The estimates of approximation by using a new type of weighted modulus of continuity

WOS: 000248144000012In this paper, we introduce a new type modulus of continuity for function f belonging to a particular weighted subspace of C (0, infinity) and show that it has some properties of ordinary modulus of continuity. We obtain some estimates of approximation of functions with respect to a suitable weighted norm via the new type moduli of continuity. Finally, we give some examples. (C) 2007 Elsevier Ltd. All rights reserved

The Stein-Weiss type inequalities for the B-Riesz potentials

We establish two inequalities of Stein-Weiss type for the Riesz potential operator I?,? (B-Riesz potential operator) generated by the Laplace-Bessel differential operator ?B in the weighted Lebesgue spaces Lp,|x|ß,?. We obtain necessary and sufficient conditions on the parameters for the boundedness of I?,? from the spaces Lp,|x|ß,? to Lq,|x|-?,?, and from the spaces L1,|x|ß,? to the weak spaces WLq,|x|-?,?. In the limiting case p=Q/? we prove that the modified B-Riesz potential operator I?,? is bounded from the spaces Lp,|x|ß,? to the weighted B-BMO spaces BMO|x|-?,?. As applications, we get the boundedness of I?,? from the weighted B-Besov spaces Bs p?,|x|ß,? to the spaces Bs q?,|x|-?,?. Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue Lp,|x|ß,? and weighted B-Besov spaces Bs p?,|x|ß,? by using the fundamental solution of the B-elliptic equation ??/2 B