On the Hollenbeck-Verbitsky conjecture and M. Riesz theorem for various function spaces

Marijan Marković, Petar Melentijević

Published 2019-12-18Version 1

Let $P_+$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider two-sided estimates of $\| ( |P_ + f | ^s + |P_- f |^s) ^{\frac {1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f \in L^p(\mathbf{T})$. For some values of parameters $0<s<\infty$ and $p$, our inequalities are sharp, while in other cases we find the family of test functions which we conjecture to be sharp. Also, we obtain the right asymptotic of the constants for large $s$ as well as the appropriate vector-valued inequalities. This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases. As a consequence of inequalities we have in the paper we get Riesz-type theorems on conjugate harmonic functions for various function spaces. In particular, slightly general version of Stout's theorem for Lumer Hardy spaces were obtained by a new approach.