Stein Weiss Inequality Research Articles

On Some Multipliers Related to Discrete Fractional Integrals

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓp→ℓq bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators.

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.

Hardy-Leray inequalities in variable Lebesgue spaces

In this paper, we prove the Hardy-Leray inequality and related inequalities in variable Lebesgue spaces. Our proof is based on a version of the Stein-Weiss inequality in variable Lebesgue spaces derived from two weight inequalities due to Melchiori and Pradolini. We also discuss an application of our results to establish an existence result for the degenerate p(⋅)-Laplace operator.

Stein-Weiss-Adams inequality on Morrey spaces

We establish Adams type Stein-Weiss inequality on global Morrey spaces on general homogeneous groups. Special properties of homogeneous norms and some boundedness results on global Morrey spaces play key roles in our proofs. As consequence, we obtain fractional Hardy, Hardy-Sobolev, Rellich and Gagliardo-Nirenberg inequalities on Morrey spaces on stratified groups. While the results are obtained in the setting of general homogeneous groups, they are new already for the Euclidean space RN.

Integral inequalities with an extended Poisson kernel and the existence of the extremals

Abstract In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.

Weighted Gagliardo-Nirenberg interpolation inequalities

In this paper, we prove weighted versions of the Gagliardo-Nirenberg interpolation inequality with Riesz as well as Bessel type fractional derivatives, generalizing the celebrated result with classical derivatives. We use a harmonic analysis approach employing several methods, including the method of domination by sparse operators, to obtain such inequalities for a general class of weights satisfying Muckenhoupt-type conditions. We also obtain improved results for some particular families of weights, including power-law weights |x|α. In particular, we prove an inequality which generalizes both the Stein-Weiss inequality and the Caffarelli-Kohn-Nirenberg inequality. However, our approach is sufficiently flexible to allow as well for non-homogeneous weights and we also prove versions of the inequalities with Japanese bracket weights 〈x〉α=(1+|x|2)α2.

Stein-Weiss inequality in 𝐿¹ norm for vector fields

In this work, we investigate the limit case p = 1 p=1 of the classical Stein-Weiss inequality for the Riesz potential. Our main result is a characterization of this inequality for a special class of vector fields associated to cocanceling operators introduced by Van Schaftingen [J. Eur. Math. Soc. (JEMS) 15 (2013), pp. 877–921]. As application, we recover and improve some fractional inequalities associated to vector fields.

An inhomogeneous radial operator and an improvement of the Stein-Weiss inequality

We give the necessary and sufficient conditions for the weighted inequality \begin{document}$ \begin{equation*} \||x|^{-b} T_{{\alpha}_{1}, \dots, {\alpha}_{m}, {\beta}_{1}, \dots, {\beta}_{m}, {\gamma}_{1}, \dots, {\gamma}_{m}}f \|_{L^q(\mathbb{R}^{n})} \leq C \||x|^{a} f\|_{L^p(\mathbb{R}^{n})} \end{equation*} $\end{document} both inside and outside the scaling regime of the singular operator$ \begin{equation*} T_{{\alpha}_{1}, \dots, {\alpha}_{m}, {\beta}_{1}, \dots, {\beta}_{m}, {\gamma}_{1}, \dots, {\gamma}_{m}}f(x): = \int_{\mathbb{R}^n} \prod\limits_{i = 1}^{m}||x|^{\alpha_i}-|y|^{\beta_i}|^{ -\gamma_i} f(y) dy, \end{equation*} $with $ a, b, \alpha_i, \beta_i>0 $, $ 0<\gamma_i<1 $, $ i = 1, \cdots, m $. As a consequence, we characterize an improvement of the Stein-Weiss inequality.

Stein-Weiss inequality for local mixed radial-angular Morrey spaces

Abstract In this article, a generalization of the well-known Stein-Weiss inequality for the fractional integral operator on functions with different integrability properties in the radial and the angular direction in local Morrey spaces is established. We find that some conditions can be relaxed for the Stein-Weiss inequality for local mixed radial-angular Morrey spaces.

Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups

Abstract In this note, we prove the reverse Stein–Weiss inequality on general homogeneous Lie groups. The results obtained extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in our proofs. Also, we prove reverse Hardy, Hardy–Littlewood–Sobolev, L p {L^{p}} -Sobolev and L p {L^{p}} -Caffarelli–Kohn–Nirenberg inequalities on homogeneous Lie groups.

Existence of Extremals of Dunkl-Type Sobolev Inequality and of Stein–Weiss Inequality for Dunkl Riesz Potential

In this paper, we prove the existence of an extremal for the Dunkl-type Sobolev inequality in the case of $$p=2$$ . Also we prove the existence of an extremal of the Stein–Weiss inequality for the Dunkl Riesz potential in the case of $$r=2$$ .

