hp(X) class of X-valued harmonic functions and applications

Abstract

The concept of t-basis (generated by the tensor product) from the exponential system E = {e(int)}(n is an element of Z) is considered for Bochner space L-p(I-0; X), 1 < p < +infinity, on I-0 = [-pi, pi), where X is a Banach space with UMD (Unconditional Martingale Difference) property. We assume that Xis endowed with the involution (*). Using the t-basicity of the system epsilon, we introduce the class h(p)(+;R) of X-valued harmonic functions in the unit ball, generated by involution (*). The *-analogues of the Cauchy-Riemann conditions are obtained, and the relations between the class h(p)(+;R) (X) and the Hardy-Bochner class H-p(X) of analytic functions are established. A new method for establishing X-valued Sokhotski-Plemelj's formulas is presented. Additionally, we establish the correctness of the Dirichlet problem for X-valued harmonic functions in the class h(p)(X).