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Register a new userHardy inequalities on metric measure spaces, II: The case $p>q$
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In this note we continue giving the characterisation of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. This is a continuation of our paper [M. Ruzhansky and D. Verma. Hardy inequalities on metric measure spaces, Proc. R. Soc. A., 475(2223):20180310, 2018] where we treated the case $pleq q$. Here the remaining range $p>q$ is considered, namely, $0<q<p$, $1<p<infty.$ We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. We note that doubling conditions are not required for our analysis.
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In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.
We find sharp conditions on the growth of a rooted regular metric tree such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.
This is a continuation of the papers [Kuryakov-Sukochev, JFA, 2015] and [Sadovskaya-Sukochev, PAMS, 2018], in which the isomorphic classification of $L_{p,q}$, for $1< p<infty$, $1le q<infty$, $p e q $, on resonant measure spaces, has been obtained. The aim of this paper is to give a complete isomorphic classification of $L_{p,q}$-spaces on general $sigma$-finite measure spaces. Towards this end, several new subspaces of $L_{p,q}(0,1)$ and $L_{p,q}(0,infty)$ are identified and studied.
We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d, formulated in terms of the Laplacian Delta and of the fractional powers D^n := (-Delta)^(n/2) with real n >= 0; we review known facts and present novel results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the L^2 case where, for all sufficiently regular f : R^d -> C, the norm || D^j f||_{L^r} is bounded in terms of || f ||_{L^2} and || D^n f ||_{L^2} for 1/r = 1/2 - (theta n - j)/d, and suitable values of j,n,theta (with j,n possibly noninteger). In the special cases theta = 1 and theta = j/n + d/2 n (i.e., r = + infinity), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general L^2 case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the Gagliardo-Nirenberg and Sobolev sharp constants are confined to quite narrow intervals. Several examples are given.
In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Rellich, Hardy-Littllewood-Sobolev, Galiardo-Nirenberg, Caffarelli-Kohn-Nirenberg and Trudinger-Moser inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. Moreover, we obtain sever
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