Besov Spaces via Poisson Integrals

In this chapter we develop the theory of Besov spacesBesov space on the Euclidean space \(\textbf{R}^{n}\) , paying particular attention to Poisson integralsPoisson integral. Besov spaces are function spaces defined in terms of the \(L^{p}\) modulus of continuity, and enter naturally in connection with boundary value problems in the framework of Sobolev spaces of \(L^{p}\) type. We prove a variety of equivalent norms for the Besov spaces on \(\textbf{R}^{n}\) via Poisson integrals (Theorems 6.4, 6.7 and 6.8). The results discussed here are adapted from Taibleson [146] and Stein [137].