Carleson measure and balayage
Sandra Pott, Alexander Volberg
一个Carleson测度的平衡当然存在于BMO中。我们证明逆命题是假的。我们还作出了双边估计的Carleson范数的一个积极措施的balayages。
The balayage of a Carleson measure lies of course in BMO. We show that the converse statement is false. We also make a two-sided estimate of the Carleson norm of a positive measure in terms of balayages.
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BMO functions and Balayage of Carleson measures in the Bessel setting
本文将 Bessel 框架下具有有界支撑的奇 BMO 函数刻画为有界函数与 Carleson 测度关于 Bessel 算子相关 Poisson 半群的 balayage 之和，推广了 Carleson 的经典结果。
Dunkl--BMO functions representable in terms of generalized balayage of Carleson measures and an application
本文刻画了Dunkl--BMO空间中紧支撑函数为有界函数与Carleson测度广义扫集之和，并由此通过主值积分建立了Dunkl框架下Hardy空间的新刻画，且结论推广至热半群。
A note on martingale Hardy spaces
研究了Carleson序列与平衡数的关系，并利用这一关系给出了极大函数与平方函数在非齐次鞅背景下的L1-模等价性的一个简单证明.
Integral representation of balayage on locally compact spaces and its application
该文在局部紧空间上针对一般函数核的内外扫除理论中，给出积分表示成立的条件，结果对经典与现代位势理论中的若干核是新的，并应用于分数阶 Green 核下测度总质量的变化分析。
We develop a theory of inner balayage of a positive Radon measure $μ$ of finite energy on a locally compact space $X$ to arbitrary $A\subset X$, generalizing Cartan's theory of Newtonian inner balayage on $\mathbb R^n$, $n\geqslant3$, to a suitable function kernel on $X$. As an application of the theory thereby established, we show that if the space $X$ is perfectly normal and of class $K_σ$, then a recent result by Bent Fuglede (Anal. Math., 2016) on outer balayage of $μ$ to quasiclosed $A$ remains valid for arbitrary Borel $A$. We give in particular various alternative definitions of inner (…
Balayage of measures: behavior near a corner
本文研究了测度在有角点的域边界上的扫除行为，推导了近角点的消失率，并在特定条件下给出了显式常数。
Carleson measures on locally finite trees
该研究刻画了局部有限树上的Carleson测度，建立其与$L^p$空间间合适Poisson积分有界性的联系，并在有界度树情形下探究了其与树边界上BMO函数的关系。
The paper deals with the theory of balayage of Radon measures $μ$ of finite energy on a locally compact space $X$ with respect to a consistent kernel $κ$ satisfying the domination principle. Such theory is now specified for the case where the topology on $X$ has a countable base, while any $f\in C_0(X)$, a continuous function on $X$ of compact support, can be approximated in the inductive limit topology on the space $C_0(X)$ by potentials $κλ:=\intκ(\cdot,y)\,dλ(y)$ of measures $λ$ of finite energy. In particular, we show that then the inner balayage can always be reduced to balayage to Borel …
On Fuglede's problem on pseudo-balayage for signed Radon measures of infinite energy
针对局部紧空间上的适当核，发展一般符号Radon测度（含无限能量）到一般集合的内（外）伪扫除理论，解决Fuglede提出的理论扩展问题，并给出在加权最小能量问题中的应用。
Infinitesimal Carleson property for weighted measures induced by analytic self-maps of the unit disk
我们证明，对于每个$α> -1$，每个解析自映射的测量值$d{\cal A}_α(z) = (α+ 1) (1 - |z|^2)^α\, d{\cal A} (z)$的拉回测量值$ϕ({\cal A}_α)$（其中${\cal A}$是单位盘$\D$上的归一化面积测量值）$ϕ\colon \D \to \D$不仅是$(α+ 2)$测量值，而且大小为$\eps h$的Carleson窗口的测量值由$\eps^{α+ 2}$乘以相应大小为$h$的窗口的测量值控制。这意味着作为$(α+ 2)$ -Carleson测度的性质在所有无穷小尺度上都是正确的。给出了加权Bergman-Orlicz空间上复合算子紧性的一个应用.
