Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space

Abstract

Poincaré-Sobolev-type inequalities involving rearrangement-invariant norms on the entire ${\mathbb{R}}^n$ are provided. Namely, inequalities of the type $\Vert u-P{\Vert }_{Y({\mathbb{R}}^n)}\le C\Vert {\nabla }^{m}u{\Vert }_{X({\mathbb{R}}^n)}$, where $X$ and $Y$ are either rearrangement-invariant spaces over ${\mathbb{R}}^n$ or Orlicz spaces over ${\mathbb{R}}^n$, $u$ is a $m-$times weakly differentiable function whose gradient is in $X$, $P$ is a polynomial of order at most $m-1$, depending on $u$, and $C$ is a constant independent of $u$, are studied. In a sense optimal rearrangement-invariant spaces or Orlicz spaces $Y$ in these inequalities when the space $X$ is fixed are found. A variety of particular examples for customary function spaces are also provided.

18 pages. Section 2 containing function-space-theoretical preliminaries overlaps with Section 2 of arXiv:1908.03384, which shares the same function-space-theoretical background

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