scholarly journals Reverse integral Hardy inequality on metric measure spaces

2021 ◽  
Vol 47 (1) ◽  
pp. 39-55
Author(s):  
Aidyn Kassymov ◽  
Michael Ruzhansky ◽  
Durvudkhan Suragan
Keyword(s):  
Hardy Inequality ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Minkowski Inequality ◽  
Hyperbolic Spaces ◽  
Metric Measure Spaces ◽  
Continuous Version ◽  
Homogeneous Groups ◽  
Measure Spaces ◽  
Necessary And Sufficient

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. In addition, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.  

scholarly journals Parameter depending almost monotonic functions and their applications to dimensions in metric measure spaces

2009 ◽  
Vol 7 (1) ◽  
pp. 61-89 ◽  
Author(s):  
Natasha Samko
Keyword(s):  
Operator Theory ◽  
Sufficient Conditions ◽  
Metric Measure Spaces ◽  
Lower And Upper Bounds ◽  
Index Numbers ◽  
Weights And Measures ◽  
Monotonic Functions ◽  
Measure Spaces ◽  
Necessary And Sufficient

In connection with application to various problems of operator theory, we study almost monotonic functionsw(x, r) depending on a parameterxwhich runs a metric measure spaceX, and the so called index numbersm(w, x),M(w, x) of such functions, and consider some generalized Zygmund, Bary, Lozinskii and Stechkin conditions. The main results contain necessary and sufficient conditions, in terms of lower and upper bounds of indicesm(w, x) andM(w, x) , for the uniform belongness of functionsw(·,r) to Zygmund-Bary-Stechkin classes. We give also applications to local dimensions in metric measure spaces and characterization of some integral inequalities involving radial weights and measures of balls in such spaces.

scholarly journals Hardy inequalities on metric measure spaces

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180310
Author(s):  
Michael Ruzhansky ◽  
Daulti Verma
Keyword(s):  
Integral Form ◽  
Hyperbolic Spaces ◽  
Hadamard Manifolds ◽  
Metric Measure Spaces ◽  
Differentiable Structure ◽  
Hardy Inequalities ◽  
Homogeneous Groups ◽  
Measure Spaces

In this note, we give several characterizations 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 Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on R n , on homogeneous groups, on hyperbolic spaces and on Cartan–Hadamard manifolds. We note that doubling conditions are not required for our analysis.

scholarly journals Hardy inequalities on metric measure spaces, II: the case p  >  q

2021 ◽  
Vol 477 (2250) ◽  
pp. 20210136
Author(s):  
Michael Ruzhansky ◽  
Daulti Verma
Keyword(s):  
Integral Form ◽  
Hyperbolic Spaces ◽  
Hadamard Manifolds ◽  
Metric Measure Spaces ◽  
Differentiable Structure ◽  
Hardy Inequalities ◽  
Homogeneous Groups ◽  
Link Type ◽  
Measure Spaces

In this paper, we continue our investigations giving the characterization 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 Hardy’s original inequality. This is a continuation of our paper (Ruzhansky & Verma 2018. Proc. R. Soc. A 475 , 20180310 ( doi:10.1098/rspa.2018.0310 )) where we treated the case p  ≤  q . Here the remaining range p  >  q is considered, namely, 0 <  q  <  p , 1 <  p  < ∞. We give several examples of the obtained results, finding conditions on the weights for integral Hardy inequalities on homogeneous groups, as well as on hyperbolic spaces and on more general Cartan–Hadamard manifolds. As in the first part of this paper, we do not need to impose doubling conditions on the metric.

scholarly journals A unified point of view on boundedness of Riesz type potentials

2017 ◽  
pp. 99-121
Author(s):  
Bibiana Iaffei ◽  
Liliana Nitti
Keyword(s):  
Geometric Property ◽  
Variable Exponent ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Point Of View ◽  
Metric Measure Spaces ◽  
Riesz Potentials ◽  
Underlying Space ◽  
Measure Spaces ◽  
Necessary And Sufficient

We introduce a natural extension of the Riesz potentials to quasi-metric measure spaces with an upper doubling measure. In particular, these operators are defined when the underlying space has components of differing dimensions. We study the behavior of the potential on classical and variable exponent Lebesgue spaces, obtaining necessary and sufficient conditions for its boundedness. The technique we use relies on a geometric property of the measure of the balls which holds both in the doubling and non-doubling situations, and allows us to present our results in a unified way.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

scholarly journals Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces

2018 ◽  
Vol 6 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Riemannian Manifolds ◽  
Short Note ◽  
Sufficient Condition ◽  
Metric Measure Spaces ◽  
Necessary And Sufficient Condition ◽  
Local Dimension ◽  
Base Points ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient

Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

scholarly journals Weighted Hardy inequality on Riemannian manifolds

2016 ◽  
Vol 18 (06) ◽  
pp. 1550072 ◽  
Author(s):  
El Hadji Abdoulaye Thiam
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Hardy Inequality ◽  
Compact Riemannian Manifold ◽  
Sufficient Conditions ◽  
Extremal Functions ◽  
Dimension Formula ◽  
Existence Of Minimizers ◽  
Necessary And Sufficient ◽  
Funct Anal

Let [Formula: see text] be a smooth compact Riemannian manifold of dimension [Formula: see text] and let [Formula: see text] to be a closed submanifold of dimension [Formula: see text]. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on [Formula: see text] within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.

N-Lusin property in metric measure spaces: A new sufficient condition

2018 ◽  
Vol 30 (6) ◽  
pp. 1475-1486
Author(s):  
Marcela Garriga ◽  
Pablo Ochoa
Keyword(s):  
Measure Zero ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Metric Measure Spaces ◽  
Type Estimate ◽  
Measure Spaces ◽  
Approximate Differentiability

Abstract In this work, we are concerned with the study of the N-Lusin property in metric measure spaces. A map satisfies that property if sets of measure zero are mapped to sets of measure zero. We prove a new sufficient condition for the N-Lusin property using a weak and pointwise Lipschitz-type estimate. Relations with approximate differentiability in metric measure spaces and other sufficient conditions for the N-Lusin property will be provided.

Dyadic nonlocal diffusions in metric measure spaces

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Marcelo Actis ◽  
Hugo Aimar
Keyword(s):  
Fractional Derivative ◽  
Haar System ◽  
Main Tool ◽  
Nonlocal Diffusion ◽  
Space Of Homogeneous Type ◽  
Metric Measure Spaces ◽  
Norm Convergence ◽  
Initial Value ◽  
Derivative Operator ◽  
Measure Spaces

AbstractIn this paper we solve the initial value problem for the nonlocal diffusion generated by the space fractional derivative induced by the dyadic tilings of M. Christ on a space of homogeneous type. We consider the problems of pointwise and norm convergence to the initial data. The main tool is the use of the Haar system induced by a dyadic tiling, which is actually the set of eigenfunctions for the fractional derivative operator.

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