We establish the theory of balayage for the Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, on $\mathbb R^n$, $n\geqslant3$, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan's ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner e…
Balayage of Radon measures of infinite energy on locally compact spaces
针对局部紧空间上不一定具有有限能量的Radon测度，发展了到任意子集的内部与外部balayage理论，证明了扫除测度的存在唯一性并推广了现有理论。
Balayage of measures: behavior near a cusp
本文研究正测度在具有向外尖点的区域边界上的扫除测度，在测度密度满足幂次渐近条件下给出尖点附近的首阶项，并推广到多角点与尖点情形，还应用于二维库仑气体。
For the Riesz kernel $κ_α(x,y):=|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, we introduce the inner pseudo-balayage $\hatω^A$ of a (Radon) measure $ω$ on $\mathbb R^n$ to a set $A\subset\mathbb R^n$ as the (unique) measure minimizing the Gauss functional \[\intκ_α(x,y)\,d(μ\otimesμ)(x,y)-2\intκ_α(x,y)\,d(ω\otimesμ)(x,y)\] over the class $\mathcal E^+(A)$ of all positive measures $μ$ of finite energy, concentrated on $A$. For quite general signed $ω$ (not necessarily of finite energy) and $A$ (not necessarily closed), such $\hatω^A$ does exist, and it maintains the basic features of …
The paper deals with the theory of inner (outer) capacities on locally compact spaces with respect to general function kernels, the main emphasis being placed on the establishment of alternative characterizations of inner (outer) capacities and inner (outer) capacitary measures for arbitrary sets. The analysis is substantially based on the close interaction between the theory of capacities and that of balayage. As a by-product, we provide a rigorous justification of Fuglede's theories of inner and outer capacitary measures and capacitability (Acta Math., 1960). The results obtained are largely…
For $0<s<1$, let $\{z_n\}$ be a sequence in the open unit disk such that $\sum_n (1-|z_n|^2)^s δ_{z_n}$ is an $s$-Carleson measure. In this paper, we consider the connections between this $s$-Carleson measure and the theory of Möbius invariant $F(p, p-2, s)$ spaces by the Volterra type operator, the reciprocal of a Blaschke product, and second order complex differential equations having a prescribed zero sequence.
We study properties of the $α$-Green kernel $g_D^α$ of order $0<α\leqslant2$ for a domain $D\subset\mathbb R^n$, $n\geqslant3$. This kernel is associated with the $α$-Riesz kernel $|x-y|^{α-n}$, $x,y\in\mathbb R^n$, in a manner particularly well known in the case $α=2$. Besides the usual principles of potential theory, we establish for the $α$-Green kernel the property of consistency. This allows us to prove the completeness of the cone of positive measures $μ$ on $D$ with finite energy $g_D^α(μ,μ):=\iint g_D^α(x,y)\,dμ(x)\,dμ(y)$ in the topology defined by the energy norm $\|μ\|_{g_D^α}=\sqrt…
Generalized Carleson Embeddings of M{ü}ntz Spaces
本文研究 Müntz 空间到加权 Lebesgue 空间的广义 Carleson 嵌入，分别在 β≥1 与 0<β<1 时刻画相应测度，并推广 Gurariy-Macaev 估计。
General theory of balayage on locally compact spaces. Applications to weighted minimum energy problems
本文在局部紧空间上针对一般 Radon 测度和一般集合建立了内（外）扫除理论，证明其存在唯一性并给出多种等价刻画，从而推广了有限能量情形下的已有工作，并改进了 Gauss 变分问题的相关结果。
We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob's maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator…
Carleson-type embeddings with closed range
研究Carleson测度μ下Hardy空间H^p在L^p(μ)中的像是否闭合，并推广至(p,q)-Carleson测度及Bergman空间。
Characterization of Carleson measures by the Hausdorff-Young property
证明了定义在正半轴上的Lp（1<p<=2）函数的拉普拉斯变换在右复半平面上满足权为正的Hausdorff-Young型不等式当且仅当权为Carleson测度.
Carleson measures on planar sets
本文研究了复平面上开子集上的Carleson测度。一个圆形域是一个连通的开子集，它的边界由许多不相交的圆组成。我们称一个域$G$为多好连通的，如果存在一个圆形域$W$和一个从$W$到$G$的共形映射$ψ$，使得$ψ$关于$\partial W$上的弧长几乎是单叶的.我们刻画了这些开子集的所有Carleson测度，使得它们的每个分量都是多良连通的，并且分量的调和测度是相互奇异的.我们的结果表明Carleson测度的扩展可能取决于这一类开子集。
We give an alternate proof of three versions of the theorem on extrapolation of Carleson measures.
