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Weak (1,1) bounded operators

Arup Maity

TL;DR

The paper constructs a natural class of Fourier multipliers with weak (1,1) boundedness but no weak (p,p) boundedness for any p>1, and proves that this endpoint behavior is sharp. It introduces a factorizable symbol class A consisting of sums m = ∑ f_i * g_i with ∑ ||f_i||_1 ||g_i||_q < ∞ and 1 ≤ q ≤ (2n+2)/(n+3), and shows T_m is weak (1,1) via kernel/log-type estimates and restriction-type arguments, while ruling out any p>1 via Marcinkiewicz interpolation. The sharpness is demonstrated by a 1D construction yielding unbounded L^1→L^{1,r} behavior for all r<∞, highlighting limitations of endpoint control for non-smooth multiplier symbols. Overall, the work delineates a boundary between endpoint weak-type control and strong L^p bounds within a natural, non-Hörmander class of multipliers.

Abstract

We construct a class of Fourier multipliers whose associated operators are weak (1,1) bounded but fail to be weak (p, p) bounded for any 1 < p \leq \infty. Moreover, we show that this result is sharp.

Weak (1,1) bounded operators

TL;DR

The paper constructs a natural class of Fourier multipliers with weak (1,1) boundedness but no weak (p,p) boundedness for any p>1, and proves that this endpoint behavior is sharp. It introduces a factorizable symbol class A consisting of sums m = ∑ f_i * g_i with ∑ ||f_i||_1 ||g_i||_q < ∞ and 1 ≤ q ≤ (2n+2)/(n+3), and shows T_m is weak (1,1) via kernel/log-type estimates and restriction-type arguments, while ruling out any p>1 via Marcinkiewicz interpolation. The sharpness is demonstrated by a 1D construction yielding unbounded L^1→L^{1,r} behavior for all r<∞, highlighting limitations of endpoint control for non-smooth multiplier symbols. Overall, the work delineates a boundary between endpoint weak-type control and strong L^p bounds within a natural, non-Hörmander class of multipliers.

Abstract

We construct a class of Fourier multipliers whose associated operators are weak (1,1) bounded but fail to be weak (p, p) bounded for any 1 < p \leq \infty. Moreover, we show that this result is sharp.

Paper Structure

This paper contains 3 sections, 4 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Weak $(1,1)$ bound
  3. Sharpness

Key Result

Theorem 2.1

Let $S: L^1(\mathbb{R}^n) \to M(\mathbb{R}^n)$ be a linear operator which is defined by where $K$ is the kernel and $M(\mathbb{R}^n)$, space of measurable functions. Then the operator $S$ is weak $(1,1)$ bounded if where $d\sigma_x$ is the surface measure of the unit sphere.

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof