Poisson integrals of Bessel potentials

Abstract

We are concerned with the growth of the harmonic function \({u_\alpha } = P[{f_\alpha }]\), i.e., the Poisson integral of a Bessel potential \({f_\alpha } = {G_\alpha }*F\) with F ∈ Lp(Rn), in the half-space R+n+1 = {(x, y): x ∈ Rn, y > 0} near the boundary ∂R+n+1 = Rn×{0}. When the order α of the Bessel kernel \({G_\alpha }\) is a positive integer, and p > 1, the corresponding space of potentials can be identified with a Sobolev space. We obtain weighted integral estimates and explicit growth estimates for \({u_\alpha }(x,y)\) as (x, y) → (x0, 0) through certain approach regions in R+n+1 which are valid for all x0 ∈ Rn outside exceptional sets of x0 of an appropriate capacity zero. We deduce some local integral results for \({f_\alpha }\) which illustrate the ‘smoothing’ effects of Bessel kernels.