Tangential Boundary Behaviour of Harmonic and Holomorphic Functions

Let K be a kernel on R n , that is, K is a non‐negative, unbounded L 1 function that is radially symmetric and decreasing. We define the convolution K * F byK * F ( x ) = ∫ R nK ( x − t ) F ( t ) d t ,and note from L p ‐capacity theory [ 11 , Theorem 3] that, if F ∈ L p , p > 1, then K * F exists as a finite Lebesgue integral outside a set A ⊂ R n with C K,p ( A ) = 0. For a Borel set A ,C K , p( A ) = inf{‖ F ‖p p : F ⩾ 0 , K * F ⩾ 1   on   A } ,where‖ F ‖ p =( ∫ R n| F | p d x )1 / p.We define the Poisson kernel forR + n + 1= {( x , y ) : x ∈ R n , y > 0} byP y ( x ) =c n y(| x | 2 + y 2 )( n + 1 ) / 2,c n = Γ (( n + 1 ) / 2 ) π −( n + 1 ) / 2,and setu ( x , y ) = ∫ R nP y ( x − t ) K * F ( t ) d t ,( x , y ) ∈ R + n + 1 .Thus u is the Poisson integral of the potential f = K * F , and we write u = P y *( K * F )= P y * f = P [ f ]. We are concerned here with the limiting behaviour of such harmonic functions at boundary points ofR + n + 1, and in particular with the tangential boundary behaviour of these functions, outside exceptional sets of capacity zero or Hausdorff content zero.