$L^p\to L^q$ norm estimates of Cauchy transforms on the Dirichlet problem and their applications

Denote by $C^{\alpha}(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are Holder continuous with the exponent $\alpha$, and denote by $C^{1, \alpha}(\mathbb{D})$ the space which consists of differentiable functions $f$ such that their derivatives are in the space $C^{\alpha}(\mathbb{D})$. Let $\mathcal{C}$ be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of $\|\mathcal{C}\|_{L^p\to L^q}$, where $3/2

Keywords:

• Differentiable function
• Nabla symbol
• Unit disk
• Mathematics
• Exponent
• Lipschitz continuity
• Hölder condition
• Dirichlet problem
• Combinatorics
• Series (mathematics)

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