Lp theory for outer measures and two themes of Lennart Carleson united
我们开发了一个理论的Lp空间的基础上外措施，而不是措施。该理论包括测度空间上的经典Lp理论作为特例。它还包括部分位势论和Carleson嵌入定理。该理论是描述经典奇异积分理论的优美语言，如仿积估计和T（1）定理，它是特别有用的推广奇异积分理论的时频分析。我们制定并证明了一个广义Carleson嵌入定理，并给出了一个相对较短的减少基本估计的双线性希尔伯特变换这个新的Carleson嵌入定理。
Relations among Besov-Type Spaces, Triebel-Lizorkin-Type Spaces and Generalized Carleson Measure Spaces
本文构造了一些反例，证明了广义Carleson测度空间与Triebel-Lizorkin型空间对某些参数不等价，这在[Taiwan J. Math.15（2011），919 - 926]中得到了证明.进一步证明了对于某些特殊参数，广义Carleson测度空间、Triebel-Lizorkin型空间和Besov型空间与某个Triebel-Lizorkin空间重合，从而回答了[Lecture Notes in Mathematics 2005，Springer-Verlag，Berlin，2010，xi +281 pp.]的注6.11（i）中提出的一个问题.
$Φ$-Carleson measures and multipliers between Bergman-Orlicz spaces of the unit ball of $\mathbb {C}^n$
定义了$Φ$ -Carleson测度的概念，其中$Φ$是凹增长函数或凸增长函数，并给出了等价的定义.然后，我们刻画了Bergman-Orlicz空间的$Φ$ -Carleson测度，并应用它们刻画了Bergman-Orlicz空间之间的乘子.
Hole probabilities and balayage of measures for planar Coulomb gases
研究二维库仑气体空穴概率，当点数\(n\to\infty\)时，证明无点在\(U\)中的概率为\(\exp(-Cn^{2}+o(n^{2}))\)，通过平衡测度等确定\(C\)，给出\(\nu\)和\(C\)显式表达式例子，推广前人结果。
Characterization of uniformly convex and smooth Banach spaces by using Carleson measures in Bessel settings
本文利用由Bessel算子和Banach值BMO函数相关Poisson积分定义的Carleson测度，给出了q一致凸与光滑Banach空间的新刻画，并通过Lusin积分的映射性质描述其一致凸性与光滑性。
We prove generalized Carleson embeddings for the continuous wave packet transform from $L^p(\mathbb{R},w)$ into an outer $L^p$ space for $2< p < \infty$ and weight $w \in A_{p/2}$. This work is a weighted extension of the corresponding Lebesgue result in arXiv:1309.0945v3 and generalizes a similar result in arXiv:1207.1150. The proof in this article relies on $L^2$ restriction estimates for the wave packet transform which are geometric and may be of independent interest.
Balayage, equilibrium measure, and Deny's principle of positivity of mass for $α$-Green potentials
本文研究$\alpha$-Green势理论，建立正Radon测度扫掠的存在唯一性与刻画，给出相关等式成立条件，得到推广意义下平衡测度的存在条件，并改进了Deny质量正性原理的已有结果。
Carleson operators on doubling metric measure spaces
本文在加倍度量测度空间上提出调制函数的一般公理化方法，证明相应Carleson算子的L^p有界性，推广了经典结果，并用Lean验证。
In this paper we formulate embedding maps into time-frequency space related to the Carleson operator and its variational counterpart. We prove bounds for these embedding maps by iterating the outer measure theory of [DT15]. Introducing iterated outer $L^p$ spaces is a main novelty of this paper. [DT15] Yen Do and Christoph Thiele. "$L^p$ theory for outer measures and two themes of Lennart Carleson united". In: Bulletin of the American Mathematical Society 52.2 (2015), pp. 249-296.
