Besma Ben Ali
The paper concerns the magnetic Schrödinger operator on $R^n$. We prove some $L^p$ estimates on the Riesz transforms and we establish some related maximal inequalities. The conditions that we arrive at, are essentially based on the control of the magnetic field by the electric potential.
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The paper concerns the magnetic Schrödinger operator on $R^n$. Under certain conditions, given in terms of the reverse Hölder inequality on the magnetic field and the electric potential, we prove some $L^p$ estimates on the Riesz transforms and we establish some related maximal inequalities.
We show various $L^p$ estimates for Schrödinger operators $-Δ+V$ on $\RR^n$ and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen \cite{Sh1}. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of $-Δ+V$ and their gradients.
We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main tool used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure.
In this paper, we prove analogues of O'Neil's inequalities for the convolution in the weighted Lebesgue spaces. We also establish the weighted two-sided norm inequalities for the potential operator.
Certain rearrangement inequalities of a type considered by Hardy, Riesz, and Brascamp-Lieb-Luttinger are studied. Subsets of the real line that extremize these inequalities are characterized. Our results apply only to special cases, and complement earlier results by Burchard treating other cases. This draft incorporates minor corrections and clarifications.
We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.
In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh-Fourier series.
We consider weighted norm inequalities for the Riesz potentials $I_α$, also referred to as fractional integral operators. First we prove mixed $A_p$-$A_\infty$ type estimates in the spirit of [13, 15, 17]. Then we prove strong and weak type inequalities in the case $p<q$ using the so-called log bump conditions. These results complement the strong type inequalities of Pérez [30] and answer a conjecture from [3]. For both sets of results our main tool is a corona decomposition adapted to fractional averages.
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $ω$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a …
We discuss weighted estimates for the squares of the Riesz transforms, R^{2}, on L^{2}(W) where W is a matrix A2 weight. We prove that if W is close to the Identity matrix Id, then the operator norm of R^{2} is close to its unweighted norm on L^{2} which is one. This is done by the use of the Bellman function technique.
In this paper, let $L=L_{0}+V$ be a Schrödinger type operator where $L_{0}$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption on $V$, the authors study the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q\leq 2$ and obtain a sharp result. Furthermore, the authors impose extra regularity assumptions on $V$ to obtain the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q>2$. As an application, the main results can be applied to the operator $L=(-Δ)^{m}…
Let $L=-Δ+V$ be a Schrödinger operator acting on $L^2(\mathbb R^n)$, $n\ge1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on $\mathbb R^n$. In this article, we will introduce weighted Hardy spaces $H^p_L(w)$ associated with $L$ by means of the area integral function and study their atomic decomposition theory. We also show that the Riesz transform $\nabla L^{-1/2}$ associated with $L$ is bounded from our new space $H^p_L(w)$ to the classical weighted Hardy space $H^p(w)$ when $\frac{n}{n+1}<p<1$ and$w\in A_1\cap RH_{(2/p)'}$.
We establish a criterion for a set of eigenfunctions of the one-dimensional Schrödinger operator with distributional potentials and boundary conditions containing the eigenvalue parameter to be a Riesz basis for $\mathscr{L}_2(0,π)$.
A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group $\mathbb{R}/\mathbb{Z}$, a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing.
Under the assumption that $V \in L^2([0,π]; dx)$, we derive necessary and sufficient conditions for (non-self-adjoint) Schrödinger operators $-d^2/dx^2+V$ in $L^2([0,π]; dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues). We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $-d^2/dx^2+V$ in $L^p([0,π]; dx)$, $p \in (1,\infty)$.
In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schrödinger operator $P=-Δ+V(x)$ on $\mathbb{R}^n, n\geq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q \geq n/2$. Let $T_1=(-Δ+V)^{-1}V,\ T_2=(-Δ+V)^{-1/2}V^{1/2}$ and $T_3=(-Δ+V)^{-1/2}\nabla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(\mathbb{R}^n)$ when $p$ ranges in a interval, where $b \in \mathbf{BMO}(\mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness.
Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the $α$ (strong) and $φ$ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized $L^1$ and $L^\infty$ space…
In this paper we establish the $L^p$-boundedness properties of the variation operators associated with the heat semigroup, Riesz transforms and commutator between Riesz transforms and multiplication by $BMO(R^n)$-functions in the Schrödinger setting.
