2023/09/05 by Bo-Yong Chen, Chen, Bo-Yong
Mathematics · #Advanced Harmonic Analysis Research
paper · pdf · doi:10.48550/arxiv.2309.01996
In this note, we extend the well-known theorems of M. Riesz and Zygmund on conjugate functions as follows. Let Ω be a domain in \mathbb Cn. Suppose that f=u+iv∈ \mathcal O(Ω) satisfies v(z0)=0 for some z0∈ Ω. Then ‖f‖p,z0 ≤ Cp ‖u‖p,z0 for 11, there exists Cα>0 such that ∫∂ Ωt (exp(\fracπ2 |f| ))/((1+|f|)α) dωz0,t ≤ Cα for any exhaustion \Ωt\ of Ω with Ωt\ni z0, where d ωz0,t is the harmonic measure of Ωt relative to z0. Analogous results for Poletsky-Stessin-Hardy spaces on hyperconvex domains are given.