Optimal domain spaces in Orlicz-Sobolev embeddings

Abstract

We deal with Orlicz-Sobolev embeddings in open subsets of R n . A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces with respect to a Frostman measure, and, in particular, for trace embeddings on the boundary.

Fractional maximal operator in Orlicz spaces

“…Such a simplification is often called reduction principle. Similar reductions in Orlicz spaces have already appeared in the literature for various types of operators, let us mention for instance the reduction principle for the Hardy-Littlewood maximal operator [15], fractional integrals and Riesz potential [4], Sobolev and Poincaré inequalities [5], Sobolev embeddings [3,9], Sobolev trace embeddings [6] or Korn type inequalities [7]. The reduction principle for fractional maximal operator under some restrictive assumptions can be found in [13].…”

“…(ii) Let us set A(t) = t, t ≥ 0, so for every t > 0 and for a suitable constant k. Another use of [9,Proposition 4.1] gives that i B > n n−γ .…”

Boundedness of the Hilbert Transform From One Orlicz Space to Another

“…Let Ω denote the set of increasing concave functions : 0,1 → 0,1 for which lim → 0 (or simply 0 0). For a function in Ω, the Lorentz space Λ 0,1 is defined by setting [7,8,9,10,11,12,13].…”

A necessary condition on domains for optimal Orlicz-Sobolev embeddings

“…We refer to [2] for the definitions of other Boyd indices and more details about them. We will need the following lemma regarding equivalency of pointwise growth, integral growth and the local upper Boyd index of a Young function, see [4] or [7] for the proof and for other equivalent conditions.…”