On M. Riesz conjugate function theorem for harmonic functions

2023/09/30 by Kalaj, David
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.00464

Abstract

Let L^p(T) be the Lesbegue space of complex-valued functions defined in the unit circle T=\z: |z|=1\⊆ ℂ. In this paper, we address the problem of finding the best constant in the inequality of the form: ‖f‖_L^p(T)≤ A_p,b ‖(|P₊ f|²+b| P_- f|²)^1/2‖_L^p(T). Here p∈[1,2], b>0, and by P_- f and P₊ f are denoted co-analytic and analytic projection of a function f∈ L^p(T). The equality is "attained" for a quasiconformal harmonic mapping. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.

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