Let K be a kernel on Rn, that is, K is a non-negative,
unbounded L1 function that is radially symmetric and decreasing.
We define the convolution K [midast ] F by
and note from Lp-capacity theory [11, Theorem 3] that, if
F ∈ Lp, p > 1, then K [midast ] F
exists as a finite Lebesgue integral outside a set A ⊂ Rn with
CK,p(A) = 0. For a Borel set A,
where
We define the Poisson kernel for Rn+1+
= {(x, y) [ratio ] x ∈ Rn, y > 0} by
and set
Thus u is the Poisson integral of the potential f = K [midast ] F, and we write
We are concerned here with the limiting behaviour of such harmonic functions at
boundary points of Rn+1+, and in particular with the tangential boundary behaviour
of these functions, outside exceptional sets of capacity zero or Hausdorff content zero.