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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}.

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.

Building on recent progress at the intersection of combinatorial optimization and deep learning, we propose an end-to-end trainable architecture for deep graph matching that contains unmodified combinatorial solvers. Using the presence of heavily optimized combinatorial solvers together with some improvements in architecture design, we advance state-of-the-art on deep graph matching benchmarks for keypoint correspondence. In addition, we highlight the conceptual advantages of incorporating solvers into deep learning architectures, such as the possibility of post-processing with a strong multi-graph matching solver or the indifference to changes in the training setting. Finally, we propose two new challenging experimental setups.