Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Endpoint Variation and jump inequalities for rough singular integrals

Ankit Bhojak, Saurabh Shrivastava

Abstract

In this article, we prove weak type $(1,1)$ bounds for the variation and jump operators corresponding to the family of truncations of singular integrals with rough kernels. This resolves an open question raised by Jones, Seeger and Wright (Trans. Amer. Math. Soc. (2008)). Moreover, as an immediate consequence of the variational estimate, we recover the weak type $(1,1)$ boundedness of the maximal truncation operator corresponding to singular integrals with rough kernels.

Endpoint Variation and jump inequalities for rough singular integrals

Abstract

In this article, we prove weak type bounds for the variation and jump operators corresponding to the family of truncations of singular integrals with rough kernels. This resolves an open question raised by Jones, Seeger and Wright (Trans. Amer. Math. Soc. (2008)). Moreover, as an immediate consequence of the variational estimate, we recover the weak type boundedness of the maximal truncation operator corresponding to singular integrals with rough kernels.
Paper Structure (10 sections, 5 theorems, 62 equations)

This paper contains 10 sections, 5 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Proof of \ref{['mainthm']} \ref{['main:jump']}
  3. Estimate for short jumps
  4. Estimate for $I_{bd}$:
  5. Estimate for $I_{in}$:
  6. Estimate for long jumps
  7. Estimate for $L_2$:
  8. Estimate for $L_1$:
  9. Microlocal estimates for rough singular integrals
  10. Estimates of Seeger:

Key Result

Theorem 1.1

Let $\Omega\in L\log L({\mathbb S}^{d-1})$. Then, for all $\alpha>0$, we have Moreover for $q>2$, we have

Theorems & Definitions (6)

  • Theorem 1.1
  • Lemma 1.2: DOP2017
  • Lemma 2.1: HLP2013
  • Lemma A.1: Seeger1996
  • Lemma A.2
  • proof