Stein Weiss Inequalities Research Articles

Stein–Weiss inequalities on local Morrey spaces with variable exponents

Stein–Weiss inequalities on local Morrey spaces with variable exponents

Stein–Weiss inequalities on Morrey spaces

Stein–Weiss inequalities on Morrey spaces

Hardy inequalities on metric measure spaces, III: the case q ≤ p ≤ 0 and applications

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy–Littlewood–Sobolev and the Stein–Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in the Euclidean space. This work further complements the ranges of p and q (namely, q ≤ p < 0 ) considered in the work of Ruzhansky & Verma (Ruzhansky & Verma 2019 Proc. R. Soc. A 475 , 20180310 ( doi:10.1098/rspa.2018.0310 ); Ruzhansky & Verma. 2021 Proc. R. Soc. A 477 , 20210136 ( doi:10.1098/rspa.2021.0136 )), which treated the cases 1 < p ≤ q < ∞ and p > q , respectively.

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 for radial local Morrey spaces

This paper establishes a generalization of the celebrated Stein–Weiss inequalities for the fractional integral operators on radial functions in local Morrey spaces. We find that some conditions can be relaxed for the Stein–Weiss inequalities on radial local Morrey spaces.

On some multilinear type integral systems

In this paper we prove some Liouville theorems for systems of integral equations related to Beckner and Stein–Weiss inequalities on RN. We consider both variational and non-variational solutions. In order to study the first type of solutions we applied Rellich type identities.

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.

Hardy—Littlewood—Sobolev Inequalities with the Fractional Poisson Kernel and Their Applications in PDEs

The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel on the upper half space $$\int_{\mathbb{R}_ + ^n} {{{\int_{\partial \mathbb{R}_ + ^n} {f\left( \xi \right)P\left( {x,\xi,\alpha } \right)g\left( x \right)d\xi dx \leq {C_{n,\alpha,p,q'}}\parallel g\parallel } }_{Lq'\left( {\mathbb{R}_ + ^n} \right)}}} \parallel f{\parallel _{Lq'\left( {\partial \mathbb{R}_ + ^n} \right),}}$$ where \(f\, \in \,{L^p}\left( {\partial \mathbb{R}_ + ^n} \right),\,g\, \in \,{L^{q'}}\left( {\mathbb{R}_ + ^n}\right)\,\text{and}\;\,p,\,q'\, \in \,\left( {1 + \infty } \right),\,2\, \leq {\alpha<n}\; \text{satisfying}\;\frac{{n - 1}}{{np}} + \frac{1}{{q'}} + \frac{{2 - \alpha }}{n} = 1.\)Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant Cn,α,p,q′. Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore, in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler-Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point \({\xi _0}\, \in \,\partial \mathbb{R}_+ ^n.\) Our results proved in this paper play a crucial role in establishing the Stein-Weiss inequalities with the Poisson kernel in our subsequent paper.

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.

Reverse Stein–Weiss Inequalities on the Upper Half Space and the Existence of Their Extremals

Abstract The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein–Weiss inequality on the upper half space: ∫ ℝ + n ∫ ∂ ⁡ ℝ + n | x | α | x - y | λ f ( x ) g ( y ) | y | β d y d x ≥ C n , α , β , p , q ′ ∥ f ∥ L q ′ ⁢ ( ℝ + n ) ∥ g ∥ L p ⁢ ( ∂ ⁡ ℝ + n ) \int_{\mathbb{R}^{n}_{+}}\int_{\partial\mathbb{R}^{n}_{+}}\lvert x|^{\alpha}|x% -y|^{\lambda}f(x)g(y)|y|^{\beta}\,dy\,dx\geq C_{n,\alpha,\beta,p,q^{\prime}}\|% f\/\|_{L^{q^{\prime}}(\mathbb{R}^{n}_{+})}\|g\|_{L^{p}(\partial\mathbb{R}^{n}_% {+})} for any nonnegative functions f ∈ L q ′ ⁢ ( ℝ + n ) {f\in L^{q^{\prime}}(\mathbb{R}^{n}_{+})} , g ∈ L p ⁢ ( ∂ ⁡ ℝ + n ) {g\in L^{p}(\partial\mathbb{R}^{n}_{+})} , and p , q ′ ∈ ( 0 , 1 ) {p,q^{\prime}\in(0,1)} , β &lt; 1 - n p ′ {\beta&lt;\frac{1-n}{p^{\prime}}} or α &lt; - n q {\alpha&lt;-\frac{n}{q}} , λ &gt; 0 {\lambda&gt;0} satisfying n - 1 n ⁢ 1 p + 1 q ′ - α + β + λ - 1 n = 2 . \frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime}}-\frac{\alpha+\beta+\lambda-1}{n}% =2. Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler–Lagrange equations of the reverse Stein–Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give a spherical form of the Stein–Weiss inequality and reverse Stein–Weiss inequality on the upper half space ℝ + n {\mathbb{R}^{n}_{+}} .

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).

Fractional integral operator for L1 vector fields and its applications

This paper studies fractional integral operator for vector fields in weighted L1. Using the estimates on fractional integral operator and Stein-Weiss inequalities, we can give a new proof for a class of Caffarelli-Kohn-Nirenberg inequalities and establish new div-curl inequalities for vector fields.

Reverse Stein–Weiss inequalities and existence of their extremal functions

Reverse Stein–Weiss inequalities and existence of their extremal functions

Two-Weight Norm, Poincaré, Sobolev and Stein–Weiss Inequalities on Morrey Spaces

We establish two-weight norm inequalities for singular integral operators and fractional integral operators on Morrey spaces. As a consequence of these inequalities, we obtain two-weight Poincaré and Sobolev inequalities on Morrey spaces. Moreover, we also establish the Stein–Weiss inequality, the Hardy inequality and the Rellich inequality on Morrey spaces.

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group

In this paper, we study two types of weighted Hardy–Littlewood–Sobolev (HLS) inequalities, also known as Stein–Weiss inequalities, on the Heisenberg group. More precisely, we prove the |u| weighted HLS inequality in Theorem 1.1 and the |z| weighted HLS inequality in Theorem 1.5 (where we have denoted u=(z,t) as points on the Heisenberg group). Then we provide regularity estimates of positive solutions to integral systems which are Euler–Lagrange equations of the possible extremals to the Stein–Weiss inequalities. Asymptotic behavior is also established for integral systems associated to the |u| weighted HLS inequalities around the origin. By these a priori estimates, we describe asymptotically the possible optimizers for sharp versions of these inequalities.

Stein–Weiss inequalities for the fractional integral operators in Carnot groups and applications

In this article we consider the fractional integral operator I α on any Carnot group 𝔾 (i.e. nilpotent stratified Lie group) in the weighted Lebesgue spaces L p,ρ(x)β (𝔾). We establish Stein–Weiss inequalities for I α, and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator I α from the spaces L p,ρ(x)β (𝔾) to L q,ρ(x)−γ (𝔾), and from the spaces L 1,ρ(x)β (𝔾) to the weak spaces WL q,ρ(x)−γ (𝔾) by using the Stein–Weiss inequalities. In the limiting case , we prove that the modified fractional integral operator is bounded from the space L p,ρ(x)β (𝔾) to the weighted bounded mean oscillation (BMO) space BMOρ(x)−γ (𝔾), where Q is the homogeneous dimension of 𝔾. As applications of the properties of the fundamental solution of sub-Laplacian ℒ on 𝔾, we prove two Sobolev–Stein embedding theorems on weighted Lebesgue and weighted Besov spaces in the Carnot group setting. As another application, we prove the boundedness of I α from the weighted Besov spaces to .