Embeddings of Sobolev-type spaces into generalized Holder spaces involving k-modulus of smoothness

Abstract

We use an estimate of the k-modulus of smoothness of a function f such that the norm of its distributional gradient $|NABLA^k f|$ belongs locally to the Lorentz space$ L^{n/k,1}(R^n)$, k in N, k less or equal n, and we prove its reverse form to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces.

Keywords

• Rearrangement-invariant Banach function space
• Modulus of smoothness
• Distributional gradient
• Lorentz space
• Sobolev-type space
• Banach lattice
• Hölder-type space
• Embeddings