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Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

Solange Mukeshimana, David Rule

Abstract

Beltran \& Cladek~\cite{BC} use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in $S^m_{ρ,δ}$ up to, but not including, the sharp end-point in decay $m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove know sparse bounds for pseudodifferential operators with symbols in $S^0_{1,δ}$ for $δ< 1$.

Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

Abstract

Beltran \& Cladek~\cite{BC} use to bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in up to, but not including, the sharp end-point in decay . We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove know sparse bounds for pseudodifferential operators with symbols in for .

Paper Structure

This paper contains 4 sections, 13 theorems, 139 equations.

Table of Contents

  1. Introduction and Main Results
  2. Sparse Bounds from Scale-Invariant $L^r$ to $L^s$ Bounds
  3. Rough and Smooth Pseudodifferential Operators
  4. Sharp Maximal Function Bounds

Key Result

Proposition 1.4

Let $Q_k$ denote a cube of radius $2^{-k}$ with $k\in{\bf Z}$ and assume that the operator $T$ is a countable sum $T = \sum_{j\in I} T^{j}$ of sublinear operators $T_j$ indexed by $j \in I$. Furthermore, assume that, for a given function $\iota \colon I \to {\bf Z}$, exponents $1\leq r,s \leq\infty$ for each $j \in I$ and $T^{j}\left(f\chi_{\frac{1}{3}Q_{\iota(j)}}\right)$ is supported in $Q_{\iot

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 20 more