William Beckner
Sharp extensions of Pitt's inequality and bounds for Stein-Weiss fractional integrals are obtained that incorporate gradient forms and vector-valued operators. Such results include Hardy-Rellich inequalities.
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In this article we study modified Kirchhoff-Schrödinger equations involving critical Stein-Weiss type nonlinearity and $p$-Laplacian
In 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.
In this note we prove the reverse Stein-Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in our proofs. Also, we show reverse Hardy, Hardy-Littlewood-Sobolev, Lp-Sobolev and Lp-Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups.
In this paper, we establish some Stein-Weiss type inequalities with general kernels on the upper half space and study the existence of extremal functions for this inequality with the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of the positive solutions of the corresponding Euler-Lagrange integral system. As an application, we finally derive some Liouville type results for the Hartree type equations in the half space.
The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-Δu =\frac{1}{|x|^α}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{α, μ}^{\ast}}}{|x-y|^μ|y|^α}dy\right) |u|^{2_{α, μ}^{\ast}-2}u,~~~x\in\mathbb{R}^{N}, $$ where the critical exponent is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, …
Multiple nodal solutions of planar Stein-Weiss equations
本文研究含 Trudinger-Moser 次临界或临界增长非线性项的平面 Stein-Weiss 问题，结合粘合方法与 Nehari 流形论证，证明对任意正整数 k 存在恰好变号 k 次的径向对称基态节点解。
Radial symmetry of positive solutions of an integral system associated with the reversed Stein-Weiss inequality
研究反向Stein-Weiss不等式极值函数所满足的Euler-Lagrange系统正解的径向对称性，采用移动平面法，证明了该系统正解具有径向对称性。
Stein-Weiss inequality on non-compact symmetric spaces
本文完整刻画了任意秩非紧对称空间上Laplace-Beltrami算子相关算子族Stein-Weiss不等式成立的充要条件，并由此得到容许指标集更大的加权泛函不等式。
Sharp $L^p$ extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. More generally, optimal constants are obtained for the full Stein-Weiss potential as a map from $L^p$ to itself, and new proofs are given for the classical Pitt and Stein-Weiss estimates.
Stein-Weiss type inequalities with partial variable weight on the upper half space and related weighted inequalities
本文在上半空间建立了一类具有部分变权函数的 Stein-Weiss 型不等式，利用加权 Hardy 不等式和集中紧性原理证明极值函数存在，并通过移动平面法等方法刻画其对称性与正则性，进而导出 Laplacian 和分数阶 Laplacian 的加权 Sobolev 不等式。
On the Stein-Weiss inequalities and higher-order Caffarelli-Kohn-Nirenberg type inequalities: sharp constants, symmetry of extremal functions
本文通过分类加权四阶方程的径向对称解，推导了多类带最佳常数与径向对称极值函数的加权不等式，包括Stein-Weiss、Caffarelli-Kohn-Nirenberg型及Hardy-Rellich等不等式。
In this work, we investigate the limit case $p=1$ of the classical Stein--Weiss inequality for the Riesz potential.We present a characterization for a special class of vector fields associated to cocanceling operators introduced by Van Schaftingen in arXiv:1104.0192. As an application, we recover some div-curl inequalities found in the literature. In addition, we discuss a two-weight inequality with general weights in the scalar case, extending the previous result of Sawyer to this case.
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 …
We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of $L^{p}$ spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be unified with their improved radial versions. A number of consequences are obtained: in particular we deduce precised versions of weighted Sobolev embeddings, Caffarelli-Kohn-Nirenberg estimates, and Strichartz estimates for the wave equation, which extend the radial improvements to the case of arbitrary functions.
