LpLp estimate of convolution transform of singular measure by approximate identity

We study the boundary properties of u(x,y)=Ky∗μ(x), defined on the half-space R∗+×RN by the convolution of an approximate identity Ky(⋅)(y>0) and the Ahlfors–David singular ss-measure μμ on RNRN. The Poisson and the Gauss–Weierstrass integral are unified as special cases in our setting. The sharp LpLp estimates of the function u(x,y)u(x,y) near the boundary are given. In particular, we show that ∫RNup(x,y)dx∼1δ(y)(N−s)(p−1)as  y→0+, where δ(y)δ(y) is a positive function on R∗+ with limy→0+δ(y)=0limy→0+δ(y)=0.