$h_p(X)$ Class of $X$-Valued Harmonic Functions and Applications

Abstract

The concept of $t$-basis (generated by the tensor product) from the exponential system $\mathcal{E}= \{e^{i n t}\}_{n \in \mathbb{Z}}$is considered for Bochner space $L_p(I_0; X)$, $1 < p < +\infty$, on $I_0 = [ -\pi, \pi )$, where $X$ is a Banach space with UMD (Unconditional Martingale Difference) property. We assume that $X$ is endowed with the involution $(*)$. Using the $t$-basicity of the system $\mathcal{E}$, we introduce the class $h_p^{+;\mathbb R}(X)$ 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 $h_p^{+;\mathbb R}(X)$ 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 $h_p(X)$.