Suppose \(00\), \(j=1,\dots,d.\) In this paper the author deals with the limit of \({\widetilde S}^j_\mu f\) \(( j=0,1,\dots,d)\) in some norm as \(\mu\to 0\) under the restriction \[ \tau^{-1}\leq {\mu_k\over\mu_j}\leq \tau,\quad j,k=1,\dots,d\quad (\tau\geq 1). \] The corresponding maximal operator \({\widetilde S}^j_*\) is defined by taking the maximum of \(|{\widetilde S}^j_\mu|\) over \(\mu\) under the same restriction. Main result can be stated as Theorem. If \({d\over d+1}\rho|\leq {C\over\rho}\|f\|_1\quad (\rho>0). \] Corresponding result of convergence in \(H_{pq}\)-norm is obtained as a corollary.