k Baseline right-parenthesis ModifyingAbove u 1 With caret left-parenthesis xi 1 right-parenthesis ellipsis ModifyingAbove u Subscript k Baseline With caret left-parenthesis xi Subscript k Baseline right-parenthesis period EndLayout

We will often apply normal upper Lamda Subscript k to k copies of the same function u in which case the dependence upon u may be suppressed in the notation: normal upper Lamda Subscript k Baseline left-parenthesis m semicolon u comma ellipsis comma u right-parenthesis may simply be written normal upper Lamda Subscript k Baseline left-parenthesis m right-parenthesis .

If m is symmetric, then normal upper Lamda Subscript k Baseline left-parenthesis m right-parenthesis is a symmetric k -linear functional.

As an example, suppose that u is an double-struck upper R -valued function. We calculate double-vertical-bar u double-vertical-bar Subscript upper L squared Superscript 2 Baseline equals integral ModifyingAbove u With caret left-parenthesis xi right-parenthesis ModifyingAbove Above ModifyingAbove u With caret With bar left-parenthesis xi right-parenthesis d xi equals integral Underscript xi 1 plus xi 2 equals 0 Endscripts ModifyingAbove u With caret left-parenthesis xi 1 right-parenthesis ModifyingAbove u With caret left-parenthesis xi 2 right-parenthesis equals normal upper Lamda 2 left-parenthesis 1 right-parenthesis period

The time derivative of a symmetric k -linear functional can be calculated explicitly if we assume that the function u satisfies a particular PDE. The following statement may be directly verified by using the KdV equation.

Proposition 1

Suppose u satisfies the KdV equation Equation1.1 and that m is a symmetric k -multiplier. Then

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel StartFraction d Over d t EndFraction normal upper Lamda Subscript k Baseline left-parenthesis m right-parenthesis equals normal upper Lamda Subscript k Baseline left-parenthesis m alpha Subscript k Baseline right-parenthesis minus i StartFraction k Over 2 EndFraction normal upper Lamda Subscript k plus 1 Baseline left-parenthesis m left-parenthesis xi 1 comma ellipsis comma xi Subscript k minus 1 Baseline comma xi Subscript k Baseline plus xi Subscript k plus 1 Baseline right-parenthesis StartSet xi Subscript k Baseline plus xi Subscript k plus 1 Baseline EndSet right-parenthesis comma EndLayout

where

StartLayout 1st Row with Label left-parenthesis 2.4 right-parenthesis EndLabel alpha Subscript k Baseline equals i left-parenthesis xi 1 cubed plus dot dot dot plus xi Subscript k Superscript 3 Baseline right-parenthesis period EndLayout

Note that the second term in Equation2.3 may be symmetrized.

3. Modified energies

Let m colon double-struck upper R long right-arrow from bar double-struck upper R be an arbitrary even double-struck upper R -valued 1-multiplier and define the associated operator by

StartLayout 1st Row with Label left-parenthesis 3.1 right-parenthesis EndLabel ModifyingAbove upper I f With caret left-parenthesis xi right-parenthesis equals m left-parenthesis xi right-parenthesis ModifyingAbove f With caret left-parenthesis xi right-parenthesis period EndLayout

We define the modified energy upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis by

upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis equals double-vertical-bar upper I u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Superscript 2 Baseline period

The name “modified energy” is in part justified since in case m equals 1 comma upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis equals double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper L squared Superscript 2 Baseline period We will show later that for m of a particular form, certain modified energies enjoy an almost conservation property. By Plancherel and the fact that m and u are double-struck upper R -valued,

upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 2 left-parenthesis m left-parenthesis xi 1 right-parenthesis m left-parenthesis xi 2 right-parenthesis right-parenthesis period

Using Equation2.3, we have

StartLayout 1st Row with Label left-parenthesis 3.2 right-parenthesis EndLabel StartFraction d Over d t EndFraction upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 2 left-parenthesis m left-parenthesis xi 1 right-parenthesis m left-parenthesis xi 2 right-parenthesis alpha 2 right-parenthesis minus i normal upper Lamda 3 left-parenthesis m left-parenthesis xi 1 right-parenthesis m left-parenthesis xi 2 plus xi 3 right-parenthesis StartSet xi 2 plus xi 3 EndSet right-parenthesis period EndLayout

The first term vanishes. We symmetrize the remaining term to get

StartFraction d Over d t EndFraction upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 3 left-parenthesis minus i left-bracket m left-parenthesis xi 1 right-parenthesis m left-parenthesis xi 2 plus xi 3 right-parenthesis left-parenthesis xi 2 plus xi 3 right-parenthesis right-bracket Subscript s y m Baseline right-parenthesis period

Note that the time derivative of upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis is a 3-linear expression. Let us denote

StartLayout 1st Row with Label left-parenthesis 3.3 right-parenthesis EndLabel upper M 3 left-parenthesis xi 1 comma xi 2 comma xi 3 right-parenthesis equals minus i left-bracket m left-parenthesis xi 1 right-parenthesis m left-parenthesis xi 2 plus xi 3 right-parenthesis StartSet xi 2 plus xi 3 EndSet right-bracket Subscript s y m Baseline period EndLayout

Observe that if m equals 1 , the symmetrization results in upper M 3 equals c left-parenthesis xi 1 plus xi 2 plus xi 3 right-parenthesis . This reproduces the Fourier proof of upper L squared -mass conservation from the introduction.

Form the new modified energy

upper E Subscript upper I Superscript 3 Baseline left-parenthesis t right-parenthesis equals upper E Subscript upper I Superscript 2 Baseline left-parenthesis t right-parenthesis plus normal upper Lamda 3 left-parenthesis sigma 3 right-parenthesis

where the symmetric 3-multiplier sigma 3 will be chosen momentarily to achieve a cancellation. Applying Equation2.3 gives

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel StartFraction d Over d t EndFraction upper E Subscript upper I Superscript 3 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 3 left-parenthesis upper M 3 right-parenthesis plus normal upper Lamda 3 left-parenthesis sigma 3 alpha 3 right-parenthesis plus normal upper Lamda 4 left-parenthesis minus i three-halves sigma 3 left-parenthesis xi 1 comma xi 2 comma xi 3 plus xi 4 right-parenthesis StartSet xi 3 plus xi 4 EndSet right-parenthesis period EndLayout

