Sobolev type inequalities in ultrasymmetric spaces with applications to Orlicz‐Sobolev embeddings

Abstract

LetDkfmean the vector composed by all partial derivatives of orderkof a functionf(x),x∈Ω⊂ℝn. Given a Banach function spaceA, we look for a possibly small spaceBsuch that‖f‖B≤c‖|Dkf|‖Afor allf∈C0k(Ω). The estimates obtained are applied to ultrasymmetric spacesA=Lφ,E,B=Lψ,E, giving some optimal (or rather sharp) relations between parameter-functionsφ(t)andψ(t)and new results for embeddings of Orlicz-Sobolev spaces.

Functional properties of rearrangement invariant spaces defined in terms of oscillations

“…More generally, if the Hardy operator P is bounded on L p (w) the method of proof of (19) given in [2] can be easily extended to the spaces S p (w) introduced in this paper so that we can readily show that…”

“…Milman and Pustylnik in the recent paper [18], (see also [19]), extending the methods developed in [2] to the case k > 1, obtained a unified method to prove the Sobolev embedding theorem and the corresponding sharp borderline cases. They started by showing that 6…”

Symmetrization inequalities and Sobolev embeddings

“…Following [23] and [26] we shall now construct the range spaces for our generalized Sobolev embedding theorem. Suppose that X and Y are r.i. spaces, and let s ∈ R. We define…”

Symmetrization inequalities in the fractional case and Besov embeddings

“…In [2] it was shown that the oscillation of the decreasing rearrangement of f, f * o (t) = f * * (t) − f * (t) can be estimated by f * o (t) c n t 1/n |∇f | * * (t), f ∈ C ∞ 0 R n , (1.1) where f * * (t) = 1 t t 0 f * (s) ds, and f * is the non-increasing rearrangement of f . The formulation of inequalities in terms of the oscillation f * o (t) leads to general forms of the Sobolev embedding theorem that are sharp up to the endpoints and particularly useful in the study of higher order Sobolev inequalities (see [2,20,22,25]). …”