Riesz Potential Research Articles

Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces

In this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces GMpθw(·). Our main result is the compactness of the commutators of the Riesz potential b,Iα in global Morrey-type spaces from GMp1θ1w1(·) to GMp2θ2w2(·). We also present new sufficient conditions for the commutator b,Iα to be bounded from GMp1θ1w1(·) to GMp2θ2w2(·). In the proof of the theorem regarding the compactness of the commutator for the Riesz potential, we primarily utilize the boundedness condition for the commutator for the Riesz potential b,Iα in global Morrey-type spaces GMpθw(·), and the sufficient conditions derived from the theorem on pre-compactness of sets in global Morrey-type spaces GMpθw(·).

Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth

In this paper, we study the following Hamilton–Choquard type elliptic system: − Δ u + u = ( I α ∗ F ( v ) ) f ( v ) , x ∈ R 2 , − Δ v + v = ( I β ∗ F ( u ) ) f ( u ) , x ∈ R 2 , where I α and I β are Riesz potentials, f : R → R possessing critical exponential growth at infinity and F ( t ) = ∫ 0 t f ( s ) d s. Without the classic Ambrosetti–Rabinowitz condition and strictly monotonic condition on f, we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates.

Sharp Trace and Korn Inequalities for Differential Operators

AbstractWe establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $$p=1$$ p = 1 , a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $$p=1$$ p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k-th order gradient estimates. Moreover, for $$1<p<\infty $$ 1 < p < ∞ , where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.

Trace principle for Riesz potentials on Herz-type spaces and applications

We establish trace inequalities for Riesz potentials on Herz-type spaces and examine the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo–Nirenberg–Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.

Asymptotic profiles for Choquard equations with combined attractive nonlinearities

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation(Pε)−Δu+εu=(Iα⁎|u|p)|u|p−2u+|u|q−2u,inRN,where N≥3 is an integer, p∈[N+αN,N+αN−2], q∈(2,2NN−2), Iα is the Riesz potential of order α∈(0,N) and ε>0 is a parameter. We show that as ε→0 (resp. ε→∞), the ground state solutions of (Pε), after appropriate rescalings dependent on parameter regimes, converge in H1(RN) to particular solutions of five different limit equations. We also establish a sharp asymptotic characterisation of such rescalings, and the precise asymptotic behaviour of uε(0), ‖∇uε‖22, ‖uε‖22, ∫RN(Iα⁎|uε|p)|uε|p and ‖uε‖qq, which depend in a non-trivial way on the exponents p, q and the space dimension N. Further, we discuss a connection of our results with a mass constrained problem, associated to (Pε) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, multiplicity and precise asymptotic behaviour of positive normalized solutions of such a problem as c→0 and c→∞.

On density functional theory models for one-dimensional homogeneous materials

This paper studies Density Functional Theory (DFT) models for homogeneous 1D materials in the 3D space. It follows the previous work [Gontier et al., Commun. Math. Phys. 388, 1475–1505 (2021)] about DFT models for homogeneous 2D materials in 3D. We show how to reduce the problem from a 3D energy functional to a 2D energy functional. The kinetic energy is treated as in the 2D material case by diagonalizing admissible states, and writing the kinetic energy as the infimum of a modified kinetic energy functional on reduced states. Besides, we treat here the Hartree interaction term in 2D, and show how to properly define the mean-field potential, through Riesz potential. We then show the well-posedness of the reduced model and present some numerical illustrations.

Sign-changing solutions for critical Choquard equation on bounded domain

We consider the existence of sign-changing solutions for the following nonlinear Choquard equation{−Δu=(Iα⁎|u|p)|u|p−2u+λu,x∈Ω,u∈H01(Ω), where N≥3,α∈(0,N),λ∈R, p∈(1,N+αN−2] and Ω⊂RN is a smooth bounded domain. Iα:RN→R is the Riesz potential. In critical case p=N+αN−2, if Ω is symmetric about the axis x1, we firstly develop the limit profiles for the symmetric Palais-Smale sequence by the concentration compactness principle. Then we conclude that the problem admits an odd solution with exactly two nodal domains for λ∈(0,λ1),N≥4andα∈(2,N). In contrast, the local Brezis-Nirenberg type problem −Δu=|u|4N−2u+λu,u∈H01(Ω) can not permit such type of odd symmetry solution.

On Trudinger-type inequalities in Musielak–Orlicz–Morrey spaces of an integral form over metric measure spaces

We establish Trudinger-type inequalities for variable Riesz potentials J_{\alpha(\cdot),\tau}f of functions f in Musielak–Orlicz–Morrey spaces of an integral form over metric measure spaces X . As an application and example, we give Trudinger’s inequality for double-phase functionals with variable exponents. Finally, we prove the result for Sobolev functions satisfying a Poincaré inequality in X .

Sobolev embeddings in Musielak-Orlicz spaces

An embedding theorem for Sobolev spaces built upon general Musielak-Orlicz norms is offered. These norms are defined in terms of generalized Young functions which also depend on the x variable. Under minimal conditions on the latter dependence, a Sobolev conjugate is associated with any function of this type. Such a conjugate is sharp, in the sense that, for each fixed x, it agrees with the sharp Sobolev conjugate in classical Orlicz spaces. Both Sobolev inequalities in the whole Rn and Sobolev-Poincaré inequalities in domains are established. Compact Sobolev embeddings are also presented. In particular, optimal embeddings for standard Orlicz-Sobolev spaces, variable exponent Sobolev spaces, and double-phase Sobolev spaces are recovered and complemented in borderline cases. A key tool, of independent interest, in our approach is a new weak type inequality for Riesz potentials in Musielak-Orlicz spaces involving a sharp fractional-order Sobolev conjugate.

