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Canonical Sobolev Projections of Weak Type \((1,1)\)
Earl Berkson, University of Illinois, Urbana, IL, Jean Bourgain, Institute for Advanced Study, Princeton, NJ, and Aleksander Pełczynski and Michał Wojciechowski, Polish Academy of Sciences, Warszawa, Poland

Memoirs of the American Mathematical Society
2001; 75 pp; softcover
Volume: 150
ISBN-10: 0-8218-2665-4
ISBN-13: 978-0-8218-2665-2
List Price: US$49
Individual Members: US$29.40
Institutional Members: US$39.20
Order Code: MEMO/150/714
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Let \(\mathcal S\) be a second order smoothness in the \(\mathbb{R}^n\) setting. We can assume without loss of generality that the dimension \(n\) has been adjusted as necessary so as to insure that \(\mathcal S\) is also non-degenerate. We describe how \(\mathcal S\) must fit into one of three mutually exclusive cases, and in each of these cases we characterize by a simple intrinsic condition the second order smoothnesses \(\mathcal S\) whose canonical Sobolev projection \(P_{\mathcal{S}}\) is of weak type \((1,1)\) in the \(\mathbb{R}^n\) setting. In particular, we show that if \(\mathcal S\) is reducible, \(P_{\mathcal{S}}\) is automatically of weak type \((1,1)\). We also obtain the analogous results for the \(\mathbb{T}^n\) setting. We conclude by showing that the canonical Sobolev projection of every \(2\)-dimensional smoothness, regardless of order, is of weak type \((1,1)\) in the \(\mathbb{R}^2\) and \(\mathbb{T}^2\) settings. The methods employed include known regularization, restriction, and extension theorems for weak type \((1,1)\) multipliers, in conjunction with combinatorics, asymptotics, and real variable methods developed below. One phase of our real variable methods shows that for a certain class of functions \(f\in L^{\infty}(\mathbb R)\), the function \((x_1,x_2)\mapsto f(x_1x_2)\) is not a weak type \((1,1)\) multiplier for \(L^1({\mathbb R}^2)\).


Graduate students and research mathematicians interested in real functions, functional analysis, and operator theory.

Table of Contents

  • Introduction and notation
  • Some properties of weak type multipliers and canonical projections of weak type \((1,1)\)
  • A class of weak type \((1,1)\) rational multipliers
  • A subclass of \(L^\infty(\mathbb{R}^2)\backslash M_1^{(w)}(\mathbb{R}^2)\) induced by \(L^\infty(\mathbb{R})\)
  • Some combinatorial tools
  • Necessity proof for the second order homogeneous case: A converse to Corollary (2.14)
  • Canonical projections of weak type \((1,1)\) in the \(\mathbb{T}^n\) model: Second order homogeneous case
  • The non-homogeneous case
  • Reducible smoothnesses of order \(2\)
  • the canonical projection of every two-dimensional smoothness is of weak type \((1,1)\)
  • References
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