Weighted estimates for bilinear fractional integral operators: a necessary and sufficient condition for power weights

2019 ◽  
Vol 71 (1) ◽  
pp. 25-37
Author(s):  
Yasuo Komori-Furuya
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Fractional Integral Operators ◽  
Weighted Estimates ◽  
Power Weights ◽  
Necessary And Sufficient

Compactness criteria for fractional integral operators

2019 ◽  
Vol 22 (5) ◽  
pp. 1269-1283 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Mieczysław Mastyło ◽  
Alexander Meskhi
Keyword(s):  
Integral Operator ◽  
Measure Space ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Bounded Domains ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Rectifiable Curves ◽  
Necessary And Sufficient

Abstract We establish necessary and sufficient conditions for the compactness of fractional integral operators from Lp(X, μ) to Lq(X, μ) with 1 < p < q < ∞, where μ is a measure on a quasi-metric measure space X. As an application we obtain criteria for the compactness of fractional integral operators defined in weighted Lebesgue spaces over bounded domains of the Euclidean space ℝn with the Lebesgue measure, and also for the fractional integral operator associated to rectifiable curves of the complex plane.

15.—The Measure of Non-compactness of Some Linear Integral Operators

1973 ◽  
Vol 71 (2) ◽  
pp. 167-179
Author(s):  
C. A. Stuart
Keyword(s):  
Compact Operator ◽  
Essential Spectrum ◽  
Differential Operators ◽  
Integral Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Set ◽  
Linear Integral ◽  
Necessary And Sufficient ◽  
Linear Integral Operators

SynopsisThe measure of non-compactness of linear integral operators on the half-line [0, ∞) of a special type is studied. In particular, a necessary and sufficient condition is established for an operator of this type to define a compact operator from L2(0, ∞) into itself. These results are then used to discuss the spectrum of second-order differential operators. A necessary and sufficient condition for the spectrum to be discrete is established together with estimates for the distance of a point in the resolvent set from the essential spectrum.

scholarly journals Some applications of generalized Ruscheweyh derivatives involving a general fractional derivative operator to a class of analytic functions with negative coefficients II

2020 ◽  
Vol 28 (1) ◽  
pp. 85-103
Author(s):  
Waggas Galib Atshan ◽  
S. R. Kulkarni
Keyword(s):  
Fractional Derivative ◽  
Linear Combination ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Integral Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Derivative Operator ◽  
Necessary And Sufficient

AbstractIn this paper, we study a class of univalent functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfying{\mathop{\rm Re}\nolimits} \left\{{{{z\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)'} \over {\left({1 - \gamma} \right){\bf{J}}_1^{\lambda,\mu}f\left(z \right) + \gamma {z^2}\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)''}}} \right\} > \beta.A necessary and sufficient condition for a function to be in the class A_\gamma ^{\lambda,\mu,\nu}\left({n,\beta} \right) is obtained. Also, our paper includes linear combination, integral operators and we introduce the subclass A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) consisting of functions with negative and fixed finitely many coefficients. We study some interesting properties of A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right).

Two-weight norm estimates for sublinear integral operators in variable exponent Lebesgue spaces

2014 ◽  
Vol 51 (3) ◽  
pp. 384-406 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Measure Space ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Integral Operators ◽  
Doubling Condition ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Norm Estimates ◽  
Variable Exponent Lebesgue Spaces ◽  
Necessary And Sufficient

Two-weight norm estimates for sublinear integral operators involving Hardy-Littlewood maximal, Calderón-Zygmund and fractional integral operators in variable exponent Lebesgue spaces are derived. Operators and the space are defined on a quasi-metric measure space with doubling condition. The derived conditions are written in terms ofLp(·)norms and are simultaneously necessary and sufficient for appropriate inequalities for maximal and fractional integral operators mainly in the case when weights are of radial type.

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