𝐻𝛾,𝑤𝑝(𝑰) Space anew Contribution to Weighted Space Theory and Smoothness Analysis

Abstract

Abstract

   The space   is a weighted Sobolev-type space consisting of functions with classical derivatives up to order m, measured through a weighted norm involving both the function and its derivatives. This space is characterized by smoothness, integrability , and the ability to apply standard differentiation rules. The study establishes that every Cauchy sequence in  converges within the space, proving that it is a complete Banach space. It also shows that  forms a closed subspace of the weighted Sobolev space . The modulus of smoothness and difference operators further describe its approximation properties.