Weyl Fractional Integral Operators Research Articles

Some Relations for the Generalized G ̃n,P ̃n Integral Transforms and Riemann-Liouville, Weyl Integral Operators

In this paper, Parseval-Goldstein type theorems involving the G ̃n-integral transform which is modified from G_2n-integral transform [7] and the -integral transform [8] are examined. Then, theorems in this paper are shown to yield a number of new identities involving several well-known integral transforms. Using these theorems and their corollaries, a number of interesting infinite integrals of elementary and special functions are presented. Generalizations of Riemann-Liouville and Weyl fractional integral operators are also defined. Some theorems relating generalized Laplace transform, generalized Widder Potential transform, generalized Hankel transform and generalized Bessel transform are obtained. Some illustrative examples are given as applications of these theorems and their results.

New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series.

Interrelationships Between Marichev–Saigo–Maeda Fractional Integral Operators, the Laplace Transform and the $${\overline{H}}$$-Function

In this article, we first evaluate the Laplace transform of Marichev–Saigo–Maeda (M–S–M) fractional integral operators whose kernel is the Appell function $$F_{3}$$ and point out its six special cases (Saigo, Erdelyi–Kober, Riemann–Liouville and Weyl fractional integral operators). Certain new and known results can be obtained as special cases of our key findings. Next, we find the image of $${\overline{H}}$$ -Function under the operators of our study. Some interesting special cases of our key result are also considered and demonstrated to be connected with certain existing ones.

Some identities on fractional integrals and integral transforms

In this paper, we introduce various theorems that associate the generalized Riemann-Liouville fractional integral operator and the generalized Weyl fractional integral operator with some well-known integral transforms including generalized Laplace transform, Widder potential transform, generalized Widder transform, Hankel transform and Bessel transform. We evaluate certain integrals of some elementary functions and some special functions as applications of these theorems and their results.

On a general theorem connecting Laplace transform and generalized Weyl fractional integral operator

The main aim of the present paper is to establish a general theorem which asserts an interesting relationship between the Laplace transform and generalized Weyl fractional integral operator involving general class of functions. Our main theorem involves a multidimensional series with essentially arbitrary sequence of complex numbers. By suitably assigning different values to the sequence, one can easily evaluate generalized Weyl fractional integral operator of special functions of several variables. An illustration for generalized Lauricella function is mentioned. Several interesting special cases of the general theorem are also mentioned.

Weighted weak-type inequalities for some fractional integral operators

For $0<\alpha<1$, let $W_{\alpha}$ and $R_{\alpha}$ denote Weyl fractional integral operator and Riemann-Liouville fractional integral operator, respectively. We establish sharp versions of Muckenhoupt-Wheeden conjecture for these operators. Specifically, we prove that for any weight $w$ on $[0,\infty)$, we have \begin{equation*} \|{W}_{α} f\|_{L^{1/(1-α),∞}(w)}≤ α^{-1}\|{f}\|_{L^{1}((M_{-}w)^{1-α})} \end{equation*} and \begin{equation*} \|{R}_{α} f\|_{L^{1/(1-α),∞}(w)}≤ α^{-1}\|{f}\|_{L^{1}((M_{+}w)^{1-α})}. \end{equation*} Here $M_{-}$, $M_{+}$ denote the one-sided Hardy-Littlewood maximal operators on $[0,\infty)$. In each of the estimates, the constant $\alpha^{-1}$ is the best possible.

Generalized Mittag-Leffler Function Associated with Weyl Fractional Calculus Operators

This paper is devoted to study further properties of generalized Mittag-Leffler functionEα,β,pγ,δ,qassociated with Weyl fractional integral and differential operators. A new integral operatorℰα,β,p,w,∞γ,δ,qdepending on Weyl fractional integral operator and containingEα,β,pγ,δ,q(z)in its kernel is defined and studied, namely, its boundedness. Also, composition of Weyl fractional integral and differential operators with the new operatorℰα,β,p,w,∞γ,δ,qis established.

Convolution integral equation of Fredholm type with certain trancedental functions

The aim of this paper is to establish a solution of a certain class of convolution integral equation of Fredholm type whose kernel involve certain product of special function by using Riemann-Liouville and Weyl fractional integral operators. Some interesting particular cases are also considered.

Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators

Let b be a BMO function, \(0 1) and weighted endpoint estimates (p = 1) for the operator \(I^{+,k}_{\alpha,b}\) and for the pairs of weights of the type (w, \({\mathcal{M}_w}\)), where w is any weight and \(\mathcal{M}\) is a suitable one-sided maximal operator. We also prove that, for \(A^{+}_{\infty}\) weights, the operator \(I^{+,k}_{\alpha,b}\) is controlled in the Lp(w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators \(I^{+,k}_{\alpha,b}\). The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral \(I^{-,k}_{\alpha,b}\)

A GENERAL THEOREM FOR THE GENERALIZED WEYL FRACTIONAL INTEGRAL OPERATOR INVOLVING THE MULTIVARIABLE H-FUNCTION

In this paper we establish a very general and useful theorem which interconnects the Laplace transform and the generalized Weyl fractional integral operator involving the multivariable H-function of related functions of several variables. Our main theorem involves a multidimensional series with essentially arbitrary sequence of complex numbers. By suitably assigning different values to these sequences, one can easily evaluate the generalized Weyl fractional integral operator of special functions of several variables. We have illustrated it for Srivastava-Daoust multivariable hypergeometric function. On account of general nature of this function a number of results involving special functions of one or more variables can be obtained merely by specializing the parameters.

