Boundedness of classical operators on rearrangement-invariant spaces

Abstract

We study the behaviour on rearrangement-invariant 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 a rearrangement-invariant 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.

38 pages

References in corpus (6)

• Pointwise Symmetrization Inequalities for Sobolev functions and applications
• The optimal range of the Calderòn operator and its applications
• Boundedness of the Hilbert transform on weighted Lorentz spaces
• Optimal domain spaces in Orlicz-Sobolev embeddings
• Fractional Maximal Operator in Orlicz spaces
• Interpolation theorem for Marcinkiewicz spaces with applications to Lorentz gamma spaces

Cited by in corpus (12)

• Embeddings of homogeneous Sobolev spaces on the entire space
• Fractional Maximal Operator in Orlicz spaces
• Reduction principle for Gaussian $K$-inequality
• On Bilinear Hardy Inequality and Corresponding Geometric Mean Inequality
• Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space
• Compactness of Sobolev embeddings and decay of norms
• Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces
• On the properties of rearrangement-invariant quasi-Banach function spaces
• Increasingly global convergence of Hermite serie
• Non-improvability of sharp endpoint estimates
• Non-strict singularity of optimal Sobolev embeddings
• Optimal function spaces in weighted Sobolev embeddings with $\alpha$-homogeneous weights