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

11 pages, to appear in JMAA11 pages, to appear in JMAAThe authors were supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK grant 5127/GF4. No new data was collected or generated during the course of this research

Sobolev Type Inequalities, Euler–Hilbert–Sobolev and Sobolev–Lorentz–Zygmund Spaces on Homogeneous Groups

We define Euler–Hilbert–Sobolev spaces and obtain embedding results on homogeneous groups using Euler operators, which are homogeneous differential operators of order zero. Sharp remainder terms of Lp and weighted Sobolev type and Sobolev–Rellich inequalities on homogeneous groups are given. Most inequalities are obtained with best constants. As consequences, we obtain analogues of the generalised classical Sobolev type and Sobolev–Rellich inequalities. We also discuss applications of logarithmic Hardy inequalities to Sobolev–Lorentz–Zygmund spaces. The obtained results are new already in the anisotropic Rn as well as in the isotropic Rn due to the freedom in the choice of any homogeneous quasi-norm. © 2018, The Author(s)

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

In this short paper, we establish a range of Caffarelli–Kohn–Nirenberg and weighted Lp-Sobolev type inequalities on stratified Lie groups. All inequalities are obtained with sharp constants. Moreover, the equivalence of the Sobolev type inequality and Hardy inequality is shown in the L2-case. © 2017, The Author(s)

Improved critical Hardy inequalities on 2-dimensional quasi-balls

In this note we obtain a remainder estimate for improved critical Hardy inequalities on a 2-dimensional quasi-ball on homogeneous Lie groups. These results are new even in the Abelian case of ℝ2 in terms of choosing any choice of homogeneous quasi-norm as well as replacing the full gradient by the radial derivative. © 2017 Author(s)

Improved critical Hardy inequalities on 2-dimensional quasi-balls

In this note we obtain a remainder estimate for improved critical Hardy inequalities on a 2-dimensional quasi-ball on homogeneous Lie groups. These results are new even in the Abelian case of ℝ2 in terms of choosing any choice of homogeneous quasi-norm as well as replacing the full gradient by the radial derivative. © 2017 Author(s)

HARDY-LITTLEWOOD, BESSEL-RIESZ, AND FRACTIONAL INTEGRAL OPERATORS IN ANISOTROPIC MORREY AND CAMPANATO SPACES

29 page

Sobolev Type Inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund Spaces on Homogeneous Groups

34 pages; The title and document style have been changed, to appear in Integral Equations and Operator Theor