Best Constant Inequalities Involving the Analytic and Co-Analytic Projection

Abstract

Let P ₊ denote the Riesz projection and P ₋=I−P+ denote the co-analytic projection where I is the identity operator. We prove

$$ \left\| {\max (\left| {P_ + f} \right|,\left| {P_ - f} \right|)} \right\|_{L^p (T)} \leqslant \csc \frac{\pi } {p}\left\| f \right\|_{L^p (T)} , 1 < p < \infty , $$

where f∈L ^p(T) is a complex-valued function, and the constant $\csc \tfrac{\pi } {p}$ p p is sharp. Our proof is based on an explicit construction of a plurisubharmonic minorant for the function $F(w,z) = \csc ^p \tfrac{\pi } {p}\left| {w + \bar z} \right|^p - \max (\left| w \right|,\left| z \right|)^p$ on C ². More generally, we discuss the best constant problem for the inequality

$$ \left\| {(\left| {P_ + f} \right|^s ,\left| {P_ - f} \right|^s )^{\tfrac{1} {s}} } \right\|_{L^p (T)} \leqslant C(p,s)\left\| f \right\|_{L^p (T)} , 1 < p < \infty , $$

where 0<s<∞, which may serve as a model problem for some vectorvalued inequalities, where the method of plurisubharmonic minorants seems to be promising.

The second author was supported in part by NSF Grant DMS-0556309.