Fractional Hardy type inequalities on homogeneous Lie groups in the case Q<spQ<sp

In this paper, we obtain a fractional Hardy inequality in the case Q < s p on homogeneous Lie groups, and as an application we show the corresponding uncertainty principle. Also, we show a fractional Hardy–Sobolev-type inequality on homogeneous Lie groups. In addition, we prove fractional logarithmic Hardy–Sobolev and fractional Nash-type inequalities on homogeneous Lie groups. We note that the case Q > s p was extensively studied in the literature, while here we are dealing with the complementary range Q < s p

Fractional Hardy-type inequalities on homogeneous Lie groups in the case Q&lt;sp

In this paper, we obtain a fractional Hardy inequality in the case Qspwas extensively studied in the literature, while here we are dealing with the complementary range Q<sp

Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups

In this paper, we prove the fractional logarithmic Sobolev inequality on homogeneous groups. Also, we establish fractional logarithmic Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups. In addition, we show blow-up results for the fractional heat equation with logarithmic nonlinearity on homogeneous groups

Geometric Hardy Inequalities on Starshaped Sets

We present geometric Hardy inequalities on starshaped sets in Carnot groups. Also, we obtain geometric Hardy inequalities on half-spaces for general vector fields

Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub- Laplacians which are operators of the form $$\mathcal{L}_p f := \sum_{i=1}^{N}X_i(|X_if|^{p_i-2}X_i f), \quad 1<p_i< \infty,$$ where $X_i, i=1,\ldots, N,$ are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups

Geometric Hardy Inequalities on Starshaped Sets

We present geometric Hardy inequalities on starshaped sets in Carnot groups. Also, we obtain geometric Hardy inequalities on half-spaces for general vector fields

CRITICAL HARDY INEQUALITIES

19 pages; new inequalities have been added yielding also new results on Rn; therefore, the title has been also change

Elements of Potential Theory on Carnot Groups

We propose and study elements of potential theory for the sub-Laplacian on homogeneous Carnot groups. In particular, we show the continuity of the single-layer potential and establish Plemelj-type jump relations for the double-layer potential. As a consequence, we derive a formula for the trace on smooth surfaces of the Newton potential for the sub-Laplacian. Using this, we construct a sub-Laplacian version of Kac’s boundary value problem

Green's Identities, Comparison Principle and Uniqueness of Positive Solutions for Nonlinearp-sub-Laplacian Equations on Stratified Lie Groups

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Isoperimetric inequalities for Schatten norms of Riesz potentials

In this note we prove that the ball is a maximiser for integer order Schatten p-norms of the Riesz potential operators among all domains of a given measure in R d . In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities