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 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 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 Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

scholarly journals A Highly D‑efficient Spring Balance Weighing Design for an Even Number of Objects

2019 ◽  
Vol 5 (344) ◽  
pp. 17-27
Author(s):  
Małgorzata Graczyk ◽  
Bronisław Ceranka
Keyword(s):  
Optimal Design ◽  
Sufficient Conditions ◽  
Optimality Criteria ◽  
Point Of View ◽  
Necessary And Sufficient Conditions ◽  
Optimal Designs ◽  
Efficient Design ◽  
Spring Balance ◽  
Experimental Errors ◽  
Necessary And Sufficient

The problem of determining unknown measurements of objects in the model of spring balance weighing designs is presented. These designs are considered under the assumption that experimental errors are uncorrelated and that they have the same variances. The relations between the parameters of weighing designs are deliberated from the point of view of optimality criteria. In the paper, designs in which the product of the variances of estimators is possibly the smallest one, i.e. D‑optimal designs, are studied. A highly D‑efficient design in classes in which a D‑optimal design does not exist are determined. The necessary and sufficient conditions under which a highly efficient design exists and methods of its construction, along with relevant examples, are introduced.

scholarly journals Variable exponent Sobolev spaces on metric measure spaces

2006 ◽  
Vol 36 (0) ◽  
pp. 79-94 ◽  
Author(s):  
Petteri Harjulehto ◽  
Peter Hästö ◽  
Mikko Pere
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Metric Measure Spaces ◽  
Variable Exponent Sobolev Spaces ◽  
Measure Spaces

scholarly journals Generalized Bessel and Riesz Potentials on Metric Measure Spaces

2009 ◽  
Vol 30 (4) ◽  
pp. 315-340 ◽  
Author(s):  
J. Hu ◽  
M. Zähle
Keyword(s):  
Metric Measure Spaces ◽  
Riesz Potentials ◽  
Measure Spaces

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.

Sobolev’s Inequality for Riesz Potentials of Functions in Musielak–Orlicz–Morrey Spaces Over Non-doubling Metric Measure Spaces

2019 ◽  
Vol 63 (2) ◽  
pp. 287-303
Author(s):  
Takao Ohno ◽  
Tetsu Shimomura
Keyword(s):  
Morrey Spaces ◽  
Variable Exponents ◽  
Metric Measure Spaces ◽  
Riesz Potentials ◽  
Sobolev’S Inequality ◽  
Double Phase ◽  
Measure Spaces

AbstractOur aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

A simple proof of a random stable limit theorem

1970 ◽  
Vol 7 (2) ◽  
pp. 502-504 ◽  
Author(s):  
Stephen R. Kimbleton
Keyword(s):  
Limit Theorem ◽  
Elementary Proof ◽  
General Theorem ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Limit ◽  
Stable Law ◽  
Sufficient Conditions For Convergence ◽  
Necessary And Sufficient

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.

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