L^p-theory for Cauchy-transform on the unit disk

2020/08/07 by David Kalaj, Kalaj, David, Petar Melentijević +3 · 1 citation
Mathematics · #Holomorphic and Operator Theory #Advanced Harmonic Analysis Research #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2008.03068

Abstract

Let \mathbbD be the unit disk and φ∈ L^p(\mathbbD, dA), where 1≤ p≤∞. For z∈\mathbbD, the Cauchy-transform on \mathbbD, denote by P, is defined as follows: P[φ](z)=-∫_\mathbbD((φ(w))/(w-z)+\fraczφ(w)1-wz)dA(w). The Beurling transform on \mathbbD, denote by H, is now defined as the z-derivative of P. In this paper, by using Hardy's type inequalities and Bessel functions, we show that ‖P‖_{L²→ L²}=α≈1.086, where α is a solution to the equation: 2J₀(2/α)-αJ₁(2/α)=0, and J₀, J₁ are Bessel functions. Moreover, for p>2, by using Taylor expansion, Parseval's formula and hypergeometric functions, we also prove that ‖P‖_{L^p→ L^∞}=2(Γ(2-q)/Γ²(2-(q)/(2)))^1/q, where q=p/(p-1) is the conjugate exponent of p, and Γ is the Gamma function. Finally, applying the same techniques developed in this paper, we show that the Beurling transform H acts as an isometry of L²(\mathbbD, dA).

Citations

• Spectral Properties of the Cauchy Operator and Its Product with Bergman's Projection on a Bounded Domain

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