Symmetry in Fourier Analysis: Heisenberg Group to Stein–Weiss Integrals

Embedded symmetry within the Heisenberg group is used to couple geometric insight and analytic calculation to obtain a new sharp Stein–Weiss inequality with mixed homogeneity on the line of duality. SL(2,R) invariance and Riesz potentials define a natural bridge for encoded information that connects distinct geometric structures. The intrinsic character of the Heisenberg group makes it the natural playing field on which to explore the laws of symmetry and the interplay between analysis and geometry on a manifold.

Stein–Weiss inequality on product spaces

We give a classification between weighted norm inequalities of strong fractional integral operators and their corresponding multi-parameter Muckenhoupt characteristics, by considering the weights to be power functions. As a result, we extend the classical Stein–Weiss theorem on product spaces.

Reversed Stein–Weiss Inequalities with Poisson-Type Kernel and Qualitative Analysis of Extremal Functions

Abstract Through conformal map, isoperimetric inequalities are equivalent to the Hardy–Littlewood–Sobolev (HLS) inequalities involved with the Poisson-type kernel on the upper half space. From the analytical point of view, we want to know whether there exists a reverse analogue for the Poisson-type kernel. In this work, we give an affirmative answer to this question. We first establish the reverse Stein–Weiss inequality with the Poisson-type kernel, finding that the range of index 𝑝, q ′ q^{\prime} appearing in the reverse inequality lies in the interval ( 0 , 1 ) (0,1) , which is perfectly consistent with the feature of the index for the classical reverse HLS and Stein–Weiss inequalities. Then we give the existence and asymptotic behaviors of the extremal functions of this inequality. Furthermore, for the reverse HLS inequalities involving the Poisson-type kernel, we establish the regularity for the positive solutions to the corresponding Euler–Lagrange system and give the sufficient and necessary conditions of the existence of their solutions. Finally, in the conformal invariant index, we classify the extremal functions of the latter reverse inequality and compute the sharp constant. Our methods are based on the reversed version of the Hardy inequality in high dimension, Riesz rearrangement inequality and moving spheres.

Stein–Weiss inequalities with the fractional Poisson kernel

In this paper, we establish the following Stein–Weiss inequality with the fractional Poisson kernel: \begin{align*} \tag{$\star$} \int_{\mathbb{R}^n_{+}} \int_{\partial\mathbb{R}^n_{+}} |\xi|^{-\alpha} f(\xi) & P(x,\xi,\gamma) g(x) |x|^{-\beta} d\xi dx \\ &\leq C_{n,\alpha,\beta,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n_{+}) \|f\|\{L^p(\partial \mathbb{R}^{n}_{+})}, \end{align*} where P(x,\xi,\gamma)=\frac{x_n}{(|x'-\xi|^2+x_n^2)^{(n+2-\gamma)/2}}, 2\le \gamma < n , f\in L^{p}(\partial\mathbb{R}^n_{+}) , g\in L^{q'}(\mathbb{R}^n_{+}) , and p,\ q'\in (1,\infty) and satisfy (n-1)/(np)+1/q'+(\alpha+\beta+2-\gamma)/{n}=1 . Then we prove that there exist extremals for the Stein–Weiss inequality ( \star ), and that the extremals must be radially decreasing about the origin. We also provide the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler–Lagrange equations of the extremals to the Stein–Weiss inequality ( \star ) with the fractional Poisson kernel. Our result is inspired by the work of Hang, Wang and Yan, where the Hardy–Littlewood–Sobolev type inequality was first established when \gamma=2 and \alpha=\beta=0 . The proof of the Stein–Weiss inequality ( \star ) with the fractional Poisson kernel in this paper uses recent work on the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel by Chen, Lu and Tao, and the present paper is a further study in this direction.

Weighted inequalities for the fractional Laplacian and the existence of extremals

In this paper, we obtain improved versions of Stein–Weiss and Caffarelli–Kohn–Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein–Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of Lieb [Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118(2) (1983) 101–116].

Weighted estimates for bilinear fractional integral operators

AbstractMoen (2016) proved weighted estimates for the bilinear fractional integrals where . We improve his results when and consider the case . As a corollary we obtain a bilinear Stein–Weiss inequality where .

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities on homogeneous Lie groups

ABSTRACTIn this note, we prove the Stein–Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms play a key role in our proofs. Also, we give a simple proof of the Hardy–Littlewood–Sobolev inequality on general homogeneous Lie groups.

Weighted estimates for bilinear fractional integral operators: a necessary and sufficient condition for power weights

We consider weighted estimates for two bilinear fractional integral operators \(I_{2,\alpha }\) and \(BI_{\alpha }\). Moen (Collect Math 60:213–238, 2009) obtained a necessary and sufficient condition for \(I_{2,\alpha }\). However we know only some sufficient conditions for \(BI_{\alpha }\) which is a variant of the bilinear Hilbert transform. Restricted to power weights we obtain a necessary and sufficient condition for \(BI_{\alpha }\). We also prove a bilinear Stein–Weiss inequality.

Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group

In this paper, we establish the existence of extremals for two kinds of Stein–Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein–Weiss inequalities with full weights in Theorem 1.1 and the Stein–Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein–Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein–Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein–Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5).