# Canonical Sobolev Projections of Weak Type \((1,1)\)

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*Earl Berkson; Jean Bourgain; Aleksander Pełczynski; Michał Wojciechowski*

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 \(L^1({\mathbb R}^2)\).

#### Table of Contents

# Table of Contents

## Canonical Sobolev Projections of Weak Type $(1,1)$

- Contents vii8 free
- Abstract viii9 free
- 1. Introduction and Notation 110 free
- 2. Some Properties of Weak Type Multipliers and Canonical Projections of Weak Type (1,1) 514 free
- 3. A Class of Weak Type (1,1) Rational Multipliers 2029
- 4. A Subclass of L∞ (R[sup(2)]) \ M[sup(w)sub(1)] (R[sup(2)] Induced by L∞ (R) 3645
- 5. Some Combinatorial Tools 4756
- 6. Necessity Proof for the Second Order Homogeneous Case: A Converse to Corollary (2.14) 5261
- 7. Canonical Projections of Weak Type (1,1) in the Tn Model: Second Order Homogeneous Case 5665
- 8. The Non–Homogeneous 5968
- 9. Reducible Smoothnesses of Order 2 6372
- 10. The Canonical Projection of Every Two-Dimensional Smoothness Is of Weak Type (1,1) 6776
- References 7483