We choose

StartLayout 1st Row with Label left-parenthesis 3.5 right-parenthesis EndLabel sigma 3 equals minus StartFraction upper M 3 Over alpha 3 EndFraction EndLayout

to force the two normal upper Lamda 3 terms in Equation3.4 to cancel. With this choice, the time derivative of upper E Subscript upper I Superscript 3 Baseline left-parenthesis t right-parenthesis is a 4-linear expression normal upper Lamda 4 left-parenthesis upper M 4 right-parenthesis where

StartLayout 1st Row with Label left-parenthesis 3.6 right-parenthesis EndLabel upper M 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis equals minus i three-halves left-bracket sigma 3 left-parenthesis xi 1 comma xi 2 comma xi 3 plus xi 4 right-parenthesis StartSet xi 3 plus xi 4 EndSet right-bracket Subscript s y m Baseline period EndLayout

Upon defining

upper E Subscript upper I Superscript 4 Baseline left-parenthesis t right-parenthesis equals upper E Subscript upper I Superscript 3 Baseline left-parenthesis t right-parenthesis plus normal upper Lamda 4 left-parenthesis sigma 4 right-parenthesis

with

StartLayout 1st Row with Label left-parenthesis 3.7 right-parenthesis EndLabel sigma 4 equals minus StartFraction upper M 4 Over alpha 4 EndFraction comma EndLayout

we obtain

StartLayout 1st Row with Label left-parenthesis 3.8 right-parenthesis EndLabel StartFraction d Over d t EndFraction upper E Subscript upper I Superscript 4 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 5 left-parenthesis upper M 5 right-parenthesis EndLayout

where

StartLayout 1st Row with Label left-parenthesis 3.9 right-parenthesis EndLabel upper M 5 left-parenthesis xi 1 comma ellipsis comma xi 5 right-parenthesis equals minus 2 i left-bracket sigma 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 plus xi 5 right-parenthesis StartSet xi 4 plus xi 5 EndSet right-bracket Subscript s y m Baseline period EndLayout

This process can clearly be iterated to generate upper E Subscript upper I Superscript n satisfying

StartFraction d Over d t EndFraction upper E Subscript upper I Superscript n Baseline left-parenthesis t right-parenthesis equals normal upper Lamda Subscript n plus 1 Baseline left-parenthesis upper M Subscript n plus 1 Baseline right-parenthesis comma n equals 2 comma 3 comma period period period period

These higher degree corrections to the modified energy upper E Subscript upper I Superscript 2 may be of relevance in studying various qualitative aspects of the KdV evolution. However, for the purpose of showing GWP in upper H Superscript s Baseline left-parenthesis double-struck upper R right-parenthesis down to s greater-than negative three-fourths and in upper H Superscript s Baseline left-parenthesis double-struck upper T right-parenthesis down to s greater-than-or-equal-to negative one-half , we will see that almost conservation of upper E Subscript upper I Superscript 4 Baseline left-parenthesis t right-parenthesis suffices.

The modified energy construction process is illustrated in the case of the Dirichlet energy

upper E Subscript upper D Superscript 2 Baseline left-parenthesis t right-parenthesis equals double-vertical-bar partial-differential Subscript x Baseline u double-vertical-bar Subscript upper L Sub Subscript x Sub Superscript 2 Subscript Superscript 2 Baseline equals normal upper Lamda 2 left-parenthesis left-parenthesis i xi 1 right-parenthesis left-parenthesis i xi 2 right-parenthesis right-parenthesis period

Define upper E Subscript upper D Superscript 3 Baseline left-parenthesis t right-parenthesis equals upper E Subscript upper D Superscript 2 Baseline left-parenthesis t right-parenthesis plus normal upper Lamda 3 left-parenthesis sigma 3 right-parenthesis , and use Equation2.3 to see

partial-differential Subscript t Baseline upper E Subscript upper D Superscript 3 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 3 left-parenthesis left-bracket i left-parenthesis xi 1 plus xi 2 right-parenthesis i xi 3 StartSet xi 1 plus xi 2 EndSet right-bracket Subscript s y m Baseline right-parenthesis plus normal upper Lamda 3 left-parenthesis sigma 3 alpha 3 right-parenthesis plus normal upper Lamda 4 left-parenthesis upper M 4 right-parenthesis comma

where upper M 4 is explicitly obtained from sigma 3 . Noting that i left-parenthesis xi 1 plus xi 2 right-parenthesis i xi 3 StartSet xi 1 plus xi 2 EndSet equals minus xi 3 cubed on the set xi 1 plus xi 2 plus xi 3 equals 0 , we know that

partial-differential Subscript t Baseline upper E Subscript upper D Superscript 3 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 3 left-parenthesis minus one-third alpha 3 right-parenthesis plus normal upper Lamda 3 left-parenthesis sigma 3 alpha 3 right-parenthesis plus normal upper Lamda 4 left-parenthesis upper M 4 right-parenthesis period

The choice of sigma 3 equals one-third results in a cancellation of the normal upper Lamda 3 terms and

upper M 4 equals left-bracket StartSet xi 1 plus xi 2 EndSet right-bracket Subscript s y m Baseline equals xi 1 plus xi 2 plus xi 3 plus xi 4

so upper M 4 equals 0 .

Therefore, upper E Subscript upper D Superscript 3 Baseline left-parenthesis t right-parenthesis equals normal upper Lamda 2 left-parenthesis left-parenthesis i xi 1 right-parenthesis left-parenthesis i xi 2 right-parenthesis right-parenthesis plus normal upper Lamda 3 left-parenthesis one-third right-parenthesis is an exactly conserved quantity. The modified energy construction applied to the Dirichlet energy led us to the Hamiltonian for KdV. Applying the construction to higher order derivatives in upper L squared will similarly lead to the higher conservation laws of KdV.

4. Pointwise multiplier bounds

This section presents a detailed analysis of the multipliers upper M 3 comma upper M 4 comma upper M 5 which were introduced in the iteration process of the previous section. The analysis identifies cancellations resulting in pointwise upper bounds on these multipliers depending upon the relative sizes of the multiplier’s arguments. These bounds are applied to prove an almost conservation property in the next section. We begin by recording some arithmetic and calculus facts.