Multiplicity of solutions for singular elliptic problems with Stein-Weiss term
本文研究全空间上带 Stein-Weiss 双重加权非局部项的奇异椭圆问题，利用 Nehari 方法和非线性 Rayleigh 商证明至少存在两个正弱解，并给出与极值 λ* 相关的不存在性结果。
Extremizers for a trilinear Stein-Weiss inequality with nonnegative weights
该研究针对$\boldsymbol{\text{R}}^n$上带非负权重的三线性Stein-Weiss不等式，在特定假设下证明极值可达性，结合对称递减重排等方法推导相关方程并分析其不变性。
This paper explores the intrinsic nature of the celebrated Stein-Weiss $H^1$-inequality $$ \|I_s u\|_{L^\frac{n}{n-s}}\lesssim \|u\|_{L^1}+\|\vec{R}u\|_{L^{1}}=\|u\|_{H^1} $$ through the tracing and duality laws based on Riesz's singular integral operator $I_s$. We discover that $f\in I_s\big([\mathring{H}^{s,1}_{-}]^\ast\big)$ if and only if $\exists\ \vec{g}=(g_1,...,g_n)\in \big(L^\infty\big)^n$ such that $f=\vec{R}\cdot\vec{g}=\sum_{j=1}^n R_jg_j$ in $\mathrm{BMO}$ (the John-Nirenberg space introduced in their 1961 {\it Comm. Pure Appl. Math.} paper \cite{JN}) where $\vec{R}=(R_1,...,R_n)$…
A Stein-Weiss type inequality on Carnot groups is established by proving the boundedness of an integral operator and the Hardy-Littlewood-Sobolev type inequality on Carnot groups is also derived.
A Generalized Choquard equation with weighted anisotropic Stein-Weiss potential on nonreflexive Orlicz-Sobolev Spaces
本文研究非自反Orlicz-Sobolev空间中带各向异性Stein-Weiss势的广义Choquard方程，利用Hardy不等式、Stein-Weiss不等式及山路引理和Nehari流形方法，证明了解与基态解的存在性，并涵盖N-函数不满足Δ2条件的情形。
Revisiting Positive Exponent power weighted Stein-Weiss inequalities in $L^1$
本文针对$L^1$中正指数幂加权的Stein-Weiss不等式，确定了新的较弱必要条件，补充了De Nápoli与Picon此前关于抵消条件的研究，完善了该类不等式的充要条件体系。
Stein-Weiss, and power weight Korn type Hardy-Sobolev Inequalities in $L^1$ norm
本文将 $L^1$ Stein-Weiss 不等式推广至正非整数幂权数据，以消失矩假设替代协同抵消条件，并在整数权情形建立更弱的 Korn 型 Hardy-Sobolev 不等式及对偶估计改进实例。
In this article 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 E. Lieb.
In this paper, we establish the following Stein-Weiss inequality with the fractional Poisson kernel (see Theorem 1.1): \begin{equation}\label{int1} \int_{\mathbb{R}^n_{+}}\int_{\partial\mathbb{R}^n_{+}}|ξ|^{-α}f(ξ)P(x,ξ,γ)g(x)|x|^{-β}dξdx\leq C_{n,α,β,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n_{+})}\|f\|_{L^p(\partial \mathbb{R}^{n}_{+})}, \end{equation} where $P(x,ξ,γ)=\frac{x_n}{(|x'-ξ|^2+x_n^2)^{\frac{n+2-γ}{2}}}$, $2\le γ<n$, $f\in L^{p}(\partial\mathbb{R}^n_{+})$, $g\in L^{q'}(\mathbb{R}^n_{+})$ and $p,\ q'\in (1,\infty)$ and satisfy $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}+\frac{α+β+2-γ}{n}=1$. Then …
Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$
本文研究$\boldsymbol{\rm R}^N$中含Stein-Weiss型卷积、临界指数增长非线性项的拟线性Schrödinger方程，在给定参数范围与位势、非线性项假设下，探讨其正解的存在性。
Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt's inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide short proofs of the Frank-Seiringer "Hardy inequalities" where fractional smoothness is measured by a Besov norm.
