Optimal Sobolev imbeddings

Abstract

The aim of this paper is to study Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. We establish the equivalence of a Sobolev imbedding to the boundedness of a certain weighted Hardy operator. This Hardy operator is then used to prove the existence of rearrangement-invariant norms that are optimal in the imbedding inequality. Our approach is to use the methods and principles of Interpolation Theory.1991 Mathematics Subject Classification: 46E35; 46E30. m f ðlÞ :¼ jfx A W : j f ðxÞj > lgj; l > 0:

Calderon-type theorems for operators of non-standard endpoint behaviour

“…Theorem 1.4, however, shows that in fact they are equivalent. Using a different, quite indirect, technique, this equivalence also follows from [7,Theorem 3.9]. Finally, we focus on operators whose endpoint behaviour is, in a certain sense, dual to (1.5).…”

“…Its particular case is somewhat hidden in the proof of [7,Theorem 3.9]; here we state it explicitly in a more general situation and give a much simpler direct proof. (ii) Let h be a nonnegative and nonincreasing function on (0, ∞) and t ∈ (0, ∞).…”

Boundary trace inequalities and rearrangements

“…In the proof of Theorems 1.1 and 1.2 we shall need the following auxiliary result in the spirit of [EKP,Theorem 4.5] and [KP,Theorem 3.2].…”

“…By a straightforward extension of [KP,Theorem 3.9], for any α ∈ (0, 1) there exists a constant C = C(α) such that…”

Mixed norm spaces and rearrangement invariant estimates

“…The embedding (4) is due to Poornima [25], and it can be also traced in the work of Peetre [24] (the case W 1 0 L p , p > 1), and Kerman and Pick [17], where a characterization of Sobolev embeddings in rearrangement invariant (r.i.) spaces was obtained.…”