4.1. Arithmetic and calculus facts

The following arithmetic facts may be easily verified:

StartLayout 1st Row with Label left-parenthesis 4.1 right-parenthesis EndLabel xi 1 plus xi 2 plus xi 3 equals 0 long right double arrow alpha 3 equals xi 1 cubed plus xi 2 cubed plus xi 3 cubed equals 3 xi 1 xi 2 xi 3 period 2nd Row with Label left-parenthesis 4.2 right-parenthesis EndLabel xi 1 plus xi 2 plus xi 3 plus xi 4 equals 0 long right double arrow alpha 4 equals xi 1 cubed plus xi 2 cubed plus xi 3 cubed plus xi 4 cubed equals 3 left-parenthesis xi 1 plus xi 2 right-parenthesis left-parenthesis xi 1 plus xi 3 right-parenthesis left-parenthesis xi 1 plus xi 4 right-parenthesis period EndLayout

A related observation for the circle was exploited by C. Fefferman Reference19 and by Carleson and Sjölin Reference11 for curves with nonzero curvature. These properties were also observed by Rosales Reference47 and Equation4.1 was used by Bourgain in Reference5.

Definition 3

Let a and b be smooth functions of the real variable xi . We say that a is controlled by b if b is nonnegative and satisfies b left-parenthesis xi right-parenthesis tilde b left-parenthesis xi prime right-parenthesis for StartAbsoluteValue xi EndAbsoluteValue tilde StartAbsoluteValue xi prime EndAbsoluteValue and

StartLayout 1st Row 1st Column a left-parenthesis xi right-parenthesis 2nd Column equals 3rd Column upper O left-parenthesis b left-parenthesis xi right-parenthesis right-parenthesis comma 2nd Row 1st Column a prime left-parenthesis xi right-parenthesis 2nd Column equals 3rd Column upper O left-parenthesis StartFraction b left-parenthesis xi right-parenthesis Over StartAbsoluteValue xi EndAbsoluteValue EndFraction right-parenthesis comma 3rd Row 1st Column a double-prime left-parenthesis xi right-parenthesis 2nd Column equals 3rd Column upper O left-parenthesis StartFraction b left-parenthesis xi right-parenthesis Over StartAbsoluteValue xi EndAbsoluteValue squared EndFraction right-parenthesis comma EndLayout

for all nonzero xi .

With this notion, we can state the following forms of the mean value theorem.

Lemma 4.1

If a is controlled by b and StartAbsoluteValue eta EndAbsoluteValue much-less-than StartAbsoluteValue xi EndAbsoluteValue , then

StartLayout 1st Row with Label left-parenthesis 4.3 right-parenthesis EndLabel a left-parenthesis xi plus eta right-parenthesis minus a left-parenthesis xi right-parenthesis equals upper O left-parenthesis StartAbsoluteValue eta EndAbsoluteValue StartFraction b left-parenthesis xi right-parenthesis Over StartAbsoluteValue xi EndAbsoluteValue EndFraction right-parenthesis period EndLayout

Lemma 4.2

If a is controlled by b and StartAbsoluteValue eta EndAbsoluteValue comma StartAbsoluteValue lamda EndAbsoluteValue much-less-than StartAbsoluteValue xi EndAbsoluteValue , then

StartLayout 1st Row with Label left-parenthesis 4.4 right-parenthesis EndLabel a left-parenthesis xi plus eta plus lamda right-parenthesis minus a left-parenthesis xi plus eta right-parenthesis minus a left-parenthesis xi plus lamda right-parenthesis plus a left-parenthesis xi right-parenthesis equals upper O left-parenthesis StartAbsoluteValue eta EndAbsoluteValue StartAbsoluteValue lamda EndAbsoluteValue StartFraction b left-parenthesis xi right-parenthesis Over StartAbsoluteValue xi EndAbsoluteValue squared EndFraction right-parenthesis period EndLayout

We will sometimes refer to our use of Equation4.4 as applying the double mean value theorem.

4.2. upper M 3 bound

The multiplier upper M 3 was defined in Equation3.3. In this section, we will generally be considering an arbitrary even double-struck upper R -valued 1-multiplier m . We will specialize to the situation when m is of the form Equation4.7 below. Recalling that xi 1 plus xi 2 plus xi 3 equals 0 and that m is even allows us to re-express Equation3.3 as

StartLayout 1st Row with Label left-parenthesis 4.5 right-parenthesis EndLabel upper M 3 left-parenthesis xi 1 comma xi 2 comma xi 3 right-parenthesis equals minus i left-bracket m squared left-parenthesis xi 1 right-parenthesis xi 1 right-bracket Subscript s y m Baseline equals minus StartFraction i Over 3 EndFraction left-bracket m squared left-parenthesis xi 1 right-parenthesis xi 1 plus m squared left-parenthesis xi 2 right-parenthesis xi 2 plus m squared left-parenthesis xi 3 right-parenthesis xi 3 right-bracket period EndLayout

Lemma 4.3

If m is even double-struck upper R -valued and m squared is controlled by itself, then, on the set xi 1 plus xi 2 plus xi 3 equals 0 comma StartAbsoluteValue xi Subscript i Baseline EndAbsoluteValue tilde upper N Subscript i Baseline (dyadic),

StartLayout 1st Row with Label left-parenthesis 4.6 right-parenthesis EndLabel StartAbsoluteValue upper M 3 left-parenthesis xi 1 comma xi 2 comma xi 3 right-parenthesis EndAbsoluteValue less-than-or-equivalent-to max left-parenthesis m squared left-parenthesis xi 1 right-parenthesis comma m squared left-parenthesis xi 2 right-parenthesis comma m squared left-parenthesis xi 3 right-parenthesis right-parenthesis min left-parenthesis upper N 1 comma upper N 2 comma upper N 3 right-parenthesis period EndLayout

Proof.

Symmetry allows us to assume upper N 1 equals upper N 2 greater-than-or-equal-to upper N 3 . In case upper N 3 much-less-than upper N 1 , the claimed estimate is equivalent to showing

m squared left-parenthesis xi 1 right-parenthesis xi 1 minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis left-parenthesis xi 1 plus xi 3 right-parenthesis plus m squared left-parenthesis xi 3 right-parenthesis xi 3 less-than-or-equal-to max left-parenthesis m squared left-parenthesis xi 1 right-parenthesis comma m squared left-parenthesis xi 3 right-parenthesis right-parenthesis upper N 3 period

But this easily follows when we rewrite the left side as left-parenthesis m squared left-parenthesis xi 1 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis right-parenthesis xi 1 minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis xi 3 plus m squared left-parenthesis xi 3 right-parenthesis xi 3 and use Equation4.3. In case upper N 3 tilde upper N 2 , Equation4.6 may be directly verified.

