An exact estimate of the boundary behavior of functions from Hardy-Sobolev classes in the critical case
The Hardy-Sobolev spaces \(H^p_q(B^n)\) in the unit ball \(B^n \subset\mathbb{C}^n\) is defined as fractional integration of the usual Hardy space \(H_p(B^n)\): \(H^p_q(B^n)= R_q(H^p(B^n)\), where \(R_qf(z)= \Sigma (m+1)^qf_m(z)\) is the operator of the fractional integration. Here \(f(z)= \Sigma f_m(z)\) is the homogeneous expansion of \(f\). If \(pq=n\), the functions from \(H^p_q(B^n)\) have a limit almost everywhere on the boundary along certain regions of exponential contact with the boundary. It is proved that the maximal operator associated with these regions is bounded as an operator from \(H^p_q(B^n)\) to \(L_p(\partial B^n)\).
scientific article; zbMATH DE number 1257138
scientific article