Interior Schauder-type estimates for m − th order elliptic operators in rearrangement-invariant Sobolev spaces

Mamedov E. M., Çetin Ş.

Turkish Journal of Mathematics, vol.48, no.4, pp.793-816, 2024 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48 Issue: 4
  • Publication Date: 2024
  • Doi Number: 10.55730/1300-0098.3541
  • Journal Name: Turkish Journal of Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.793-816
  • Keywords: Banach function space, elliptic operator, rearrangement-invariant spaces, Schauder type estimate, Sobolev spaces
  • Istanbul Gelisim University Affiliated: No

Abstract

In this study, we investigate the m-th order elliptic operators on n-dimensional bounded domain Ω ⊂ Rn with discontinuous coefficients in the rearrangement-invariant Sobolev space WXm (Ω). In general, the considered rearrangement-invariant spaces are not separable, so the use of classical methods in these spaces requires substantial modification of classical methods and a lot of preparation, concerning correctness of substitution operator, problems related to the extension operator in such spaces, etc. For this purpose, the corresponding separable subspaces of these spaces, in which the set of compact supported infinitely differentiable functions is dense, are introduced based on the shift operator. We establish interior Schauder-type estimates in the above subspaces. Note that Lebesgue spaces Lp (Ω), grand-Lebesgue spaces, Marcinkiewicz spaces, weak-type Lwp spaces, etc. are also covered by such spaces.