Loukas Grafakos, Atanas Stefanov
Convolution type Calderón-Zygmund singular integral operators with rough kernels $\pv \Om(x)/|x|^n$ are studied. A condition on $\Om$ implying that the corresponding singular integrals and maximal singular integrals map $L^p \to L^p$ for $1<p<\nf$ is obtained. This condition is shown to be different from the condition $\Om\in H^1(\sn)$.
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In this paper, we establish BMO estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.
In this paper, we give the boundedness of some parabolic multilinear commutators generated by a class of parabolic maximal and linear operators with rough kernel and parabolic local Campanato functions on the parabolic generalized local Morrey spaces, respectively. Indeed, the results in this paper are extensions of some known results.
We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in [9] and [10]
We obtain weak type (1,1) estimates for the inverses of truncated discrete rough Hilbert transform. We include an ex- ample showing that our result is sharp. One of the ingredients of the proof are regularity estimates for convolution of singular measure associated with the sequence $[m^α]$.
The aim of this paper is to get the product Lp-estimates, weighted estimates and two-weighted estimates for rough multilinear fractional integral operators and rough multi-sublinear fractional maximal operators, respectively. The author also studies two-weighted weak type estimate on product Lp (Rn) for rough multi-sublinear fractional maximal operators. In fact, this article is the rough kernel versions of [4, 5]' s results.
We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.
In this paper, we consider the boundedness of a class of sublinear operators and their commutators by with rough kernels associated with Calderon-Zygmund operator, Hard-Littlewood maximal operator, fractional integral operator, fractional maximal operator by with rough kernels both on vanishing generalized Morrey spaces and vanishing Morrey spaces, respectively.
We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.
Endpoint and Interpolation Estimates for Higher-Order Commutators of Rough Fractional Maximal Operators with Variable Kernels on Variable Exponent Morrey Spaces
研究变指数Morrey空间中粗糙分数次极大算子高阶交换子，在变指数和Morrey型权重条件下建立有界性结果，分析端点边界情况，还开发插值框架连接端点和内部估计。
Weak type $(1,1)$ estimates for noncommutative rough homogeneous singular integrals
该研究将Seeger的端点定理推广到非交换情形，针对满足LlogL条件的算子值粗糙齐次奇异积分，通过非对角分析建立了弱型(1,1)端点估计，避免了额外对数损失。
Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.
In this paper, we show the strong and weak type boundedness of $T_{Ω,α}^A$ and $M_{Ω,α}^A$, the multilinear fractional integral operators and the corresponding fractional maximal operators, on the two weights weighted Morrey space. Furthermore, we also obtain the mulitilinear singular integral operators $T_Ω^A$ and the corresponding maximal operators $M_Ω^A$ are strong bounded operators on the same spaces.
In this paper, we first give some new characterizations of Muckenhoupt type weights through establishing the boundedness of maximal operators on the weighted Lorentz and Morrey spaces. Secondly, we establish the boundedness of sublinear operators including many interesting in harmonic analysis and its commutators on the weighted Morrey spaces. Finally, as an application, the boundedness of strongly singular integral operators and commutators with symbols in BMO space are also given.
Rough spectral asymptotics for commutators of general singular integrals
本文证明了度量测度空间上一般奇异积分交换子奇异值的谱渐近公式（用上下极限代替精确极限），并提出了Marcinkiewicz插值定理的新渐近版本及临界指标处交换子上界的新方法。
The aim of this paper is to get the boundedness of rough sublinear operators generated by fractional integral operators on vanishing generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. Also, rough fractional integral operator and a related rough fractional maximal operator which satisfy the conditions of our main result can be considered as some examples.