Two weight Sobolev norm inequalities for fractional vector Riesz transforms and doubling weights
本文建立分数阶向量 Riesz 变换在两加权 Sobolev 空间间的 T1 定理，以 A_2 条件和两个对偶测试条件刻画有界性，并证明倍增测度下多种加权 Sobolev 范数等价。
This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hytönen-Naor \cite{HN}. We also improve the constant in $L^1…
We derive bounds and asymptotics for the maximum Riesz polarization quantity $$M_n^p(A) := \max_{{\bold x}_1, {\bold x}_2, \ldots, {\bold x}_n \in A} {\min_{{\bold x} \in A}{\sum_{j=1}^n{\frac{1}{|{\bold x} - {\bold x}_j|^{p}}}}}$$ (which is $n$ times the Chebyshev constant) for quite general sets $A \subset {\Bbb R}^m$ with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarizatio…
Sharp pointwise gradient estimates for Riesz potentials with a bounded density
本文在$\mathbb{R}^{n}$中建立了具有有界密度的Riesz位势及其梯度的锐点态不等式，并说明其在位势分析、矩理论等领域的应用价值。
Let $M$ be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and $L^q$ scaled Poincaré inequalities on suitable remote balls for some $q<2$. We prove the inequality $\left\Vert Δ^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ for all $p\in (q,2]$, which generalizes previous results due to Auscher and Coulhon. Our conclusion applies, in particular, when $M$ has a finite number of Euclidean ends. The proof strongly relies on Hardy inequalities, which are also new in this context and of independent interest. The second part of this work deal…
Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures
本文刻画分数阶向量 Riesz 变换在加倍测度下的两权 L^{p} 有界性，并将 Hytönen–Vuorinen 三元测试条件简化为局部测试、偏移 Muckenhoupt 与弱有界性条件。
Weak type $(1,1)$ estimates for Riesz transforms of elliptic operators and magnetic Schrödinger operators
该研究针对完备黎曼流形上的磁薛定谔算子与$\boldsymbol{\textit{R}}^d$粗糙区域的对称椭圆算子，证明其对应里斯变换的弱型(1,1)估计，拓展了相关算子的$L^p$有界性结论。
We prove the existence of $L^2$ solutions to the Helmholtz equation $(-Δ- 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(-Δ-1)u = -Vu + f$, are connected to the Limiting Absorption Principle for Schrödinger operators. The same techniques are then used to prove that a wide range of $L^p \mapsto L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their action on the clo…
Stability of Weighted Norm Inequalities
本文研究 Riesz 变换加权范数不等式在变量替换下的稳定性，证明单个及奇数阶迭代 Riesz 变换在 doubling 权重上对满测度旋转不稳定，而在 $A_{\infty}$ 权重上对双 Lipschitz 替换稳定。
Fix an integer $ n$ and number $d$, $ 0< d\neq n-1 \leq n$, and two weights $ w$ and $ σ$ on $ \mathbb R ^{n}$. We two extra conditions (1) no common point masses and (2) the two weights separately are not concentrated on a set of codimension one, uniformly over locations and scales. (This condition holds for doubling weights.) Then, we characterize the two weight inequality for the $ d$-dimensional Riesz transform on $ \mathbb R ^{n}$, \begin{equation*} \sup_{0< a < b < \infty}\left\lVert \int_{a < \lvert x-y\rvert < b} f (y) \frac {x-y} {\lvert x-y\rvert ^{d+1}} \; σ(dy) \right\rVert_{L ^{2}…
This paper is the continuation of the study on discrete harmonic analysis related to Jacobi expansions initiated in [1]. Considering the operator $\mathcal{J}^{(α,β)}=J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we focus on the study of weighted inequalities for the Riesz transform associated with it.
The classical Erdős-Turán inequality on the distribution of roots for complex polynomials can be equivalently stated in a potential theoretic formulation, that is, if the logarithmic potential generated by a probability measure on the unit circle is close to $0$, then this probability measure is close to the uniform distribution. We generalize this classical inequality from $d=1$ to higher dimensions $d>1$ with the class of Riesz potentials which includes the logarithmic potential as a special case. In order to quantify how close a probability measure is to the uniform distribution in a genera…
In this paper we prove that the variation operators associated with the heat semigroup and Riesz transforms related to the Schrödinger operator are bounded on the suitable $BMO$ type space.
