Boundedness of classical operators on rearrangement-invariant spaces

Abstract

We study the behaviour on rearrangement-invariant (r.i.) spaces of such classical operators of interest in harmonic analysis as the Hardy-Littlewood maximal operator (including the fractional version), the Hilbert and Stieltjes transforms, and the Riesz potential. The focus is on sharpness questions, and we present characterisations of the optimal domain (or range) partner spaces when the range (domain) is fixed. When an r.i. partner space exists at all, a complete characterisation of the situation is given. We illustrate the results with a variety of examples of sharp particular results involving customary function spaces.and α 0 ≥ 1 and α ∞ ∈ [−1, 0], then the optimal (smallest possible) r.i. range space Y such thatis the space whose associate space has normSuch results have not been available before, and the latter norm cannot be identified with any customary known one. We get analogous sets of examples for other operators, too. For example in the case of the fractional maximal operator we essentially improve some results from earlier papers such as [33,35,36,60]. PreliminariesIn this section we collect all the background material that will be used in the paper. We start with the operation of the nonincreasing rearrangement of a measurable function.Throughout this section, let (R, µ) be a σ-finite nonatomic measure space. We setis a µ−measurable function on R with values in [−∞, ∞]}, M 0 (R, µ) = {f ∈ M(R, µ) : f is finite µ-a.e. on R} and M + (R, µ) = {f ∈ M(R, µ) : f ≥ 0}.

Embeddings of homogeneous Sobolev spaces on the entire space

“…which was shown in the proof of [20,Theorem 4.5]. Now, we shall prove that T α is bounded on X ′ (0, ∞) in the remaining cases, that is, p = ∞ and q ∈ [1, ∞), or p = q = ∞ and α ∞ < 0.…”

“…The first inequality was proved in [14, Theorem 3.4] for (0, 1) instead of (0, ∞). However, the proof works just as well for (0, ∞) when combined with the argument from the proof of [20,Lemma 4.10]. For the sake of brevity, the details are omitted.…”

“…Fortunately, it can be simplified significantly in many customary situations. This is the content of the following statement, which follows from theorems 4.2 and 4.7 in [21]. We shall need the operator T α defined for any fixed α ∈ (0, 1) by…”

“…However, this time there is no 'localization', unlike the situation in which the full gradient is considered on the right-hand side, so the integral and supremal operators that we need to handle to obtain not only the optimal rearrangement-invariant spaces on either side of (1.3), but also various interesting examples of such spaces act on unbounded intervals, which complicates the analysis of their behaviour. In order to overcome that, we need to suitably utilize recent results on such operators obtained in [21,33]. Careful synthesis of new as well as well-established results with sharp estimates enables us to thoroughly analyse (1.3) and provide sharp results without any unnecessary assumptions on function norms.…”

Compactness of Sobolev embeddings and decay of norms

“…for every g ∈ M(0, ν(Ω)), where the last but one inequality follows from a simple modification of [17,Lemma 4.10]. Combining these two chains of inequalities with (4.8) and (4.7), we find that T is bounded on Y ′ (0, ν(Ω)).…”