Durvudkhan Suragan

On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups ☆

Abstract In this paper, we present a version of horizontal weighted Hardy–Rellich type and Caffarelli–Kohn–Nirenberg type inequalities on stratified groups and study some of their consequences. Our results reflect on many results previously known in special cases. Moreover, a new simple proof of the Badiale–Tarantello conjecture [2] on the best constant of a Hardy type inequality is provided. We also show a family of Poincare inequalities as well as inequalities involving the weighted and unweighted p -sub-Laplacians.

Isoperimetric inequalities for the logarithmic potential operator

Abstract In this paper we prove that the disc is a maximiser of the Schatten p-norm of the logarithmic potential operator among all domains of a given measure in R 2 , for all even integers 2 ≤ p ∞ . We also show that the equilateral triangle has the largest Schatten p-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh–Faber–Krahn or Polya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

Abstract We give sharp remainder terms of L p and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy–Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.

Anisotropic L2-weighted Hardy and L2-Caffarelli–Kohn–Nirenberg inequalities

We establish sharp remainder terms of the L2-Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli–...

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

In this note we prove an analogue of the Rayleigh–Faber–Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere

On Green functions for Dirichlet sub-Laplacians on H-type groups ☆

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Caffarelli–Kohn–Nirenberg and Sobolev type inequalities on stratified Lie groups

Sn and on the real hyperbolic space