In this paper we establish the product Hardy spaces associated with the Bessel Schrödinger operator introduced by Muckenhoupt and Stein, and provide equivalent characterizations in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions, and Littlewood--Paley theory, which are consistent with the classical product Hardy space theory developed by Chang and Fefferman. Moreover, in this specific setting, we also provide another characterization via the Telyakovskií transform, which further implies that the product Hardy space associated with this Bessel Schrödinger opera…
In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the $A_2$ conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal operator. As a consequence we are able to prove a number of sharp one and two weight norm inequalities for the Riesz potential.
Pointwise estimates for rough operators with applications to Sobolev inequalities
本文研究粗糙算子的Sobolev不等式，证明其满足被Riesz势作用于梯度的逐点界，并由此推导出左侧带算子的新Sobolev型不等式。
Weighted norm inequalities of higher-order Riesz transforms associated with Laguerre expansions
该研究针对与拉盖尔展开相关的高阶里斯变换，通过分析微分算子L_ν及偏导数δ_{ν_j}，探究其加权估计，补全了高阶里斯变换在全范围参数下有界性的描述。
In this paper, the mean value formula depends on the Bessel generalized shift operator corresponding to the solutions of the boundary value problem related to the Bessel operator are studied. In addition to, Riesz Bessel transforms related to the Bessel operators are studied. Since Bessel generalized shift operator is translation operator corresponding to the Bessel operator, we construct a family of RBxj by using Bessel generalized shift operator. Finally, we analysis weighted inequalities involving Riesz Bessel transforms .
We obtain convolution inequalities in Lebesgue and Lorentz spaces with power weights when the functions involved are assumed to be radially symmetric. We also present applications of these results to inequalities for Riesz potentials of radial functions in weighted Lorentz spaces and embedding theorems for radial Besov spaces with power weights.
The main aim of this paper is to investigate $\left( H_{p},L_{p}\right) $ and $\left( H_{p},L_{p,\infty }\right) $ type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin-Fourier series.
Unique Continuation Inequalities for the Schrödinger equations associated with the Special Hermite operators
本文研究了与特殊 Hermite 算子相关的 Schrödinger 方程解的唯一延拓不等式，证明了若解在两个不同时间点的有限测度集外保持微小，则其在整个空间亦保持微小，并探讨了 Hermite-Schrödinger 方程的类似结果。
Weighted inequalities for variation operators associated with fractional Schrödinger semigroups
本文研究了与分数阶 Schrödinger 半群相关的变差算子，在势能属于逆向 Hölder 类的条件下，建立了其加权有界性，包括加权 $L^{p}-L^{q}$ 定量不等式和混合弱型不等式。
Sharp commutator estimates of all order for Coulomb and Riesz modulated energies
本研究拓展改进了库仑与Riesz调制能量沿输运迭代导数的泛函不等式，得到任意阶、可局部化的精确估计，发展了交换子局部正则性理论，可应用于平均场极限等问题的收敛速率分析。
Let $\mathcal{L}=-Δ+\mathit{V}(x)$ be a Schrödinger operator, where $Δ$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse Hölder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón-Zygmund operatos and Littlewood-Paley functions. These results generalize substantially some well-know results.
Weighted mixed inequalities for commutators of Schrödinger type operators
本文研究了薛定谔背景下奇异积分算子交换子的加权混合不等式，针对 $(s,δ)$ 型算子和 $\text{BMO}(ρ)$ 符号给出估计，并推广了先前结果，同时引入 $A_p^ρ$ 权作为 Muckenhoupt 类的扰动。
We prove that Calderón-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on a wide family of function spaces.
In this paper we introduce Stein's square function associated with bilinear Bochner-Riesz means and investigate its $L^p$ boundedness properties. Further, we discuss several applications of the square function in the context of bilinear multipliers. In particular, we obtain results for maximal function associated with generalised bilinear Bochner-Riesz means. This extends the results proved in~\cite{JS}. Another application concerns the $L^p$ estimates for bilinear fractional Schrödinger multipliers. Finally, we improve upon a result of Grafakos, He and Honzik~\cite{GHH} in the context of bili…
In these lecture notes we describe some recent work on two weight norm inequalities for fractional integral operators, also known as Riesz potentials, and for commutators of fractional integrals. These notes are based on three lectures delivered at the 6th International Course of Mathematical Analysis in Andalucia, held in Antequera, Spain, September 8-12, 2014. They are, however, greatly expanded to include both new results and many details that I did not present in my lectures due to time constraints.
We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-$λ$ method that does not use any size or smoothness estimates for the kernels.
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