Riesz Potential Operator Research Articles

Weighted Rellich and Hardy inequalities in L p(·) spaces

Abstract In this note we prove weighted Rellich–Sobolev and Hardy–Sobolev inequalities in variable exponent Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} defined on stratified homogeneous groups 𝔾 {\mathbb{G}} . To derive the main results, we rely on weighted estimates for the Riesz potential operators in L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} , where 𝔾 {{\mathbb{G}}} is a general homogeneous group. The results are new even for the Abelian (Euclidean) case 𝔾 = ( ℝ d , + ) {\mathbb{G}=(\mathbb{R}^{d},+)} and the Heisenberg groups 𝔾 = ℍ n {\mathbb{G}={\mathbb{H}}^{n}} . The main statements are obtained for variable exponents satisfying the condition that the Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue spaces. We also give some quantitative estimates for the norms of integral operators involved in derived estimates.

The Fractional Maximal and Riesz Potential Operators Involving the Lebedev–Skalskaya Transform

The Fractional Maximal and Riesz Potential Operators Involving the Lebedev–Skalskaya Transform

An approach to elliptic equations with nonlinear gradient terms via a modulation framework

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] is the nonlinearity, [Formula: see text] the potential and [Formula: see text] is a forcing term. Some examples of nonlinearities dealt with are [Formula: see text], [Formula: see text] and [Formula: see text], covering large values of [Formula: see text], and particularly variational supercritical powers for [Formula: see text] and super-[Formula: see text] ones for [Formula: see text] (superquadratic if [Formula: see text]). Moreover, we are able to consider some exponential growths, [Formula: see text] belonging to certain classes of power series, or [Formula: see text] satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

The Boundedness of Commutators of Sublinear Operators on Herz Triebel–Lizorkin Spaces with Variable Exponent

In this paper, the authors first discuss the characterization of Herz Triebel–Lizorkin spaces with variable exponent via two families of operators. By this characterization, the authors prove that the Lipschitz commutators of sublinear operators is bounded from Herz spaces with variable exponent to Herz Triebel–Lizorkin spaces with variable exponent. As applications, the corresponding boundedness estimates for the commutators of maximal operator, Riesz potential operator and Calderón–Zygmund operator are established.

CHARACTERIZATIONS OF COMMUTATORS OF SINGULAR INTEGRAL OPERATORS ON MORREY TRIEBEL-LIZORKIN

In this paper, we obtain the characterizations of Morrey Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Morrey TriebelLizorkin spaces, it is proved that b is a Lipschitz function if and only if the commutator [b, T] is bounded from Morrey spaces to Morrey Triebel-Lizorkin spaces, where T is singular integral operator or Riesz potential operator.

Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension

We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on d-dimensional ball domains, providing a substantial generalization of the work started in Gutleb et al. (Math Comput 9:2247–2281, 2022) for the one-dimensional case based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension d, in stark constrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support.

Boundedness of Riesz Potential Operator on Grand Herz-Morrey Spaces

In this paper, we introduce grand Herz–Morrey spaces with variable exponent and prove the boundedness of Riesz potential operators in these spaces.

Strictly singular non-compact operators between $L_p$ spaces

We study the structure of strictly singular non-compact operators between L_p spaces. Answering a question raised in earlier work on interpolation properties of strictly singular operators, it is shown that there exist operators T , for which the set of points (1/p,1/q)\in(0,1)\times (0,1) such that T\colon L_p\rightarrow L_q is strictly singular but not compact contains a line segment in the triangle \{(1/p,1/q):1<p<q<\infty\} of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular (i.e., difference of positive) operators defined on subspaces of L_p is also explored.

Nonhomogeneous central Morrey-type spaces in Lp(⋅) and weak estimates for the maximal and Riesz potential operators

Our aim in this paper is to discuss the weak estimate for the maximal and potential operators in the nonhomogeneous central Morrey-type space Mp(⋅),q,ν(Rn), which consists of all measurable functions f on Rn satisfying ‖f‖Mp(⋅),q,ν(Rn)=(∫1∞(r−ν‖f‖ Lp(⋅)(B(0,r)))qdr r)1∕q<∞, where ν>0, 0<q<∞ and p(⋅) is a variable exponent satisfying the log-Hölder condition at infinity. When q=∞, we apply a necessary modification.

Гладкостные свойства оператора типа потенциала Рисса с логарифмической характеристикой

Marcel Riesz kernels generalize the ones of classical theory, and convolution with them implements the negative frac-tional powers of the Laplace operator. Along with hypersingular integrals, the Riesz potential type operators arise in new areas of analysis and its applications, for example, in various problems of mathematical physics. In the study of such problems, a significant role is played by the conditions of solvability of the corresponding multidimensional integral equations in certain function spaces, often non-classical but postulating the required analytical properties of solutions. In the presented paper, the Hardy-Littlewood type theorem is proved, considering the boundedness con-ditions for the potential with a power-logarithmic kernel and a density, integrable by Lebesgue with a specific power weight. It is shown that for the higher orders of potential, the image of a function from this class belongs to the weighted generalized Hölder space with a power-logarithmic characteristic. The action of this operator in the gener-alized Hölder spaces with a weight from the scale of power functions is also considered, including the isomorphisms of these spaces in special cases. Stereographic projection and theorems proved for the spherical Riesz potential type operators are applied. Consequently, the solvability conditions of a Poisson-type equation with a negative fractional power of the Laplace operator are obtained.

Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline cases

AbstractWe define the grand Lebesgue space corresponding to the case and similar grand spaces for Morrey and Morrey type spaces, also for , on open sets in . We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases for Lebesgue spaces and for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.

A Faber–Krahn inequality for the Riesz potential operator for triangles and quadrilaterals

We prove an analog of the Faber–Krahn inequality for the Riesz potential operator. The proof is based on Riesz’s inequality under Steiner symmetrization and the continuity of the first eigenvalue of the Riesz potential operator with respect to the convergence, in the complementary Hausdorff distance, of a family of uniformly bounded non-empty convex open sets.

Two-type estimates for the boundedness of generalized Riesz potential operator in the generalized weighted local Morrey spaces

In this paper, we prove the Spanne-type boundedness of the generalized Riesz potential operator $I_{\rho}$ on the generalized weighted local Morrey spaces with $w^{q} \in A_{1+\frac{q}{p'}}$ for $1 \le p \le q \le \infty $ and including its the weak estimates with $w \in A_{1,q}$ for $1\le q\le\infty$. We also prove the Adams-type boundedness of the operator $I_{\rho}$ on the generalized weighted Morrey spaces with $ w \in A_{p,q}$ for $1\le p\le q \le \infty$ and also including its the weak estimates with $w \in A_{1,q}$ for $1\le q\le \infty$.

Sobolev embeddings in grand Morrey spaces

AbstractThe aim in this paper is to show that the Riesz potential operator embeds from grand Morrey spaces to grand Campanato–Morrey spaces. We are also concerned with Sobolev type inequalities for Riesz potentials in grand Morrey spaces.

Addendum to “On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space”, Math Meth Appl Sci. 2020; 1–8

In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order α(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and λ(x) of the Morrey space. Assumptions on the exponents were different depending on whether takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range . We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper.

Weak Estimates for the Maximal and Riesz Potential Operators in Central Herz–Morrey Spaces on the Unit Ball

Morrey spaces are the powerful tool for the study of partial differential equations. Recently, weak Morrey spaces and weak Herz spaces are known to be useful for the study of Navier–Stokes equations. In this paper we introduce weak central Herz–Morrey spaces W\mathcal H^{p(\cdot),q,\omega}(\mathbf B) and establish the weak estimate for the maximal and Riesz potential operators in the central Herz–Morrey space \mathcal H^{p(\cdot),q,\omega}(\mathbf B) . We also treat generalized Riesz potential operators. Further we obtain the strong estimate for Sobolev functions.

Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework

We prove sharp power-weighted Lp, weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator Bν in the exotic range of the parameter −∞<ν<1. Moreover, in the same framework, we characterize basic mapping properties for other fundamental harmonic analysis operators, including the heat semigroup based vertical g-function and fractional integrals (Riesz potential operators).

Fractional Hardy-Sobolev L1-embedding per capacity-duality

This paper explores essence of the fractional ( 0 < s < 1 ) Hardy-Sobolev L 1 -embedding ‖ u ‖ L n n − s ≲ ‖ ∇ + s u ‖ L 1 + ‖ ∇ − s u ‖ L 1 for all u ∈ I s ( C c ∞ ∩ H 1 ) via both capacity and duality based on the Riesz potential operator I s and the fractional differential couple ∇ ± s . Naturally, we discover that f ∈ I s ( [ H ˚ − s , 1 ] ⁎ ) if and only if there exists g → = ( g 1 , . . . , g n ) ∈ ( L ∞ ) n such that f = R → ⋅ g → = ∑ j = 1 n R j g j in the John-Nirenberg space BMO where R → = ( R 1 , . . . , R n ) is the vector-valued Riesz transform - this equivalence characterizes R → ⋅ ( L ∞ ) n in the Fefferman-Stein decomposition: BMO = L ∞ + R → ⋅ ( L ∞ ) n and indicates that I s ( [ H ˚ − s , 1 ] ⁎ ) is a solution to the Bourgain-Brezis problem under n ≥ 2 : “What are the function spaces X , W 1 , n ⊂ X ⊂ BMO , such that every F ∈ X has a decomposition F = ∑ j = 1 n R j Y j where Y j ∈ L ∞ ?”.

Some inequalities for homogeneous $B_n$-potential type integrals on $H^{p}_{\Delta_{\nu}}$ Hardy spaces

We prove the norm inequalities for potential operators and fractional integrals related to generalized shift operator defined on spaces of homogeneous type. We show that these operators are bounded from $H^{p}_{\Delta_{\nu}}$ to $H^{q}_{\Delta_{\nu}}$, for $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{Q}$, provided $0&amp;lt;\alpha&amp;lt;\frac{1}{2}$, and $\alpha&amp;lt;\beta\leq 1$ and $\frac{Q}{Q+\beta}&amp;lt;p\leq\frac{Q}{Q+\alpha}$. By applying atomic-molecular decomposition of $H^{p}_{\Delta_{\nu}}$ Hardy space, we obtain the boundedness of homogeneous fractional type integrals which extends the Stein-Weiss and Taibleson-Weiss's results for the boundedness of the $B_n$-Riesz potential operator on $H^{p}_{\Delta_{\nu}}$ Hardy space.

Trace Inequalities for Fractional Integrals in Mixed Norm Grand Lebesgue Spaces

Abstract D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.