Garsia-Rodemich Spaces: Bourgain-Brezis-Mironescu space, embeddings and\n rearrangement invariant spaces

2016/08/28 by Mario Milman, Milman, Mario
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1608.07849

openalex publication_date 2016/08/28 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We extend the construction of Garsia-Rodemich spaces in different directions.\nWe show that the new space \B, introduced by Bourgain-Brezis-Mironescu\n citebbm, can be described via a suitable scaling of the Garsia-Rodemich\nnorms. As an application we give a new proof of the embeddings BMO\⊂\n\B \⊂ L(n^\′,\∞). We then generalize the\nGarsia-Rodemich construction and introduce the GaRo_X spaces associated\nwith a rearrangement invariant space X, in such a way that GaRo_X=X, for\na large class of rearrangement invariant spaces. The underlying inequality for\nthis new characterization of rearrangement invariant spaces is an extension of\nthe rearrangement inequalities of citemilbmo. We introduce Gagliardo\nseminorms adapted to rearrangement invariant spaces and use our generalized\nGarsia-Rodemich construction to prove Fractional Sobolev inequalities in this\ncontext.\n

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