â– 

In the particular case when the multiplier m left-parenthesis xi right-parenthesis is smooth, monotone, and of the form

StartLayout 1st Row with Label left-parenthesis 4.7 right-parenthesis EndLabel m left-parenthesis xi right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 comma 2nd Column StartAbsoluteValue xi EndAbsoluteValue less-than upper N comma 2nd Row 1st Column upper N Superscript negative s Baseline StartAbsoluteValue xi EndAbsoluteValue Superscript s Baseline comma 2nd Column StartAbsoluteValue xi EndAbsoluteValue greater-than 2 upper N comma EndLayout EndLayout

we have

StartLayout 1st Row with Label left-parenthesis 4.8 right-parenthesis EndLabel StartAbsoluteValue upper M 3 left-parenthesis xi 1 comma xi 2 comma xi 3 right-parenthesis EndAbsoluteValue less-than-or-equal-to min left-parenthesis upper N 1 comma upper N 2 comma upper N 3 right-parenthesis period EndLayout

4.3. upper M 4 bound.

This subsection establishes the following pointwise upper bound on the multiplier upper M 4 .

Lemma 4.4

Assume m is of the form Equation4.7. In the region where StartAbsoluteValue xi Subscript i Baseline EndAbsoluteValue tilde upper N Subscript i Baseline comma StartAbsoluteValue xi Subscript j Baseline plus xi Subscript k Baseline EndAbsoluteValue tilde upper N Subscript j k Baseline for upper N Subscript i Baseline comma upper N Subscript j k Baseline dyadic,

StartLayout 1st Row with Label left-parenthesis 4.9 right-parenthesis EndLabel StartAbsoluteValue upper M 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis EndAbsoluteValue less-than-or-equivalent-to StartFraction StartAbsoluteValue alpha 4 EndAbsoluteValue m squared left-parenthesis min left-parenthesis upper N Subscript i Baseline comma upper N Subscript j k Baseline right-parenthesis right-parenthesis Over left-parenthesis upper N plus upper N 1 right-parenthesis left-parenthesis upper N plus upper N 2 right-parenthesis left-parenthesis upper N plus upper N 3 right-parenthesis left-parenthesis upper N plus upper N 4 right-parenthesis EndFraction period EndLayout

We begin by deriving two explicit representations of upper M 4 in terms of m . These identities are then analyzed in cases to prove Equation4.9.

Recall that,

StartLayout 1st Row with Label left-parenthesis 4.10 right-parenthesis EndLabel upper M 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis equals c left-bracket sigma 3 left-parenthesis xi 1 comma xi 2 comma xi 3 plus xi 4 right-parenthesis left-parenthesis xi 3 plus xi 4 right-parenthesis right-bracket Subscript s y m Baseline comma EndLayout

where sigma 3 equals minus StartFraction upper M 3 Over alpha 3 EndFraction and

StartLayout 1st Row with Label left-parenthesis 4.11 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper M 3 left-parenthesis x 1 comma x 2 comma x 3 right-parenthesis 2nd Column equals minus i left-bracket m left-parenthesis x 1 right-parenthesis m left-parenthesis x 2 plus x 3 right-parenthesis left-parenthesis x 2 plus x 3 right-parenthesis right-bracket Subscript s y m Baseline 2nd Row 1st Column Blank 2nd Column equals minus StartFraction i Over 3 EndFraction left-bracket m squared left-parenthesis x 1 right-parenthesis x 1 plus m squared left-parenthesis x 2 right-parenthesis x 2 plus m squared left-parenthesis x 3 right-parenthesis x 3 right-bracket comma EndLayout EndLayout

and alpha 3 left-parenthesis x 1 comma x 2 comma x 3 right-parenthesis equals x 1 cubed plus x 2 cubed plus x 3 cubed equals 3 x 1 x 2 x 3 . We shall ignore the irrelevant constant in Equation4.10. Therefore,

StartLayout 1st Row with Label left-parenthesis 4.12 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper M 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis 2nd Column equals minus one-half left-bracket StartFraction m squared left-parenthesis xi 1 right-parenthesis xi 1 plus m squared left-parenthesis xi 2 right-parenthesis xi 2 plus m squared left-parenthesis xi 3 plus xi 4 right-parenthesis left-parenthesis xi 3 plus xi 4 right-parenthesis Over 3 xi 1 xi 2 EndFraction right-bracket Subscript s y m Baseline 2nd Row 1st Column Blank 2nd Column equals minus one-half left-bracket StartFraction 2 m squared left-parenthesis xi 1 right-parenthesis xi 1 plus m squared left-parenthesis xi 3 plus xi 4 right-parenthesis left-parenthesis xi 3 plus xi 4 right-parenthesis Over 3 xi 1 xi 2 EndFraction right-bracket Subscript s y m Baseline period EndLayout EndLayout

Recall also from Equation4.2 that

StartLayout 1st Row with Label left-parenthesis 4.13 right-parenthesis EndLabel StartLayout 1st Row 1st Column alpha 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis 2nd Column equals xi 1 cubed plus xi 2 cubed plus xi 3 cubed plus xi 4 cubed 2nd Row 1st Column Blank 2nd Column equals 3 left-parenthesis xi 1 xi 2 xi 3 plus xi 1 xi 2 xi 4 plus xi 1 xi 3 xi 4 plus xi 2 xi 3 xi 4 right-parenthesis 3rd Row 1st Column Blank 2nd Column equals 3 left-parenthesis xi 1 plus xi 2 right-parenthesis left-parenthesis xi 1 plus xi 3 right-parenthesis left-parenthesis xi 1 plus xi 4 right-parenthesis period EndLayout EndLayout

