Spaces generated by smooth blocks (Q1192590)

The first author, M. H. Taibleson and G. Weiss have shown that if \(f\) belongs to the block space \(B_ q(\mathbb{R}^ n)\), \(1< q\leq \infty\), then the Bochner-Riesz multiplier \(S_ R^{(n-1)/2} f(x)\) converges a.e. to \(f(x)\). The authors introduce smooth blocks of the \((q,\lambda)\) type, by means of replacing the usual \(L^ q\) condition by an \(L^ q_ \lambda\) condition, where \(L^ q_ \lambda\) denotes the corresponding Bessel potential space, and show that the corresponding block spaces generated can be used to characterize the rate of convergence in the previously stated result. Their main result states that: \(S_ R^{(n-1)/2} f(x)- f(x)=o(1/R^ \lambda)\), a.e., if \(f\in B^ \lambda_ q(\mathbb{R}^ n)\), \(0<\lambda< 2\).