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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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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

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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