Titlebook: Real Analysis Methods for Markov Processes; Singular Integrals a Kazuaki Taira Book 2024 The Editor(s) (if applicable) and The Author(s), u

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Unique Solvability of the Homogeneous Dirichlet Problemlation (VMO) coefficients in the framework of Sobolev spaces of . style. We prove an existence and uniqueness theorem for the Dirichlet problem (Theorem .). Our proof is based on some interior and boundary . estimates for the solutions of problem (15.2) (Theorems 12.1 and 12.2). Both the interior an

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Calderón–Zygmund Kernels and Their Commutatorstegral operators with non-smooth kernels provide a powerful tool to deal with smoothness of solutions of partial differential equations, with minimal assumptions of regularity on the coefficients (see [26, 28, 107]). The results discussed here are adapted from Coifman–Rochberg–Weiss [39] and Bramanti–Cerutti [17].

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Unique Solvability of the Homogeneous Dirichlet Problemd boundary . estimates are consequences of explicit representation formulas (13.1) and (14.1) for the solutions of problem (15.2) (Theorems 13.1 and 14.1) and also of the .-boundedness of Calderón–Zygmund singular integral operators and boundary commutators appearing in those representation formulas (Theorems 14.2 and 14.5).

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Sobolev and Besov SpacesThis chapter is devoted to the precise definitions and statements of function spaces of . type with some detailed proofs.

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