Even Fourier multipliers and martingale transforms in infinite dimensions

TL;DR

This paper establishes sharp bounds for the norms of even Fourier multipliers and martingale transforms in infinite-dimensional UMD Banach spaces, linking these bounds to the symbols' ranges and introducing new analytical tools.

Contribution

It provides the first sharp lower bounds for Fourier multiplier norms in infinite dimensions and introduces UMD$_p^A$ constants and $A$-weak differential subordination techniques.

Findings

Sharp lower bounds for Fourier multiplier norms in UMD spaces.

Sharp upper bounds for Ba ilde{n}uelos-Bogdan type multipliers.

Introduction of UMD$_p^A$ constants and $A$-weak differential subordination.

Abstract

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in $\mathcal L(L^p(\mathbb R^d; X))$ for $1<p<\infty$ and for a UMD Banach space $X$ in terms of the range of the corresponding symbol. For example, if the range contains $a_1,\ldots,a_N \in \mathbb C$, then the norm of the multiplier exceeds $\|a_1R_1^2 + \cdots + a_NR_N^2\|_{\mathcal L(L^p(\mathbb R^N; X))}$, where $R_n$ is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Ba\~{n}uelos-Bogdan type multipliers in terms of the range of the functions involved. The main tools that we exploit are $A$-weak differential subordination of martingales and UMD$_p^A$ constants, which are introduced here.