In this paper, we introduce the Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth's $φ-$transform in the variable exponent settings, we show that the dual space of the variable Hardy space $H^{p(\cdot)}$ is $CMO^{p(\cdot)}$. As applications, we obtain that Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$, Campanato space with variable exponent $\mathfrak{L}_{q,p(\cdot),d}$ and Hölder-Zygmund spaces with variable exponents $\mathcal {\dot{H}}_d^{p(\cdot)}$ coincide as sets and th…
Carleson measures, trees, extrapolation, and $T(b)$ theorems
在调和分析中，关于Carleson测度、停时参数和原子分解的理论已经很好地建立起来。最近的理论是相空间分析的角度来看，波包的瓦片，树的选择算法，和树的大小估计。本文的目的是证明这两个理论实际上是密切相关的，采取现有的结果，并在一个统一的设置重新证明他们。特别是，我们给出了一个二元版本的外推Carleson措施，有两个单独的证明，以及一个双边的本地二元$T(b)$定理，推广早期的$T(b)$定理的大卫，奥纳，塞姆斯和基督。
Carleson embeddings and pointwise multipliers between Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper half-plane
本文给出涉及凹或凸增长函数的 Carleson 测度的一般刻画，并据此建立上半平面 Hardy-Orlicz 空间与 Bergman-Orlicz 空间之间的连续嵌入与逐点乘子集合的刻画。
Carleson Measures and Logvinenko-Sereda sets on compact manifolds
给定一个维数为$m\geq 2$的紧致黎曼流形$M$，研究了由$M$上Laplace-Beltrami算子的本征值小于$L\geq 1$的本征函数生成的$L^2(M)$的函数空间.在这些空间上，我们给出了Carleson测度和Logvinenko-Sereda集的特征。
Carleson--Buckley measures beyond the scope of $A_\infty$ and their applications
Carleson测度在调和分析中是普遍存在的。1991年，Fückman-Kenig-Pipher的论文中，为了解决椭圆型偏微分方程的正则性问题，引入了一类有趣的Carleson测度。这些Carleson指标与$A_\infty$权重相关。在离散设置（我们需要确切的离散设置在这里），他们研究了巴克利的，在那里他们与二元$A\infty^d$。我们在这里的目标是表明，这样的Carleson-Buckley措施（在离散设置）存在几乎任何积极的功能（重量）。当然需要一些修改，因为已知巴克利测度的Carleson性质等价于$A_\infty^d$中的权。然而，一个非常自然的概括这些事实存在的权重更一般的$A_\infty$，当然，在一个特殊的情况下，$A_\infty^d$它给巴克利的结果。我们对Buckley不等式的推广超出了$A_\infty$的范围，从而证明了所谓的bump猜想.
In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding embedding operator is continuous. Restricting to the case of light cones of dimension three, we prove that by freezing one or two variables, the problem of embedding derivatives of the Hilbert-Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert-Bergman spaces of the upper-half plane or the product of two …
Maximal function and Carleson measures in Békollé-Bonami weights
设$ω$为Békollé-Bonami权。本文给出了正测度$μ$的一个完整刻画，使得$$\int_{\mathcal H}|M_ωf(z)|^qdμ(z)\le C\left(\int_{\mathcal H}|f(z)|^pω(z)dV(z)\right)^{q/p}$$和$$μ\left(\{z\in \mathcal H: Mf(z)>λ\}\right)\le \frac{C}{λ^q}\left(\int_{\mathcal H}|f(z)|^pω(z)dV(z)\right)^{q/p}$$，其中$M_ω$是上半平面$\mathcal H$上的加权Hardy-Littlewood极大函数，而$1\le p,q<\infty$ .
We give in this paper some equivalent definitions of the so called $ρ$-Carleson measures when $ρ(t)=(\log(4/t))^p(\log\log(e^4/t))^q$, $0\le p,q<\infty$. As applications, we characterize the pointwise multipliers on $LMOA(\mathbb S^n)$ and from this space to $BMOA(\mathbb S^n)$. Boundedness of the Cesàro type integral operators on $LMOA(\mathbb S^n)$ and from $LMOA(\mathbb S^n)$ to $BMOA(\mathbb S^n)$ is considered as well.
Let $Ω$ be a domain in $\mathbb{R}^{d+1}$, $d \geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $Ω$ satisfies a corkscrew condition and if $\partial Ω$ is $d$-Ahlfors regular, i.e. Hausdorff measure $\mathcal{H}^d(B(x,r) \cap \partial Ω) \sim r^d$ for all $x \in \partial Ω$ and $0 < r < {\rm diam}(\partial Ω)$, then $\partial Ω$ is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on $Ω$ or (b) an $\varepsilon$-approximation property for all $0 < \varepsilon <1$ for every such function. Here we…
Carleson Potentials and the Reproducing Kernel Thesis for Embedding Theorems
本文给出了单位圆盘和单位球上Carleson嵌入定理的一个新证明。证明这一事实的唯一技术工具是格林公式。出发点是每个Carleson测度都产生一个有界的下调和函数。使用这个函数，我们构造了一个新的相关Carleson措施，允许一个简单的嵌入。在圆盘的情况下，这给出了已知的最佳常数，而先前的最佳常数由N给出。尼可斯基
Carleson Measures, Vanishing Mean Oscillation and Critical Points
本文利用有界外函数将有限正 Borel 测度转化为消失 Carleson 测度，并据此构造具有 VMO 边界值且保持 Hardy 空间函数临界点及重数的有界解析函数。
Weak Carleson conditions in uniformly rectifiable metric spaces: the WCD and alpha numbers
该研究通过弱Carleson条件刻画一致可求长度度量空间，证明UR空间满足弱常数密度条件，引入度量空间版alpha数并证明其弱Carleson条件可刻画度量一致可求长度性，且两结果均为更一般抽象结论的推论。
