scholarly journals Criteria of General Weak Type Inequalities for Integral Transforms with Positive Kernels

1994 ◽  
Vol 1 (1) ◽  
pp. 9-29
Author(s):  
I. Genebashvili ◽  
A. Gogatishvili ◽  
V. Kokilashvili
Keyword(s):  
Weak Type ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Integral Transforms ◽  
Lorentz Spaces ◽  
Quasiconvex Functions ◽  
Homogeneous Type ◽  
Necessary And Sufficient ◽  
Positive Kernels ◽  
Homogeneous Type Space

Abstract Necessary and sufficient conditions are derived in order that an inequality of the form be fulfilled for some positive c independent of λ and a ν-measurable nonnegative function ƒ : X → R 1, where k : X × X × [0, ∞) → R 1 is a nonnegative measurable kernel, (X, d, μ) is a homogeneous type space, φη and ψ are quasiconvex functions, ψ ∈ Δ2, and t –α θ(t) is a decreasing function for some α, 0 < α < 1. A similar problem was solved in Lorentz spaces with weights.

scholarly journals CRITERIA FOR THE BOUNDEDNESS AND COMPACTNESS OF INTEGRAL TRANSFORMS WITH POSITIVE KERNELS

2001 ◽  
Vol 44 (2) ◽  
pp. 267-284 ◽  
Author(s):  
A. Meskhi
Keyword(s):  
Borel Measure ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Integral Transforms ◽  
Necessary And Sufficient Conditions ◽  
Subject Classification ◽  
Necessary And Sufficient ◽  
Positive Kernels ◽  
Primary 46B50

AbstractThe necessary and sufficient conditions that guarantee the boundedness and compactness of integral operators with positive kernels from $L^p(a,b)$ to $L^q_{\nu}(a,b)$, where $p,q\in(1,\infty)$ or $0lt q\leq1lt plt\infty$, for a non-negative Borel measure $\nu$ on $(a,b)$ are found.AMS 2000 Mathematics subject classification: Primary 46B50; 47B34; 47B38

Weighted Reverse Weak Type Inequality for General Maximal Functions

1995 ◽  
Vol 2 (3) ◽  
pp. 277-290
Author(s):  
J. Genebashvili
Keyword(s):  
Weak Type ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Maximal Functions ◽  
Homogeneous Type ◽  
Reverse Inequality ◽  
Weak Type Inequality ◽  
Type Spaces ◽  
Necessary And Sufficient ◽  
Homogeneous Type Spaces

Abstract Necessary and sufficient conditions are found to be imposed on a pair of weights, for which a weak type two-weighted reverse inequality holds, in the case of general maximal functions defined in homogeneous type spaces.

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

scholarly journals Some results of Watson, Plancherel type integral transforms related to the Hartley, Fourier convolutions and applications

2021 ◽  
Author(s):  
Tuan Trinh
Keyword(s):  
Differential Equations ◽  
Type Theorem ◽  
Sufficient Conditions ◽  
Integral Transforms ◽  
System Of Differential Equations ◽  
Symmetric Form ◽  
Original Function ◽  
Type Integral ◽  
Necessary And Sufficient ◽  
Sequence Of Functions

In this work, we study the Watson-type integral transforms for the convolutions related to the Hartley and Fourier transformations. We establish necessary and sufficient conditions for these operators to be unitary in the L 2 (R) space and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulated the Plancherel-type theorem for the aforementioned operators and prove a sequence of functions that converge to the original function in the defined L 2 (R) norm. Next, we study the boundedness of the operators (T k ). Besides, showing the obtained results, we demonstrate how to use it to solve the class of integro-differential equations of Barbashin type, the differential equations, and the system of differential equations. And there are numerical examples given to illustrate these.

scholarly journals Norm inequalities in multidimensional Lorentz spaces

2008 ◽  
Vol 103 (2) ◽  
pp. 278
Author(s):  
Boris Simonov ◽  
Sergey Tikhonov
Keyword(s):  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Double Trigonometric Series ◽  
Norm Inequalities ◽  
Necessary And Sufficient

In this paper we obtain necessary and sufficient conditions for double trigonometric series to belong to generalized Lorentz spaces, not symmetric in general. Estimates for the norms are given in terms of coefficients.

scholarly journals H-function with complex parameters I: existence

2001 ◽  
Vol 25 (9) ◽  
pp. 571-586
Author(s):  
Fadhel A. Al-Musallam ◽  
Vu Kim Tuan
Keyword(s):  
Sufficient Conditions ◽  
Integral Transforms ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Type Integral ◽  
Necessary And Sufficient

AnH-function with complex parameters is defined by a Mellin-Barnes type integral. Necessary and sufficient conditions under which the integral defining theH-function converges absolutely are established. Some properties, special cases, and an application to integral transforms are given.

Two-Weighted Inequalities for Integral Operators in Lorentz Spaces Defined on Homogeneous Groups

1999 ◽  
Vol 6 (1) ◽  
pp. 65-82
Author(s):  
V. Kokilashvili ◽  
A. Meskhi
Keyword(s):  
Singular Integrals ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Hardy Operator ◽  
Weighted Inequalities ◽  
Homogeneous Groups ◽  
Necessary And Sufficient

Abstract The optimal sufficient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In some particular case the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces defined on homogeneous groups.

scholarly journals Indices, convexity and concavity of Calderón-Lozanovskii spaces

2003 ◽  
Vol 92 (1) ◽  
pp. 141 ◽  
Author(s):  
A. Kamińska ◽  
L. Maligranda ◽  
L. E. Persson
Keyword(s):  
Banach Space ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Boyd Indices ◽  
Type And Cotype ◽  
Convexity And Concavity ◽  
Necessary And Sufficient

In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space $E_\varphi$, generated by a quasi-Banach space $E$ and an increasing Orlicz function $\varphi$. We give estimations of convexity and concavity indices of $E_\varphi$ in terms of Matuszewska-Orlicz indices of $\varphi$ as well as convexity and concavity indices of $E$. In the case when $E_\varphi$ is a rearrangement invariant space we also provide some estimations of its Boyd indices. As corollaries we obtain some necessary and sufficient conditions for normability of $E_\varphi$, and conditions on its nontrivial type and cotype in the case when $E_\varphi$ is a Banach space. We apply these results to Orlicz-Lorentz spaces receiving estimations, and in some cases the exact values of their convexity, concavity and Boyd indices.

Imbeddings Between Weighted Orlicz–Lorentz Spaces

1997 ◽  
Vol 4 (2) ◽  
pp. 117-128
Author(s):  
M. Krbec ◽  
J. Lang
Keyword(s):  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Abstract We establish necessary and sufficient conditions for imbeddings of weighted Orlicz–Lorentz spaces.

Riesz potential in the local Morrey–Lorentz spaces and some applications

2020 ◽  
Vol 27 (4) ◽  
pp. 557-567
Author(s):  
Vagif S. Guliyev ◽  
Abdulhamit Kucukaslan ◽  
Canay Aykol ◽  
Ayhan Serbetci
Keyword(s):  
Maximal Operator ◽  
Riesz Potential ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Analytic Semigroups ◽  
Fractional Powers ◽  
Marcinkiewicz Operator ◽  
Fractional Maximal Operator ◽  
Necessary And Sufficient

AbstractIn this paper, the necessary and sufficient conditions are found for the boundedness of the Riesz potential {I_{\alpha}} in the local Morrey–Lorentz spaces {M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})}. This result is applied to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups on the local Morrey–Lorentz spaces {M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})}.

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