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Banach algebras of weakly differentiable functions

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Abstract

The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an n-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also established. The special cases of Orlicz-Sobolev and Lorentz-Sobolev spaces are discussed in detail. New results for classical Sobolev spaces on possibly irregular domains follow as well.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50137, Firenze, Italy

    Andrea Cianchi

  2. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Luboš Pick & Lenka Slavíková

Authors
  1. Andrea Cianchi
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  2. Luboš Pick
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  3. Lenka Slavíková
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Correspondence to Andrea Cianchi.

Additional information

This research was partly supported by the Research Project of Italian Ministry of University and Research (MIUR) “Elliptic and parabolic partial differential equations: geometric aspects, related inequalities, and applications” 2012, by GNAMPA of Italian INdAM (National Institute of High Mathematics), and by the grant P201-13-14743S of the Grant Agency of the Czech Republic.

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Cianchi, A., Pick, L. & Slavíková, L. Banach algebras of weakly differentiable functions. JAMA 138, 473–511 (2019). https://doi.org/10.1007/s11854-019-0030-x

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