Let $Ω$ be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When $Ω$ is connected in some quantitative form one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting these equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, there is no Fefferman-Kenig-Pipher perturbation result available. In this paper, …
We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $dμ_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions $u$ of such systems satisfying the said Carleson measure condition. These imply that BMO can be characterized as the collection of nontangential pointwise traces of smooth null-s…
Bergman-type Singular Operators and the Characterization of Carleson Measures for Besov--Sobolev Spaces on the Complex Ball
本文的目的有两个方面。首先，我们将Nazarov，Treil和Volberg的非齐次调和分析方法推广到处理“Bergman型”奇异积分算子。这种算子的典型例子是单位圆盘上的Beurling变换。其次，我们利用本文的方法解决了$\mathbb{C}^d$的复球上解析函数$B^σ_2$的索伯列夫空间的Carleson测度的刻画这一重要的公开问题。特别地，我们证明了对于任何$σ> 0$，Carleson措施的空间的特点是“T1条件”。这些结果的证明方法是Nazarov，Treil和第一作者的工作的扩展和另一个应用。
The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. We split our proof on two main ste…
The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. This paper…
We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-{\rm div}(A\nabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_\infty$ class…
Let $Ω\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partialΩ$ is uniformly $n$-rectifiable if every bounded harmonic function on $Ω$ is $\varepsilon$-approximable or if every bounded harmonic function on $Ω$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $Ω=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we …
Inner Riesz balayage in minimum energy problems with external fields
本文研究带外场的 Riesz 核最小能量问题，证明在拟闭集上内 balayage 具有有限能量时，解存在的充要条件为内容量有限或 balayage 总质量不小于1，并给出解的替代刻画与支集分析。
The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiene…
The $p$--Bloch--Carleson measure problem for $0<p<\infty$: A complete characterization via Carleson tents
本文通过二进容量条件完整刻画了单位圆盘上的Bloch-Carleson测度，给出了嵌入有界性与紧性的特征，证明基于Bergman投影表示与核算子的二进离散化。
Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition
考虑半空间$\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$上的散度型椭圆型方程$Lu:=\nabla\cdot(A\nabla u)=0$，其系数矩阵$A$是复椭圆的、有界的、可测的.此外，我们还假设$A$在横截边界的方向上满足一些额外的正则性，即$A(x,t) -A(x,0)$满足Fundaman-Kenig-Pipher型Carleson测度条件，且Carleson范数较小.在这些条件下，我们建立了$L^p$，Hardy，Sobolev，BMO和Hölder空间中双层和单层势的全范围有界性结果。此外，我们证明$L$的狄利克雷问题的可解性，与数据在$L^p(\mathbb{R}^n)$，$BMO(\mathbb{R}^n)$，和$C^α(\mathbb{R}^n)$，和诺依曼和正则性问题的可解性，与数据分别在空间$L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n)$和$L^p_1(\mathbb{R}^n)/H^{1,p}(\mathbb{R}^n)$，与适当的限制指数，假设可逆性层电位的$t$ -独立运营商$L_0:= -\nabla\cdot(A(\cdot,0)\nabla)$。
Beurling-Carleson sets, inner functions and a semi-linear equation
本文综述了Beurling-Carleson集的定义与性质，利用corona型分解揭示了奇异测度的多项性质与该集可数并集的等价关系，并证明了其与半线性方程近最大解的双射对应。
In the setting of tube domains over symmetric cones, $T_Ω$, we study the characterization of the positive Borel measures $μ$ for which the Hardy space $H^p$ is continuously embedded into the Lebesgue space $L^q (T_Ω, dμ)$, $0<p<q<\infty.$ Extending a result due to Blasco for the unit disc, we reduce the problem to standard measures. We obtain that a Hardy space $H^{p}$, $1\leq p < \infty,$ embeds continuously in weighted Bergman spaces with larger exponents. Finally we use this result to characterize multipliers from $H^{2m}$ to Bergman spaces for every positive integer $m$.
Littlewood-Paley and Carleson measure characterizations of Lipschitz spaces adapted to Schrödinger operators
针对非负且属于逆Hölder类RH_q（q>n/2）的势V的薛定谔算子L=-Δ+V，定义了适应L的Lipschitz空间Λ_L^α(R^n)，并利用Littlewood-Paley分解和Carleson测度给出了该空间在0<α<2-n/q范围内的刻画。
$L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures on compact manifolds
本文刻画了带或不带边界紧流形上所有$1<p<\infty$的$L^p$-Logvinenko-Sereda集与$L^p$-Carleson测度，进一步研究了无边界流形上特征函数的相应性质，并在标准球面上给出完整刻画与猜想，为相关开放问题提供了新进展。
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