We can now rewrite the first term in Equation4.12

StartLayout 1st Row with Label left-parenthesis 4.14 right-parenthesis EndLabel StartLayout 1st Row 1st Column left-bracket StartFraction 2 m squared left-parenthesis xi 1 right-parenthesis xi 1 xi 3 xi 4 Over 3 xi 1 xi 2 xi 3 xi 4 EndFraction right-bracket Subscript s y m 2nd Column equals two-ninths left-bracket StartFraction m squared left-parenthesis xi 1 right-parenthesis left-parenthesis xi 1 xi 2 xi 3 plus xi 1 xi 2 xi 4 plus xi 1 xi 3 xi 4 plus xi 2 xi 3 xi 4 minus xi 2 xi 3 xi 4 right-parenthesis Over xi 1 xi 2 xi 3 xi 4 EndFraction right-bracket Subscript s y m Baseline 2nd Row 1st Column Blank 2nd Column equals one-fifty-fourth left-bracket m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis plus m squared left-parenthesis xi 3 right-parenthesis plus m squared left-parenthesis xi 4 right-parenthesis right-bracket StartFraction alpha 4 Over xi 1 xi 2 xi 3 xi 4 EndFraction 3rd Row 1st Column Blank 2nd Column minus one-eighteenth left-bracket StartFraction m squared left-parenthesis xi 1 right-parenthesis Over xi 1 EndFraction plus StartFraction m squared left-parenthesis xi 2 right-parenthesis Over xi 2 EndFraction plus StartFraction m squared left-parenthesis xi 3 right-parenthesis Over xi 3 EndFraction plus StartFraction m squared left-parenthesis xi 4 right-parenthesis Over xi 4 EndFraction right-bracket period EndLayout EndLayout

The second term in Equation4.12 is rewritten, using xi 1 plus xi 2 plus xi 3 plus xi 4 equals 0 , and the fact the m is even,

StartLayout 1st Row with Label left-parenthesis 4.15 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column left-bracket StartFraction minus m squared left-parenthesis xi 1 plus xi 2 right-parenthesis left-parenthesis xi 1 plus xi 2 right-parenthesis xi 3 xi 4 Over 3 xi 1 xi 2 xi 3 xi 4 EndFraction right-bracket Subscript s y m 2nd Row 1st Column Blank 2nd Column equals minus one-eighteenth left-brace StartFraction m squared left-parenthesis xi 1 plus xi 2 right-parenthesis left-parenthesis xi 1 xi 3 xi 4 plus xi 2 xi 3 xi 4 right-parenthesis plus m squared left-parenthesis xi 3 plus xi 4 right-parenthesis left-parenthesis xi 1 xi 2 xi 3 plus xi 1 xi 3 xi 4 right-parenthesis Over xi 1 xi 2 xi 3 xi 4 EndFraction 3rd Row 1st Column Blank 2nd Column plus StartFraction m squared left-parenthesis xi 1 plus xi 3 right-parenthesis left-parenthesis xi 1 xi 2 xi 4 plus xi 2 xi 3 xi 4 right-parenthesis plus m squared left-parenthesis xi 2 plus xi 4 right-parenthesis left-parenthesis xi 1 xi 2 xi 3 plus xi 1 xi 3 xi 4 right-parenthesis Over xi 1 xi 2 xi 3 xi 4 EndFraction 4th Row 1st Column Blank 2nd Column plus StartFraction m squared left-parenthesis xi 1 plus xi 4 right-parenthesis left-parenthesis xi 1 xi 2 xi 3 plus xi 2 xi 3 xi 4 right-parenthesis plus m squared left-parenthesis xi 2 plus xi 3 right-parenthesis left-parenthesis xi 1 xi 2 xi 4 plus xi 1 xi 3 xi 4 right-parenthesis Over xi 1 xi 2 xi 3 xi 4 EndFraction right-brace 5th Row 1st Column Blank 2nd Column equals minus one-fifty-fourth StartFraction alpha 4 Over xi 1 xi 2 xi 3 xi 4 EndFraction left-bracket m squared left-parenthesis xi 1 plus xi 2 right-parenthesis plus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis plus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-bracket period EndLayout EndLayout

We record two identities for upper M 4 .

Lemma 4.5

If m is even and double-struck upper R -valued, the following two identities for upper M 4 are valid:

StartLayout 1st Row with Label left-parenthesis 4.16 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper M 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis 2nd Column equals minus StartFraction 1 Over 108 EndFraction StartFraction alpha 4 Over xi 1 xi 2 xi 3 xi 4 EndFraction left-bracket m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis plus m squared left-parenthesis xi 3 right-parenthesis plus m squared left-parenthesis xi 4 right-parenthesis 2nd Row 1st Column Blank 2nd Column minus m squared left-parenthesis xi 1 plus xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-bracket 3rd Row 1st Column Blank 2nd Column plus one-thirty-sixth StartSet StartFraction m squared left-parenthesis xi 1 right-parenthesis Over xi 1 EndFraction plus StartFraction m squared left-parenthesis xi 2 right-parenthesis Over xi 2 EndFraction plus StartFraction m squared left-parenthesis xi 3 right-parenthesis Over xi 3 EndFraction plus StartFraction m squared left-parenthesis xi 4 right-parenthesis Over xi 4 EndFraction EndSet 4th Row 1st Column Blank 2nd Column colon equals upper I plus upper I upper I period EndLayout EndLayout

StartLayout 1st Row with Label left-parenthesis 4.17 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column upper M 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis equals minus one-thirty-sixth StartFraction 1 Over xi 1 xi 2 xi 3 xi 4 EndFraction times 2nd Row 1st Column Blank 2nd Column left-brace xi 1 xi 2 xi 3 left-bracket m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis plus m squared left-parenthesis xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-bracket 3rd Row 1st Column Blank 2nd Column plus xi 1 xi 2 xi 4 left-bracket m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis plus m squared left-parenthesis xi 4 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-bracket 4th Row 1st Column Blank 2nd Column plus xi 1 xi 3 xi 4 left-bracket m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 3 right-parenthesis plus m squared left-parenthesis xi 4 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-bracket 5th Row 1st Column Blank 2nd Column plus xi 2 xi 3 xi 4 left-bracket m squared left-parenthesis xi 2 right-parenthesis plus m squared left-parenthesis xi 3 right-parenthesis plus m squared left-parenthesis xi 4 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-bracket right-brace period EndLayout EndLayout

Proof.

The identity Equation4.16 was established above. The identity Equation4.17 follows from Equation4.16 upon expanding alpha 4 and writing the second term in Equation4.16 on a common denominator.

â– 

Proof of Lemma Equation4.4.

