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

The concept of t-basis (generated by the tensor product) from the exponential system E = { e^{{int}} }n∈ Z is considered for Bochner space L_p ( I_0 ;X), 1 < p < +∞, on I_0 = [-π, π), 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 E, we introduce the class h_p+;Rp (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+;Rp (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).