We prove that the class of convolution-type kernels satisfying suitable decay conditions of the Fourier transform, appearing in the works of Christ, Christ-Rubio de Francia, and Duoandikoetxea-Rubio de Francia gives rise to maximally truncated singular integrals satisfying a sparse bound by $(1+\varepsilon,1+\varepsilon)$-averages for all $\varepsilon>0$, with linear growth in $\varepsilon^{-1}$. This is an extension of the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to maximally truncated singular integrals. Our results cover the rough homogeneous singular int…
Weak type $(1,1)$ boundedness of noncommutative singular integral operators with rough kernels
本文证明了具有粗糙核的非交换奇异积分算子的弱型 $(1,1)$ 有界性，通过发展核的正交性处理好函数，并利用微局部分解和 $TT^*$ 论证处理坏函数。
A new pointwise inequality for rough operators and applications
本文提出了一种针对粗糙奇异积分算子的新点估计，并应用该估计推导出多种泛函空间中的Sobolev型不等式。
Sparse pointwise bounds for maximal truncations of rough singular integrals and Sobolev-type inequalities
本文证明粗糙奇异积分极大截断的稀疏逐点控制，导出双权Sobolev不等式及Hedberg型估计，推广至多种加权空间。
Characterization of weights for the variable fractional maximal operator and weighted inequalities for variable fractional rough operators
本文研究了可变分数极大算子在可变Lebesgue空间上的权重刻画，扩展了已有结果，并引入了满足新变量Hörmander条件的核类，证明了Coifman-Fefferman不等式和加权不等式。
We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\mathcal{L}^{p(\cdot), q(\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\to 0$ and $s\to\infty$. To prove…
Compactness of commutators of rough singular integrals
本文刻画粗糙奇异积分算子 T_Ω 交换子在双权非对角情形下的紧性，并利用矩阵权 Kolmogorov-Riesz 定理得到矩阵权紧性结果。
In this paper, we study the boundedness theory for maximal Calderón-Zygmund operators acting on noncommutative $L_p$-spaces. Our first result is a criterion for the weak type $(1,1)$ estimate of noncommutative maximal Calderón-Zygmund operators; as an application, we obtain the weak type $(1,1)$ estimates of operator-valued maximal singular integrals of convolution type under proper {regularity} conditions. These are the {\it first} noncommutative maximal inequalities for families of linear operators that can not be reduced to positive ones. For homogeneous singular integrals, the strong type …
Maximal regularity for evolution equations with critical singular perturbations
本文研究了演化方程中由临界奇异扰动引起的最大正则性问题，提出了一种加权最大正则性理论，扩展了之前的研究成果。
We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For $2m$-th order systems with $VMO$ regularity in space, we obtain $L^{p}(L^{q})$ estimates for all $p>2$ and $q\geq 2$, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition,…
Rough differential equations are solved for signals in general Besov spaces unifying in particular the known results in Hölder and p-variation topology. To this end the paracontrolled distribution approach, which has been introduced by Gubinelli, Imkeller and Perkowski ["Paracontrolled distribution and singular PDEs", Forum of Mathematics, Pi (2015)] to analyze singular stochastic PDEs, is extended from Hölder to Besov spaces. As an application we solve stochastic differential equations driven by random functions in Besov spaces and Gaussian processes in a pathwise sense.
In this paper we pursue the study of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calderón-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the $L^2(\Rn)$ norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal…
We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calderón-Zygmund singular integral operator $T^Φ$ is of weak type $(r,r)$ for all $r\in(1,\infty)$, then $T^Φ$ extends to a bounded operator on $X(\mathbb{R}^n)$. This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ under natural assumptions on the variable exponent $p:\mathbb{R}\to(1,\infty)$. Applications of the above r…
Let $(S, d, ρ)$ be the affine group $\mathrm{R}^n \ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian metric $d$ and the right Haar measure $ρ$, which is of exponential growth at infinity. In this paper, for any linear operator $T$ on $(S, d, ρ)$ associated with a kernel $K$ satisfying certain integral size condition and Hörmander's condition, the authors prove that the following four statements regarding the corresponding maximal singular integral $T^\ast$ are equivalent: $T^\ast$ is bounded from $L_c^\infty$ to $\mathrm{BMO}$, $T^\ast$ is bounded on $L^p$ for all $p\in(1, \infty…
Let $M_{Ω,\a}$ and $I_{Ω,\a}$ be the fractional maximal and integral operators with rough kernels, where $0 < \a < n$. In this paper, we shall study the continuity properties of $M_{Ω,\a}$ and $I_{Ω,\a}$ on the generalized local Morrey spaces $LM_{p,φ}^{x_0}$. The boundedness of their commutators with local Campanato functions is also obtained.
In this work, we develop $L^p$ boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the $x$ variable. Moreover, the $B(L^p)$ operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimates are shown to be sharp with respect to the required smoothness in the $ξ$ variable. As a corollary, we obtain $L^p$ bounds for (smoothed out versions of) the maximal directional Hilbert transform and the Carleson operator.
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