Riesz Potential Operator Research Articles

Weak estimates for the maximal and Riesz potential operators on non-homogeneous central Morrey type spaces in L^1 over metric measure spaces

Weak estimates for the maximal and Riesz potential operators on non-homogeneous central Morrey type spaces in L^1 over metric measure spaces

Campanato–Morrey spaces for the double phase functionals with variable exponents

Our aim in this paper is to show that the Riesz potential operator Iα(⋅) of variable order α(⋅) embeds from variable exponent Morrey spaces Lp(⋅),ν(⋅)(G) to Campanato–Morrey spaces in the case α(x)p(x)=ν(x). Our result extends the recent work of Rafeiro and Samko (2019) and the authors (Mizuta et al. 2020). We show that Iα(⋅) embeds from Morrey spaces LΦ,ν(⋅)(G) of the double phase functionals Φ(x,t)=tp(x)+(b(x)t)q(x) to Campanato–Morrey spaces, where p(⋅) and q(⋅) satisfy log-Hölder conditions, p(x)<q(x) and b(⋅) is nonnegative, bounded and Hölder continuous of order θ∈(0,1]. We also discuss the continuity of Riesz potentials Iα(⋅)f of functions in LΦ,ν(⋅)(G) and show that Iα(⋅) embeds from LΦ,ν(⋅)(G) to vanishing Campanato–Morrey spaces.

Essential norm estimates for multilinear singular and fractional integrals

We derive lower two-weight estimates for the essential norm (measure of noncompactness) for multilinear Hilbert and Riesz transforms, and Riesz potential operators in Banach function lattices. As a corollary we have appropriate results in weighted Lebesgue spaces. From these statements we conclude that there is no \((m+1)\)-tuple of weights \((v,w_1, \dots, w_m)\) for which these operators are compact from \(L^{p_1}_{w_1} \times \dots \times L^{p_m}_{w_m}\) to \(L^q_v\).

On the boundedness of fractional maximal and Riesz potential operators associated with Kontorovich–Lebedev transform

ABSTRACT We define the maximal and fractional maximal operators associated with the Kontorovich–Lebedev transform. We study the boundedness of these operators in the space . Using these results we establish the Hardy–Littlewood–Sobobev theorem for the Riesz potential operator associated with Kontorovich–Lebedev transform.

On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators

We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $$C_0^\infty ({\mathbb {R}^n})$$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy–Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.

Duals of Hardy Amalgam Spaces and Norm Inequalities

We characterize the dual spaces of the generalized Hardy spaces defined by replacing Lebesgue quasi-norms by Wiener amalgam ones. In these generalized Hardy spaces we prove that some classical linear operators such as Calderon–Zygmund, convolution and Riesz potential operators are bounded.

Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries

ABSTRACTWe introduce a class of Morrey-type spaces , which includes the classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem for such spaces. This theorem is then applied to obtaining an analogue of O'Neil's inequality for convolutions and to proving the boundedness in the introduced Morrey-type spaces of the Riesz potential and singular integral operators.

Weighted Estimates for Maximal Operators, Riesz Potential Operators and Commutators on p-Adic Lebesgue and Morrey Spaces

In this paper, we establish the boundedness of p-adic Hardy-Littlewood maximal operators and p-adic Riesz potential operators on the weighted Lebsegue and Morrey spaces. Moreover, the boundedness for the commutators of p-adic Riesz potential operators on the weighted Morrey spaces with symbols in central BMO space is also given.

Matrix weighted Poincare inequalities and applications to degenerate elliptic systems

We prove Poincar\'e and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincar\'e inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A${}_p$ weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar A${}_p$ weight. In addition, we prove Cacciopoli and reverse H\"older inequalities for weak solutions of the degenerate systems. As a means to prove the Poincar\'e inequalities we prove that the Riesz potential and fractional maximal function operators are bounded on matrix weighted $L^p$ spaces and go on to develop an entire matrix A${}_{p, q}$ theory.

Weak estimates for the maximal and Riesz potential operators in non-homogeneous central Herz-Morrey spaces

ABSTRACTOur aim in this paper is to discuss the weak estimate for the maximal and Riesz potential operators in the non-homogeneous central Morrey type space . Further we obtain the strong estimate for Sobolev functions.

