Even Fourier multipliers and martingale transforms in infinite dimensions

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in L(L^p(R^d;X)) for 1<p<∞and for a UMD Banach space X in terms of the range of the corresponding symbol. For example, if the range contains a₁,…,a_N∈C, then the norm of the multiplier exceeds ‖a₁R₁ ²+⋯+a_NR_N ²‖_{L(L^p(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.