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 $a_1,\ldots,a_N \in \mathbb C$, 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.