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

On the boundedness of $B$-Riesz potential and its commutators on generalized weighted $B$-Morrey spaces

In the present paper, we shall investigate a characterization for the boundedness of the $B$-Riesz potential and its commutators on the generalized weighted $B$-Morrey spaces. We also give a characterization for the generalized weighted $B$-Morrey spaces via the boundedness of the Riesz potential and its commutators generated by generalized translate operators associated with Laplace-Bessel differential operator.

Compact Embeddings for Fractional Super and Sub Harmonic Functions with Radial Symmetry

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.

Nonlinear aggregation-diffusion equations with Riesz potentials

We consider an aggregation-diffusion model, where the diffusion is nonlinear of porous medium type and the aggregation is governed by the Riesz potential of order s. The addition of a quadratic diffusion term produces a more precise competition with the aggregation term for small s, as they have the same scaling if s=0. We prove existence and uniqueness of stationary states and we characterize their asymptotic behavior as s goes to zero. Moreover, we prove existence of gradient flow solutions to the evolution problem by applying the JKO scheme.

Continuity estimates for Riesz potentials on polygonal boundaries

Continuity estimates for Riesz potentials on polygonal boundaries

Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle

For the Riesz and logarithmic potentials, we consider greedy energy sequences (an)n=0∞ on the unit circle S1, constructed in such a way that for every n≥1, the discrete potential generated by the first n points a0,…,an−1 of the sequence attains its minimum value (say Un) at an. We obtain asymptotic formulae that describe the behavior of Un as n→∞, in terms of certain bounded arithmetic functions with a doubling periodicity property. As previously shown in [8], after properly translating and scaling Un, one obtains a new sequence (Fn) that is bounded and divergent. We find the exact value of liminfFn (the value of limsupFn was already given in [8]), and show that the interval [liminfFn,limsupFn] comprises all the limit points of the sequence (Fn).

On the Boundedness of the Fractional Maximal Operator, the Riesz Potential, and Their Commutators in Orlicz Spaces

On the Boundedness of the Fractional Maximal Operator, the Riesz Potential, and Their Commutators in Orlicz Spaces

Asymptotic decay of solutions for sublinear fractional Choquard equations

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation (−Δ)su+μu=(Iα∗F(u))f(u)onRNwhere s∈(0,1), N≥2, α∈(0,N), μ>0, Iα denotes the Riesz potential and F(t)=∫0tf(τ)dτ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than ∼1|x|N+2s. The result is new even for homogeneous functions f(u)=|u|r−2u, r∈[N+αN,2), and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D’Avenia et al. (2015). Differently from the local case s=1 in Moroz and Van Schaftingen (2013), new phenomena arise connected to a new “s-sublinear” threshold that we detect on the growth of f. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers.

$B-$ Riezs potential in $B-$ local Morrey-Lorentz spaces

In this paper, the Riesz potential (B−Riesz potential) which are generated by the Laplace-Bessel differential operator will be studied. We obtain the necessary and sufficient conditions for the boundedness of the B−Riesz potential $I_{\gamma }^{\alpha }$ in the B-local Morrey-Lorentz spaces $M_{p,q,\lambda,\gamma }^{loc}(\mathbb{R}_{k,+}^{n})$ with the use of the rearrangement inequalities and boundedness of the Hardy operators $H_{\upsilon }^{\beta }$ and $\mathcal{H}_{\upsilon}^{\beta }$ with power weights.

Compactness of Commutators for Riesz Potential on Generalized Morrey Spaces

In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces Mpw(·). This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spaces Lp,p>0. As an application, we prove the compactness of the commutator of the Riesz potential [b,Iα] in generalized Morrey spaces, where b∈VMO (VMO(Rn) denote the BMO-closure of C0∞(Rn)). We prove auxiliary statements regarding the connection between the norm of average functions and the norm of the difference of functions in the generalized Morrey spaces. Such results are also of independent interest.

Inductive methods for Hardy inequality on trees

We study the Hardy inequality on at most countable rooted tree. The main known criteria in the lower - triangle case for this inequality are two Arcozzi - Rochberg - Sawyer criteria and the capacity criterion. In the survey we show that these two criteria are connected with the criteria for the Hardy inequalities for the sequences, for the Hardy inequality on an interval of the real axis and for the trace inequalities with the Riesz potentials. We provide the examples from the literature, when the trace inequality or another statement is characterized in terms of the validity of the Hardy inequality on the tree. We simplify two known proofs of the Arcozzi - Rochberg - Sawyer criteria, which are based on the Marcinkiewicz interpolation theorem and on the capacity criterion. We provide new proofs for Arcozzi - Rochberg - Sawyer criteria, which are based on the induction in the tree, the inductive formula for the capacity and the formula of integration by parts. The latter of the proofs is written for the Hardy inequality on the tree with a boundary and for the Hardy inequality over the family of all binary cubes. In the diagonal case this proof provides an optimal constant $p$, which coincides with the Bennett constant in the Hardy inequality for the sequences. In the general case we provide a few new inductive criteria for the validity of the Hardy inequality in terms of the existence of a family of functions satisfying an inductive relation. One of these criteria is applied in the proof of a theorem containing additional equivalent conditions for the validity of the Hardy inequality on the trees in the diagonal case.