Embeddings of homogeneous Sobolev spaces on the entire space

Abstract

We completely characterize the validity of the inequality $\Vert u{\Vert }_{Y({\mathbb{R}}^n)}\le C\Vert {\nabla }^{m}u{\Vert }_{X({\mathbb{R}}^n)}$, where $X$ and $Y$ are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

arXiv comment25 pages

References in corpus (5)

• Boundedness of classical operators on rearrangement-invariant spaces
• Optimal domain spaces in Orlicz-Sobolev embeddings
• Fractional Maximal Operator in Orlicz spaces
• Interpolation theorem for Marcinkiewicz spaces with applications to Lorentz gamma spaces
• Reduction principle for a certain class of kernel-type operators

Cited by in corpus (3)

• Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space
• Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces
• On the properties of rearrangement-invariant quasi-Banach function spaces