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Banach algebras of weakly differentiable functions

  • Andrea CianchiEmail author
  • Luboš Pick
  • Lenka Slavíková
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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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Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Andrea Cianchi
    • 1
    Email author
  • Luboš Pick
    • 2
  • Lenka Slavíková
    • 2
  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic

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