Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

TL;DR

This paper extends sparse bound techniques to rough and smooth pseudodifferential operators, providing new pointwise bounds and alternative proofs for existing results.

Contribution

It develops methods to obtain sparse bounds for rough pseudodifferential operators and offers alternative proofs avoiding geometric decay arguments.

Findings

Established pointwise sparse bounds for rough pseudodifferential operators.

Provided sufficient conditions for sparse form bounds to hold.

Reproved known sparse bounds for operators with symbols in S^0_{1,δ}.

Abstract

Beltran \& Cladek~\cite{BC} use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with H\"ormander symbols in $S^m_{\rho,\delta}$ 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 $S^0_{1,\delta}$ for $\delta < 1$.