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We obtain the fractional Hardy-Sobolev inequality on homogeneous Lie groups. We prove a fractional version of the logarithmic Hardy-Sobolev inequality on homogeneous Lie groups, which seems new even in the Euclidean setting. As an application, we establish the existence of a weak solution for fractional sub-Laplacians with Hardy potential. Bibliography: 23 titles., UDC 517.9 1 Introduction 1.1. Fractional Hardy-Sobolev inequality. Let [OMEGA] [subset] RN be a measurable set. Then the well-known (classical) Sobolev inequality is formulated as [Please download the PDF to [...]

By using the Laplace transform method, we revisit the multivariate Mittag-Leffler function as an effective tool to construct a solution for some classes of fractional differential equations with constant coefficients. To support our results, we discuss several particular cases related to classical fractional differential operators. The techniques are not only restricted to fractional derivative operators but also can be applied to general constant coefficient differential equations, including high-order ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy--Littlewood--Sobolev and the Stein--Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in the Euclidean space. This work further complements the ranges of p and q (namely, q ≤ p<0) considered in the work of Ruzhansky & Verma (Ruzhansky & Verma 2019 Proc. R. Soc. A 475, 20180310 (doi:10.1098/rspa.2018.0310); Ruzhansky & Verma. 2021Proc. R. Soc. A 477, 20210136 (doi:10.1098/rspa.2021.0136)), which treated the cases 1

q, respectively. [ABSTRACT FROM AUTHOR]

It is proved that, of all the domains with identical measure, it is the ball that maximizes the first eigenvalue of the Riesz potential. It is shown that the sum of the squares of all the eigenvalues is also maximized in the ball among all the domains with identical measure. [ABSTRACT FROM AUTHOR]

Under study are the so-called permeable potential boundary conditions for the Laplace-Beltrami operator defined in a domain Ω on the unit sphere S in ℝ. The permeability of boundary conditions means that a solution to a boundary value problem in Ω coincides with a solution to the Laplace-Beltrami equation on the whole sphere in absence of any boundary conditions. [ABSTRACT FROM AUTHOR]

In a bounded connected domain, we obtain boundary conditions for the volume potential for the polyharmonic equation. [ABSTRACT FROM AUTHOR]

By using a special decomposition of the fundamental solution, we construct a new representation of the Green function of the Dirichlet problem for the polyharmonic equation in a ball. [ABSTRACT FROM AUTHOR]

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg-Kennard type and Heisenberg-Pauli-Weyl type uncertainty inequalities, as well as Caffarelli-Kohn-Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic [Formula: see text], and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings., Competing Interests: We declare we have no competing interests.