Hollenbeck–Verbitsky conjecture on best constant inequalities for analytic and co-analytic projections

Abstract

In this paper we address the problem of finding the best constants in inequalities of the form:

$$\begin{aligned} \big \Vert \big (|P_+f|^s+|P_-f|^s\big )^{\frac{1}{s}}\big \Vert _{L^p({{\mathbb {T}}})}\le A_{p,s} \Vert f\Vert _{L^p({{\mathbb {T}}})}, \end{aligned}$$

where $P_+f$ and $P_-f$ denote analytic and co-analytic projection of a complex-valued function $f \in L^p({{\mathbb {T}}}),$ for $p \ge 2$ and all $s>0$, thus proving Hollenbeck–Verbitsky conjecture from (Oper Theory Adva Appl 202:285–295, 2010). We also prove the same inequalities for $1<p\le \frac{4}{3}$ and $s\le \sec ^2\frac{\pi }{2p}$ and confirm that $s=\sec ^2\frac{\pi }{2p}$ is the sharp cutoff for s. The proof uses a method of plurisubharmonic minorants and an approach of proving the appropriate “elementary” inequalities that seems to be new in this topic. We show that this result implies best constants inequalities for the projections on the real-line and half-space multipliers on ${\mathbb {R}}^n$ and an analog for analytic martingales. A remark on an isoperimetric inequality for harmonic functions in the unit disk is also given.