Boundary values, integral transforms, and growth of vector valued Hardy functions

Banach space valued Hardy functions H^p, 0 < p ≤ ∞, are defined with the functions having domain in tubes T^C = R^n + iC ⊂ C^n; H² functions with values in Hilbert space are characterized as Fourier-Laplace transforms of functions which satisfy a certain norm growth property. These H² functions are shown to equal a Cauchy integral when the base C of the tube T^C is specialized. For certain Banach spaces and certain bases C of the tube T^C , all H^p functions, 1 ≤ p ≤ ∞, are shown to equal the Poisson integral of L^p functions, have boundary values in L^p norm on the distinguished boundary R^n + i{0} of the tube T^C , and have pointwise growth properties. For H² functions with values in Hilbert space we show the existence of L² boundary values on the topological boundary R^n + i ∂C of the tube T^C