Advanced Nonlinear Studies (Mar 2024)

Sharp affine weighted L 2 Sobolev inequalities on the upper half space

  • Dou Jingbo,
  • Hu Yunyun,
  • Yue Caihui

DOI
https://doi.org/10.1515/ans-2023-0117
Journal volume & issue
Vol. 24, no. 2
pp. 415 – 435

Abstract

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We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.

Keywords