New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
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
 SEARCH THIS BOOK:
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$52 Individual Members: US$31.20
Institutional Members: US\$41.60
Order Code: MEMO/150/714

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.

• 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