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 ∈ L ^p(R ⁿ), in the half-space R ⁿ⁺¹₊ = {(x, y): x ∈ R ⁿ, y > 0} near the boundary ∂ R ⁿ⁺¹₊ = R ⁿ×{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) → (x ₀, 0) through certain approach regions in R ⁿ⁺¹₊ which are valid for all x ₀ ∈ R ⁿ outside exceptional sets of x ₀ of an appropriate capacity zero. We deduce some local integral results for ${f_\alpha }$ which illustrate the ‘smoothing’ effects of Bessel kernels.