The proof consists of a case-by-case analysis pivoting on the relative sizes of upper N Subscript i Baseline comma upper N Subscript j k Baseline . Symmetry properties of upper M 4 permit us to assume that StartAbsoluteValue xi 1 EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue xi 2 EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue xi 3 EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue xi 4 EndAbsoluteValue . Consequently, we assume upper N 1 greater-than-or-equal-to upper N 2 greater-than-or-equal-to upper N 3 greater-than-or-equal-to upper N 4 period Since m squared left-parenthesis xi right-parenthesis equals 1 for xi less-than StartFraction upper N Over 2 EndFraction , a glance at Equation4.12 shows that upper M 4 vanishes when StartAbsoluteValue xi 1 EndAbsoluteValue less-than StartFraction upper N Over 4 EndFraction period We may therefore assume that StartAbsoluteValue xi 1 EndAbsoluteValue greater-than-or-equivalent-to upper N . Since xi 1 plus xi 2 plus xi 3 plus xi 4 equals 0 , we must also have StartAbsoluteValue xi 2 EndAbsoluteValue greater-than-or-equivalent-to upper N .

From Equation4.13, we know that we can replace alpha 4 on the right side of Equation4.9 by upper N 12 upper N 13 upper N 14 . Suppose upper N 12 less-than StartFraction upper N 1 Over 2 EndFraction comma upper N 13 less-than StartFraction upper N 1 Over 2 EndFraction comma upper N 14 less-than StartFraction upper N 1 Over 2 EndFraction period Then, xi 1 tilde minus xi 2 comma xi 1 tilde minus xi 3 and xi 1 tilde minus xi 4 so xi 1 plus xi 2 plus xi 3 plus xi 4 tilde minus 2 xi 1 not-equals 0 . Thus, at least one of upper N 12 comma upper N 13 comma upper N 14 must be at least of size comparable to upper N 1 . The right side of Equation4.9 may be re-expressed as

StartLayout 1st Row with Label left-parenthesis 4.18 right-parenthesis EndLabel StartFraction upper N 12 upper N 13 upper N 14 m squared left-parenthesis min left-parenthesis upper N Subscript i Baseline comma upper N Subscript j k Baseline right-parenthesis right-parenthesis Over upper N 1 squared left-parenthesis upper N plus upper N 3 right-parenthesis left-parenthesis upper N plus upper N 4 right-parenthesis EndFraction period EndLayout

Case 1. StartAbsoluteValue upper N 4 EndAbsoluteValue greater-than-or-equivalent-to StartFraction upper N Over 2 EndFraction .

Term upper I in Equation4.16 is bounded by StartFraction upper N 12 upper N 13 upper N 14 Over upper N 1 squared upper N 3 upper N 4 EndFraction m squared left-parenthesis min left-parenthesis upper N Subscript i Baseline comma upper N Subscript j k Baseline right-parenthesis right-parenthesis , and therefore, after cancelling max left-parenthesis upper N 12 comma upper N 13 comma upper N 14 right-parenthesis with one of the upper N 1 , satisfies Equation4.9. Term upper I upper I is treated next. In case upper N 12 comma upper N 13 comma upper N 14 greater-than-or-equivalent-to upper N 1 , Equation4.18 is an upper bound of StartFraction upper N 1 Over upper N 3 upper N 4 EndFraction m squared left-parenthesis upper N 4 right-parenthesis greater-than-or-equal-to StartFraction m squared left-parenthesis upper N 4 right-parenthesis Over upper N 4 EndFraction and the triangle inequality gives StartAbsoluteValue upper I upper I EndAbsoluteValue less-than-or-equivalent-to StartFraction m squared left-parenthesis upper N 4 right-parenthesis Over upper N 4 EndFraction since StartFraction m squared left-parenthesis dot right-parenthesis Over left-parenthesis dot right-parenthesis EndFraction is a decreasing function. If upper N 12 greater-than-or-equivalent-to upper N 1 comma upper N 13 much-less-than upper N 1 and upper N 14 greater-than-or-equivalent-to upper N 1 , we rewrite

StartAbsoluteValue upper I upper I EndAbsoluteValue tilde StartSet StartFraction m squared left-parenthesis xi 1 right-parenthesis Over xi 1 EndFraction plus StartFraction m squared left-parenthesis minus xi 1 plus left-parenthesis xi 1 plus xi 3 right-parenthesis right-parenthesis Over left-parenthesis minus xi 1 plus left-parenthesis xi 1 plus xi 3 right-parenthesis right-parenthesis EndFraction plus StartFraction m squared left-parenthesis xi 2 right-parenthesis Over xi 2 EndFraction plus StartFraction m squared left-parenthesis minus xi 2 plus left-parenthesis xi 2 plus xi 4 right-parenthesis right-parenthesis Over left-parenthesis minus xi 2 plus left-parenthesis xi 2 plus xi 4 right-parenthesis right-parenthesis EndFraction EndSet period

Applying the mean value theorem and using xi 1 plus xi 2 plus xi 3 plus xi 4 equals 0 gives StartAbsoluteValue upper I upper I EndAbsoluteValue less-than-or-equivalent-to left-parenthesis StartFraction m squared left-parenthesis xi 1 overTilde right-parenthesis Over xi 1 overTilde EndFraction right-parenthesis prime left-parenthesis xi 1 plus xi 3 right-parenthesis less-than-or-equivalent-to StartFraction upper N 13 Over upper N 1 squared EndFraction m squared left-parenthesis upper N 1 right-parenthesis since xi 1 overTilde equals xi 1 plus upper O left-parenthesis upper N 13 right-parenthesis and upper N 13 much-less-than upper N 1 , so this subcase is fine. If upper N 12 much-less-than upper N 1 comma upper N 13 much-less-than upper N 1 and upper N 14 greater-than-or-equivalent-to upper N 1 , the double mean value theorem Equation4.4 applied to term upper I upper I gives the bound

StartAbsoluteValue upper I upper I EndAbsoluteValue tilde left-parenthesis StartFraction m squared left-parenthesis xi 1 right-parenthesis Over xi 1 cubed EndFraction right-parenthesis double-prime left-parenthesis xi 1 plus xi 2 right-parenthesis left-parenthesis xi 1 plus xi 3 right-parenthesis period

Our assumptions on upper N 12 comma upper N 13 give the bound StartAbsoluteValue upper I upper I EndAbsoluteValue less-than-or-equivalent-to StartFraction upper N 12 upper N 13 Over upper N 1 cubed EndFraction m squared left-parenthesis upper N 1 right-parenthesis which is smaller than Equation4.18.

