Even Fourier multipliers and martingale transforms in infinite dimensions

• Published in: Indagationes mathematicae Vol. 29; no. 5; pp. 1290 - 1309
• Main Author: Yaroslavtsev, Ivan S.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.10.2018; Elsevier Science Ltd
• Subjects: Banach spaces; Even Fourier multipliers; Fourier transforms; Lagrange multiplier; Lower bounds; Martingale transforms; Martingales; Multipliers; Norms; Subordinates; UMD Banach spaces; Upper bounds; Weak differential subordination; Martingale transforms; UMD Banach spaces; Even Fourier multipliers; Weak differential subordination
• ISSN: 0019-3577; 1872-6100
• DOI: 10.1016/j.indag.2018.05.014

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in L(Lp(Rd;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,…,aN∈C, then the norm of the multiplier exceeds ‖a1R12+⋯+aNRN2‖L(Lp(RN;X)), where Rn 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 UMDpA constants, which are introduced here.