On Weyl fractional integral operators

In this paper, we first establish an interesting theorem exhibiting a relationship existing between the Laplace transform and Weyl fractional integral operator of related functions. This theorem is sufficiently general in nature as it contains $n$ series involving arbitrary complex numbers $ \Omega(r_1,\ldots r_n) $. We have obtained here as applications of the theorem, the Weyl fractional integral operators of Kamp'e de F'eriet function, Appell's functions $ F_1 $, $ F_4 $, Humbert's function $ \Psi_1$ and Lauricella's, triple hypergeometric series $ F_E $. References of known results which follow as special cases of our theorem are also cited. Finally, we obtain some transformations of $ F^{(3)}$ and Kamp'e de F'eriet function with the application of our main theorem .

A GENERAL THEOREM FOR THE GENERALIZED WEYL FRACTIONAL INTEGRAL OPERATOR INVOLVING THE MULTIVARIABLE H-FUNCTION

In this paper we establish a very general and useful theorem which interconnects the Laplace transform and the generalized Weyl fractional integral operator involving the multivariable H-function of related functions of several variables. Our main theorem involves a multidimensional series with essentially arbitrary sequence of complex numbers. By suitably assigning different values to these sequences, one can easily evaluate the generalized Weyl fractional integral operator of special functions of several variables. We have illustrated it for Srivastava-Daoust multivariable hypergeometric function. On account of general nature of this function a number of results involving special functions of one or more variables can be obtained merely by specializing the parameters.

Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces

In this paper we introduce the one-sided weighted spaces L−w (β), −1 <β< 1. The purpose of this definition is to obtain an extension of the Weyl fractional integral operator I+α from Lp w into a suitable weighted space. Under certain condition on the weight w, we have that L−w (0) coincides with the dual of the Hardy space H1 −(w). We prove for 0 <β< 1, that L− w (β) consists of all functions satisfying a weighted Lipschitz condition. In order to give another characterization of L− w (β), 0 ≤ β < 1, we also prove a one-sided version of John-Nirenberg Inequality. Finally, we obtain necessary and sufficient conditions on the weight w for the boundedness of an extension of I+ α from Lp w into L− w (β), −1 <β< 1, and its extension to a bounded operator from L− w (0) into L− w (α).

Weighted inequalities for some one-sided operators

We give a characterization of the pairs of weights (u, v) such that the Weyl fractional integral operator maps Lp(vdx) into weak Lq(udx), 1 < p ≤ q < ∞ or p = 1 < q < ∞. For the case p < q we give necessary and sufficient conditions for the weak type of a maximal operator that includes as particular cases the Weyl fractional integral, the dual of the Hardy operator and the fractional one-sided maximal operator. As a consequence we give a new characterization of the pairs of weights for which the fractional one-sided maximal operator is bounded.

Two weight norm inequalities for fractional one-sided maximal operators

In this paper we introduce a new maximal function, the dyadic one-sided maximal function. We prove that this maximal function is equivalent to the one-sided maximal function studied by the authors and Ortega in Weighted inequalities for one-sided maximal functions (Trans. Amer. Math. Soc. 319 (1990)) and by Sawyer in Weighted inequalities for the one-sided Hardy-Littlewood maximal functions (Trans. Amer. Math. Soc. 297 (1986)), but our function, being dyadic, is much easier to deal with, and it allows us to study fractional maximal operators. In this way we obtain a geometric proof of the characterization of the good weights for fractional maximal operators, answering a question raised by Andersen and Sawyer in Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators (Trans. Amer. Math. Soc. 308 (1988)). Our methods, avoiding complex interpolation, give also the case of different weights for the fractional maximal operator, which was an open problem.

Weighted Norm Inequalities for the Riemann-Liouville and Weyl Fractional Integral Operators

Weighted Norm Inequalities for the Riemann-Liouville and Weyl Fractional Integral Operators

Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators

The weight functions u ( x ) u(x) for which R α {R_\alpha } , the Riemann-Liouville fractional integral operator of order α &gt; 0 \alpha &gt; 0 , is bounded from L p ( u p d x ) {L^p}({u^p}\,dx) to L q ( u q d x ) {L^q}({u^q}\,dx) , 1 &gt; p &gt; 1 / α 1 &gt; p &gt; 1/\alpha , 1 / q = 1 / p − α 1/q = 1/p - \alpha , are characterized. Further, given p p , q q with 1 / q ⩾ 1 / p − α 1/q \geqslant 1/p - \alpha , the weight functions u &gt; 0 u &gt; 0 a.e. (resp. v &gt; ∞ v &gt; \infty a.e.) for which there is v &gt; ∞ v &gt; \infty a.e. (resp. u &gt; 0 u &gt; 0 a.e.) so that R α {R_\alpha } is bounded from L p ( v p d x ) {L^p}({v^p}\,dx) to L q ( u q d x ) {L^q}({u^q}\,dx) are characterized. Analogous results are obtained for the Weyl fractional integral. The method involves the use of complex interpolation of analytic families of operators to obtain similar results for fractional "one-sided" maximal function operators which are of independent interest.