In this work we are concerned with Fefferman-Stein type inequalities. More precisely, given an operator $T$ and some $p$, $1<p<\infty$, we look for operators $\mathcal{M}$ such that the inequality $$\int |Tf|^pw\leq C\int |f|^p \mathcal{M}w$$ holds true for any weight $w$. Specifically, we are interested in the case of $T$ being any first or second order Riesz transform associated to the Schrödinger operator $L=-Δ+ V$, with $V$ a non-negative function satisfying an appropriate reverse-Hölder condition. For the Riesz-Schrödinger transforms $\nabla L^{-1/2}$ and $\nabla^2 L^{-1}$ we make use of …
Some inequalities for the Fourier transform and their limiting behaviour
本文提出一族以球面限制猜想为极限情形的 Fourier 变换单参数不等式，并借助 Stein 复插值方法在 $L^2$ 目标空间情形加以证明，从而在极限中恢复 Tomas–Stein 定理。
Stein-Weiss problems via nonlinear Rayleigh quotient for concave-convex nonlinearities
本文通过非线性Rayleigh商与Nehari方法，研究全空间上带凹凸非线性的Stein-Weiss型非局部椭圆问题，证明了存在阈值λ*使得每个λ∈(0,λ*)时问题至少有两个正解，并给出Brezis-Lieb型引理与正则性结果。
Fefferman--Stein type inequalities via area and maximal functions for Schrödinger operators with applications
本文在分层李群上建立 Schrödinger 算子的 Fefferman--Stein 不等式，并推广到张量积情形，通过原子分解证明面积积分与 Riesz 变换的弱型端点估计，扩展了经典结果。
Reversed inequality of the Herbst-type and the related Euler-Lagrange system
本文关注反向斯坦 - 韦斯不等式的临界情况，给出反向赫布斯特不等式，证明其极值函数存在性，研究相关欧拉 - 拉格朗日系统，得到正解存在必要条件及\(|x|\)趋于\(0\)和\(\infty\)时的可积性与渐近行为。
Epiperimetric inequalities in the obstacle problem for the fractional Laplacian
本文利用周长度量不等式方法研究了分数阶拉普拉斯算子的障碍问题，证明了Weiss能量的正、负周长度量不等式及对数不等式，推导出频率间隙与爆破刻画，并给出了自由边界点的正则性另证与结构描述。
Functional inequalities on symmetric spaces of noncompact type and applications
本文系统研究非紧型对称空间上的泛函不等式，通过精细估计建立Stein-Weiss不等式并推导多种重要不等式，进而应用于证明半线性波动方程的小数据全局存在性。
Isoperimetric and geometric inequalities in quantitative form: Stein's method approach
该研究通过引入带斜边界条件的椭圆型PDE解决边界项处理难题，将Stein方法几何化，得到Brock-Weinstock不等式、Steklov特征值约束下等周不等式及加权与未加权周长组合的稳定性结果，均基于α-Zolotarev距离表述。
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 $\pp$-Laplace operator.
This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy-Littlewood maximal inequali…
The goal of this review is to explain some recent results regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).
Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds
本文建立了具有显式下界的Hardy-Littlewood-Sobolev不等式及其分数阶Sobolev不等式的稳定性，推广了已有结果，证明基于竞争对称性和对偶稳定性理论。
We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of Lp spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be unified with their improved radial versions. A number of consequences are obtained: in particular we deduce precised versions of weighted Sobolev embeddings, Caffarelli-Kohn-Nirenberg estimates, and Strichartz estimates for the wave equation, which extend the radial improvements to the case of arbitrary functions. Then we apply this technology in order to giv…
This paper studies fractional integral operator for vector fields in weighted $L^1$. 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 $\divg$-$\curl$ inequalities for vector fields.
In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12], where they considered the analogues on classical Heisenberg group. First, we give the sharp Hardy-Littlewood-Sobolev inequalities, both on quaternionic Heisenberg group and its equivalent on quaternionic sphere for exponent bigger than 4. The extremizer, as we guess, is almost uniquely constant function on sphere. Then their dual form, sharp conformally-invaria…
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