# Best Constant Inequalities Involving the Analytic and Co-Analytic Projection

• Brian Hollenbeck
• Igor E. Verbitsky
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

## 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 fL 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.

## Keywords

Analytic projection Hilbert transform best constants plurisubharmonic functions

## Mathematics Subject Classification (2000)

Primary 42A50 47B35 Secondary 31C10 32A35

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## Authors and Affiliations

• Brian Hollenbeck
• 1
• Igor E. Verbitsky
• 2
1. 1.Department of Mathematics, Computer Science, and EconomicsEmporia State UniversityEmporiaUSA
2. 2.Department of MathematicsUniversity of MissouriColumbiaUSA

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