Riesz potential and its commutators on Orlicz spaces

In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator I_{alpha} on Orlicz spaces. Cianchi (J. Lond. Math. Soc. 60(1):247-286, 2011) found necessary and sufficient conditions on general Young functions Φ and Ψ ensuring that this operator is of weak or strong type from L^{Phi} into L^{Psi}. Our characterizations for the boundedness of the above-mentioned operator are different from the ones in (Cianchi in J. Lond. Math. Soc. 60(1):247-286, 2011). As an application of these results, we consider the boundedness of the commutators of Riesz potential operator [b,I_{alpha }] on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.

Choquard equations under confining external potentials

We consider the nonlinear Choquard equation $$ -\Delta u+V u=(I_\alpha \ast \vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N $$ where $N\geq 1$, $I_\alpha$ is the Riesz potential integral operator of order $\alpha \in (0, N)$ and $p > 1$. If the potential $ V \in C (\mathbb{R}^N; [0,+\infty)) $ satisfies the confining condition $$ \liminf\limits_{\vert x\vert \to +\infty}\frac{V(x)}{1+\vert x\vert ^{\frac{N+\alpha}{p}-N}}=+\infty, $$ and $\frac{1}{p} > \frac{N - 2}{N + \alpha}$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when $p \ge 2$ the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. The growth assumption is sharp in view of a Poho\v{z}aev identity that we establish.

Isoperimetric inequalities for Schatten norms of Riesz potentials

In this note we prove that the ball is a maximiser for integer order Schatten p-norms of the Riesz potential operators among all domains of a given measure in Rd. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh–Faber–Krahn and Hong–Krahn–Szegö inequalities.

Continuity of Riesz potential operator in the supercritical case on unbounded domain

The aim of this paper is to prove continuity of the Riesz potential operator $R^{s}:E\mapsto{\mathcal {C}}H$ in optimal couple E, ${\mathcal {C}}H$ , for the supercritical case on unbounded domain, where E is a rearrangement invariant function space and ${\mathcal {C}}H$ is the generalized Holder-Zygmund space generated by a function space H. We also construct optimal domain and target quasi-norms for $R^{s}$ on unbounded domain.

Two-Weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces

Abstract We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.

A fast method for a generalized nonlocal elastic model

We develop a numerical method for a generalized nonlocal elastic model, which is expressed as a composition of a Riesz potential operator with a fractional differential operator, by composing a collocation method with a finite difference discretization.By carefully exploring the structure of the coefficient matrix of the numerical method, we develop a preconditioned fast Krylov subspace method, which reduces the computations to (Nlog⁡N) per iteration and the memory to O(N). The use of the preconditioner significantly reduces the number of iterations, and the preconditioner can be inverted in O(Nlog⁡N) computations. Numerical results show the utility of the method.

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

Abstract The non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

Boundedness of Toeplitz type operators associated to Riesz potential operator with general kernel on Orlicz space

Abstract In this paper, the boundedness properties for some Toeplitz type operators associated to the Riesz potential and general integral operators from Lebesgue spaces to Orlicz spaces are proved. The general integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts

It is known that radial symmetry is preserved by the Riesz potential operators and also by the hypersingular Riesz fractional derivatives typically used for inversion. In this paper, we collect properties, asymptotics, and estimates for the radial and spherical parts of Riesz potentials and for solutions to the Riesz potential equation of order one. Sharp estimates for spherical functions are provided in terms of seminorms, and a careful analysis of the radial part of a Riesz potential is carried out in elementary terms. As an application, we provide a two weight estimate for the inverse of the Riesz potential operator of order one acting on spherical functions.

A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions

We establish necessary and sufficient conditions on a weight pair governing the boundedness of the Riesz potential operator defined on a homogeneous group G from to , where is the Lebesgue space defined for non-negative radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels defined on a product of two homogeneous groups . In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces. To get the main results we use Sawyer-type duality theorems (which are also discussed in this paper) and two-weight Hardy-type inequalities on G and , respectively. MSC:42B20, 42B25.