The remaining subcases have either precisely one element of the set StartSet upper N 12 comma upper N 13 comma upper N 14 EndSet much smaller than upper N 1 or precisely two elements much smaller than upper N 1 . In the case of just one small upper N Subscript 1 j , we apply the mean value theorem as above. When there are two small upper N Subscript 1 j , we apply the double mean value theorem as above.

Case 2. StartAbsoluteValue upper N 4 EndAbsoluteValue much-less-than StartFraction upper N Over 2 EndFraction period

Certainly, m squared left-parenthesis min left-parenthesis upper N Subscript i Baseline comma upper N Subscript j k Baseline right-parenthesis right-parenthesis equals 1 in this region. It is not possible for both upper N 12 less-than StartFraction upper N 1 Over 4 EndFraction and upper N 13 less-than StartFraction upper N 1 Over 4 EndFraction in this region. Indeed, we find then that xi 1 tilde minus xi 2 and xi 1 tilde minus xi 3 which with xi 1 plus xi 2 plus xi 3 plus xi 4 equals 0 implies xi 4 tilde xi 1 but StartAbsoluteValue xi 4 EndAbsoluteValue much-less-than StartFraction upper N Over 2 EndFraction while StartAbsoluteValue xi 1 EndAbsoluteValue tilde upper N 1 greater-than-or-equivalent-to upper N . We need to show upper M 4 less-than-or-equal-to StartFraction upper N 12 upper N 13 Over upper N 1 left-parenthesis upper N plus upper N 3 right-parenthesis upper N EndFraction .

Case 2A. StartFraction upper N 1 Over 4 EndFraction greater-than upper N 12 greater-than-or-equivalent-to StartFraction upper N Over 2 EndFraction comma upper N 13 tilde upper N 1 .

Since upper N 4 much-less-than StartFraction upper N Over 2 EndFraction and xi 1 plus xi 2 plus xi 3 plus xi 4 equals 0 , we must have upper N 12 tilde upper N 3 . So upper N plus upper N 3 tilde upper N 3 and our goal is to show upper M 4 less-than-or-equivalent-to StartFraction upper N 12 Over upper N 3 upper N EndFraction tilde StartFraction 1 Over upper N EndFraction period The last three terms in Equation4.17 are all upper O left-parenthesis StartFraction 1 Over upper N EndFraction right-parenthesis , which is fine. The first term in Equation4.17 is

StartFraction 1 Over 18 xi 4 EndFraction left-parenthesis m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis plus m squared left-parenthesis xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-parenthesis period

Replacing xi 1 plus xi 2 by minus left-parenthesis xi 3 plus xi 4 right-parenthesis and xi 1 plus xi 3 by minus left-parenthesis xi 2 plus xi 4 right-parenthesis , we identify three differences poised for the mean value theorem. We find this term equals

StartFraction 1 Over 18 xi 4 EndFraction left-bracket left-parenthesis m squared left-parenthesis xi 1 overTilde right-parenthesis right-parenthesis prime plus left-parenthesis m squared left-parenthesis xi 2 overTilde right-parenthesis right-parenthesis prime plus left-parenthesis m squared left-parenthesis xi 3 overTilde right-parenthesis right-parenthesis prime right-bracket xi 4

with xi Subscript i Baseline overTilde equals xi Subscript i Baseline plus upper O left-parenthesis upper N 4 right-parenthesis for i equals 1 comma 2 comma 3 so StartAbsoluteValue xi Subscript i Baseline overTilde EndAbsoluteValue tilde upper N Subscript i . This expression is also upper O left-parenthesis StartFraction 1 Over upper N EndFraction right-parenthesis .

Case 2B. upper N 12 much-less-than StartFraction upper N Over 2 EndFraction comma upper N 13 tilde upper N 1 .

Since upper N 12 equals upper N 34 and upper N 4 much-less-than StartFraction upper N Over 2 EndFraction , we must have upper N 3 much-less-than StartFraction upper N Over 2 EndFraction . We have upper N 13 tilde upper N 1 and upper N 14 tilde upper N 1 here so our desired upper bound is StartFraction upper N 12 Over upper N squared EndFraction . We recall Equation4.16 and evaluate m squared when we can to find

StartLayout 1st Row with Label left-parenthesis 4.19 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper M 4 left-parenthesis xi 1 comma xi 2 comma xi 3 comma xi 4 right-parenthesis 2nd Column equals StartFraction alpha 4 Over 54 xi 1 xi 2 xi 3 xi 4 EndFraction left-parenthesis m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis plus 2 minus 1 minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column minus one-eighteenth left-parenthesis StartFraction m squared left-parenthesis xi 1 right-parenthesis Over xi 1 EndFraction plus StartFraction m squared left-parenthesis xi 2 right-parenthesis Over xi 2 EndFraction plus StartFraction xi 3 plus xi 4 Over xi 3 xi 4 EndFraction right-parenthesis period EndLayout EndLayout

The last term is dangerous so we isolate a piece of the first term to cancel it out. Expanding alpha 4 equals 3 left-parenthesis xi 1 plus xi 2 right-parenthesis left-parenthesis xi 1 plus xi 3 right-parenthesis left-parenthesis xi 1 plus xi 4 right-parenthesis , we see that

StartLayout 1st Row 1st Column StartFraction alpha 4 Over 54 xi 1 xi 2 xi 3 xi 4 EndFraction 2nd Column equals 3rd Column one-eighteenth StartFraction left-parenthesis xi 3 plus xi 4 right-parenthesis Over xi 3 xi 4 EndFraction StartFraction left-parenthesis xi 2 plus xi 4 right-parenthesis left-parenthesis xi 1 plus xi 4 right-parenthesis Over xi 1 xi 2 EndFraction 2nd Row 1st Column Blank 2nd Column equals 3rd Column one-eighteenth StartFraction left-parenthesis xi 3 plus xi 4 right-parenthesis Over xi 3 xi 4 EndFraction left-parenthesis 1 plus StartFraction xi 4 left-parenthesis xi 1 plus xi 2 plus xi 4 right-parenthesis Over xi 1 xi 2 EndFraction right-parenthesis 3rd Row 1st Column Blank 2nd Column equals 3rd Column one-eighteenth StartFraction left-parenthesis xi 3 plus xi 4 right-parenthesis Over xi 3 xi 4 EndFraction left-parenthesis 1 minus StartFraction xi 4 xi 3 Over xi 1 xi 2 EndFraction right-parenthesis period EndLayout

