scholarly journals A note on the trace inequality for Riesz potentials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Giorgi Imerlishvili ◽  
Alexander Meskhi
Keyword(s):  
Integrable Function ◽  
Riesz Potential ◽  
Sufficient Condition ◽  
Homogeneous Type ◽  
Trace Inequality ◽  
Spaces Of Homogeneous Type ◽  
Necessary And Sufficient Condition ◽  
Riesz Potentials ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

AbstractWe establish a necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality\lVert I_{\alpha}f\rVert_{L^{p}_{v}(\mathbb{R}^{n})}\leq C\lVert f\rVert_{L^{p% ,1}(\mathbb{R}^{n})}for the Riesz potential {I_{\alpha}}, where {L^{p,1}(\mathbb{R}^{n})} is the Lorentz space. The same problem is studied for potentials defined on spaces of homogeneous type.

Scattering Length and the Spectrum of –Δ + V

2006 ◽  
Vol 49 (1) ◽  
pp. 144-151 ◽  
Author(s):  
Michael Taylor
Keyword(s):  
Discrete Spectrum ◽  
Integrable Function ◽  
Scattering Length ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

AbstractGiven a non-negative, locally integrable functionV on ℝn, we give a necessary and sufficient condition that –Δ + V have purely discrete spectrum, in terms of the scattering length ofV restricted to boxes.

scholarly journals A trace inequality for generalized potentials in Lebesgue spaces with variable exponent

2004 ◽  
Vol 2 (1) ◽  
pp. 55-69 ◽  
Author(s):  
David E. Edmunds ◽  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Variable Exponent ◽  
Homogeneous Type ◽  
Trace Inequality ◽  
Spaces Of Homogeneous Type ◽  
Weighted Inequality ◽  
Euclidean Spaces ◽  
Riesz Potentials ◽  
Lebesgue Spaces ◽  
Fractal Sets

A trace inequality for the generalized Riesz potentialsIα(x)is established in spacesLp(x)defined on spaces of homogeneous type. The results are new even in the case of Euclidean spaces. As a corollary a criterion for a two-weighted inequality in classical Lebesgue spaces for potentialsIα(x)defined on fractal sets is derived.

scholarly journals Syarat Perlu dan Cukup untuk Keterbatasan Potensial Riesz di Ruang Morrey Klasik

2013 ◽  
Vol 14 (3) ◽  
pp. 227
Author(s):  
Mohammad Imam Utoyo ◽  
Basuki Widodo ◽  
Toto Nusantara ◽  
Suhariningsih Suhariningsih
Keyword(s):  
Characteristic Function ◽  
Morrey Space ◽  
Necessary Conditions ◽  
Riesz Potential ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Research Results ◽  
Necessary And Sufficient

This script was aimed to determine the necessary conditions for boundedness of Riesz potential in the classical Morrey space. If these results are combined with previous research results will be obtained the necessary and sufficient condition for boundedness of Riesz potential. This necessary condition is obtained through the use of characteristic function as one member of the classical Morrey space.

On the Nørlund Summability of a Class of Fourier Series

1970 ◽  
Vol 22 (1) ◽  
pp. 86-91 ◽  
Author(s):  
Badri N. Sahney
Keyword(s):  
Fourier Series ◽  
Integrable Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lebesgue Integrable Function ◽  
Summability Of Fourier Series ◽  
Lebesgue Set ◽  
Necessary And Sufficient

1. Our aim in this paper is to determine a necessary and sufficient condition for N∅rlund summability of Fourier series and to include a wider class of classical results. A Fourier series, of a Lebesgue-integrable function, is said to be summable at a point by N∅rlund method (N, pn), as defined by Hardy [1], if pn → Σpn → ∞, and the point is in a certain subset of the Lebesgue set. The following main results are known.

Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces

2018 ◽  
Vol 25 (2) ◽  
pp. 235-248
Author(s):  
Vagif S. Guliyev ◽  
Elman J. Ibrahimov
Keyword(s):  
Differential Operator ◽  
Morrey Space ◽  
Riesz Potential ◽  
Morrey Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Abstract In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator G_{\lambda}=(x^{2}-1)^{\frac{1}{2}-\lambda}\frac{d}{dx}(x^{2}-1)^{\lambda+% \frac{1}{2}}\frac{d}{dx},\quad x\in(1,\infty),\,\lambda\in\Bigl{(}0,\frac{1}{2% }\Bigr{)}. We prove that the G-Riesz potential {I_{G}^{\alpha}} , {0<\alpha<2\lambda+1} , is bounded from the G-Morrey space {L_{p,\lambda,\gamma}} to {L_{q,\lambda,\gamma}} if and only if \frac{1}{p}-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma},\quad 1<p<\frac{2% \lambda+1-\gamma}{\alpha}. Also, we prove that the G-Riesz potential {I_{G}^{\alpha}} is bounded from the G-Morrey space {L_{1,\lambda,\gamma}} to the weak G-Morrey space {WL_{q,\lambda,\gamma}} if and only if 1-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma}.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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