Even Fourier multipliers and martingale transforms in infinite dimensions

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 $a_1,\dots ,a_N\in \mathbb{C}$, then the norm of the multiplier exceeds $\Vert a_1{R}_1^2+\cdots +a_N{R}_N^2{\Vert }_{\mathcal{L}(L^p({\mathbb{R}}^N;X))}$, where $R_n$ is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Bañuelos-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.