Local cone multipliers and Cauchy-Szego projections in bounded symmetric domains

TL;DR

This paper investigates the boundedness properties of cone multipliers and Cauchy-Szeg"o projections in bounded symmetric domains, establishing limitations on their $L^p$-$L^q$ bounds and answering a longstanding question negatively.

Contribution

It adapts Fefferman's proof to show the cone multiplier's bounds are only trivial and resolves a question about the $L^p$-$L^q$ continuity of Cauchy-Szeg"o projections in higher-rank symmetric domains.

Findings

Cone multiplier bounds only hold in trivial $L^p$-$L^q$ ranges.

Cauchy-Szeg"o projections are not continuous from $L^p$ to $L^q$ in certain symmetric domains.

Negative answer to Békollé and Bonami's question on projection continuity.

Abstract

We show that the cone multiplier satisfies local $L^p$-$L^q$ bounds only in the trivial range $1\leq q\leq 2\leq p\leq\infty$. To do so, we suitably adapt to this setting the proof of Fefferman for the ball multiplier. As a consequence we answer negatively a question by B\'ekoll\'e and Bonami (Colloq. Math. 68, 1995, 81-100), regarding the continuity from $L^p\to L^q$ of the Cauchy-Szeg\"o projections associated with a class of bounded symmetric domains in $\mathbb{C}^n$ with rank $r\geq2$.