The first piece cancels with minus one-eighteenth StartFraction xi 3 plus xi 4 Over xi 3 xi 4 EndFraction in Equation4.19 and the second piece is of size StartFraction upper N 12 Over upper N 1 squared EndFraction , which is fine. It remains to control

StartLayout 1st Row with Label left-parenthesis 4.20 right-parenthesis EndLabel StartFraction alpha 4 Over 54 xi 1 xi 2 xi 3 xi 4 EndFraction left-parenthesis m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-parenthesis minus one-eighteenth left-parenthesis StartFraction m squared left-parenthesis xi 1 right-parenthesis Over xi 1 EndFraction plus StartFraction m squared left-parenthesis xi 2 right-parenthesis Over xi 2 EndFraction right-parenthesis comma EndLayout

by StartFraction upper N 12 Over upper N squared EndFraction period Expand alpha 4 using Equation4.13 to rewrite this expression as

StartLayout 1st Row with Label left-parenthesis 4.21 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column StartFraction 3 left-parenthesis xi 1 xi 2 xi 3 plus xi 1 xi 2 xi 4 right-parenthesis Over 54 xi 1 xi 2 xi 3 xi 4 EndFraction left-parenthesis m squared left-parenthesis xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 3 right-parenthesis minus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column minus StartFraction 3 left-parenthesis xi 1 xi 3 xi 4 plus xi 2 xi 3 xi 4 right-parenthesis Over 54 xi 1 xi 2 xi 3 xi 4 EndFraction left-parenthesis m squared left-parenthesis xi 1 plus xi 3 right-parenthesis plus m squared left-parenthesis xi 1 plus xi 4 right-parenthesis right-parenthesis 3rd Row 1st Column Blank 2nd Column plus StartFraction 1 Over 18 xi 1 xi 2 EndFraction left-bracket xi 1 m squared left-parenthesis xi 1 right-parenthesis plus xi 2 m squared left-parenthesis xi 2 right-parenthesis right-bracket period EndLayout EndLayout

(The second term in Equation4.20 cancelled with part of the first.) The second and third terms in Equation4.21 are upper O left-parenthesis StartFraction upper N 12 Over upper N squared EndFraction right-parenthesis and may therefore be ignored. We rewrite the first term in Equation4.21 using the fact that m squared is even as

StartFraction 3 left-parenthesis xi 1 xi 2 xi 3 plus xi 1 xi 2 xi 4 right-parenthesis Over 54 xi 1 xi 2 xi 3 xi 4 EndFraction left-parenthesis m squared left-parenthesis minus xi 1 right-parenthesis plus m squared left-parenthesis xi 2 right-parenthesis minus m squared left-parenthesis minus left-parenthesis xi 1 plus xi 3 right-parenthesis right-parenthesis minus m squared left-parenthesis minus left-parenthesis xi 1 plus xi 4 right-parenthesis right-parenthesis right-parenthesis period

Since minus xi 1 plus xi 2 plus left-parenthesis xi 1 plus xi 3 right-parenthesis plus left-parenthesis xi 1 plus xi 4 right-parenthesis equals 0 , we can apply the double mean value theorem to obtain

equals StartFraction 3 left-parenthesis xi 1 xi 2 xi 3 plus xi 1 xi 2 xi 4 right-parenthesis Over 54 xi 1 xi 2 xi 3 xi 4 EndFraction left-parenthesis m squared left-parenthesis minus xi 1 overTilde right-parenthesis right-parenthesis double-prime xi 3 xi 4

with minus xi 1 overTilde equals minus xi 1 plus upper O left-parenthesis upper N 3 right-parenthesis plus upper O left-parenthesis upper N 4 right-parenthesis long right double arrow StartAbsoluteValue minus xi 1 overTilde EndAbsoluteValue tilde upper N 1 . Therefore, this term is bounded by

StartFraction xi 1 xi 2 xi 3 plus xi 1 xi 2 xi 4 Over xi 1 xi 2 xi 3 xi 4 EndFraction StartFraction m squared left-parenthesis minus xi 1 overTilde right-parenthesis Over left-parenthesis minus xi 1 overTilde right-parenthesis squared EndFraction xi 3 xi 4 equals upper O left-parenthesis StartFraction upper N 12 Over upper N 1 squared EndFraction m squared left-parenthesis xi 1 overTilde right-parenthesis right-parenthesis comma

which is smaller than StartFraction upper N 12 Over upper N 1 squared EndFraction as claimed.

Case 2C. StartFraction upper N 1 Over 4 EndFraction greater-than upper N 13 greater-than-or-equivalent-to StartFraction upper N Over 2 EndFraction comma upper N 12 tilde upper N 1 .

This case follows from a modification of Case 2A.

Case 2D. upper N 13 much-less-than StartFraction upper N Over 2 EndFraction comma upper N 12 tilde upper N 1 period

This case does not occur because upper N 13 tilde upper N 24 but upper N 4 is very small which forces upper N 2 to also be small, which is a contradiction.

â– 

4.4. upper M 5 bound

The multiplier upper M 5 was defined in Equation3.9, with sigma 4 equals minus StartFraction upper M 4 Over alpha 4 EndFraction period Our work on upper M 4 above showed that upper M 4 vanishes whenever alpha 4 vanishes so there is no denominator singularity in upper M 5 . Moreover, we have the following upper bound on upper M 5 in the particular case when m is of the form Equation4.7.

Lemma 4.6

If m is of the form Equation4.7, then

StartLayout 1st Row with Label left-parenthesis 4.22 right-parenthesis EndLabel StartAbsoluteValue upper M 5 left-parenthesis xi 1 comma ellipsis comma xi 5 right-parenthesis EndAbsoluteValue less-than-or-equivalent-to left-bracket StartFraction m squared left-parenthesis upper N Subscript asterisk 45 Baseline right-parenthesis upper N 45 Over left-parenthesis upper N plus upper N 1 right-parenthesis left-parenthesis upper N plus upper N 2 right-parenthesis left-parenthesis upper N plus upper N 3 right-parenthesis left-parenthesis upper N plus upper N 45 right-parenthesis EndFraction right-bracket Subscript s y m Baseline comma EndLayout

where

upper N Subscript asterisk 45 Baseline equals min left-parenthesis upper N 1 comma upper N 2 comma upper N 3 comma upper N 45 comma upper N 12 comma upper N 13 comma upper N 23 right-parenthesis period