Norberto Laghi, Neil lyall
In this article we obtain $L^2$ bounds for strongly singular integrals along curves in $\R^d;$ our results both generalise and extend to higher dimensions those obtained by Chandarana in the plane. Moreover, we show that the operators in question are bounded from $L\log L$ to weak $L^1$ at the critical exponent $α=0.$
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We prove sharp L^p-L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature assumption.
We prove certain $L^p$ estimates ($1<p<\infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels.
This is an expository paper on the characterization of the even (or odd) smooth homogeneous convolution Calderón-Zygmund operators in R^n such that the maximal singular integral can be controlled in the L^2 norm by the singular integral. We focuss attention on special cases of the general statements to convey the main ideas of the proofs in a transparent way, as free as possible of the technical complications inherent to the general case. Particular attention is devoted to higher Riesz transforms. The exposition is based on two papers by J.Mateu, J.Orobitg, C.Pérez and J.Verdera published in 2…
We establish a quantitative weighted inequality for the bilinear rough singular integral, where the bound is controlled by the cube of the weight constant.
We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a sharp condition for the kernels. Also, we prove some weighted $L^p$ inequalities for the operators.
We prove the existence of a $(d-2)$-dimensional purely unrectifiable set upon which a family of \emph{even} singular integral operators is bounded.
We prove a T(1) Theorem to completely characterize compactness of Calderon-Zygmund operators. The result provides sufficient and necessary conditions for the compactness of singular integral operators acting on L^p(R).
Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.
We prove a sharp Lp estimate for a singular Radon transform according to a size condition of its kernel, which is useful for extrapolation.
In this paper, the behavior for commutators of a class of bilinear singular integral operator associated with non-smooth kernels on the products of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO functions, compactness of the commutators is proved.
A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized $L^p$ spaces with variable exponent.
We consider mapping properties of the iterated Stieltjes transform, establishing its new relations with the iterated Hilbert transform (a singular integral) on the half-axis and proving the corresponding convolution and Titchmarsh's type theorems. Moreover, the obtained convolution method is applied to solve a new class of singular integral equations.
Asymptotic expansion of oscillatory integrals with singular phases
本文研究相位具有奇异性的一维振荡积分，给出其奇异性的渐近展开描述，并将类似结果推广到拉普拉斯积分。
In this paper we introduce a new class of Gaussian singular integrals, the general alternative Gaussian singular integrals and study the boundedness of them in Lp, p >1 and its weak (1,1) boundedness with respect to the Gaussian measure following the works of S. Pérez and H. Aimar, L. Forzani and R. Scotto respectively.
The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. We introduce a class of singular integral operators associated with Zygmund dilations and show the boundedness for these operators on $L^p, 1<p<\infty$, which covers those studied by Ricci--Stein \cite{RS} and Nagel--Wainger \cite{NW}
In generalized Lebesgue spaces L^{p(.)} with variable exponent p(.) defined on the real axis, we obtain several inequalities of approximation by integral functions of finite degree. Approximation properties of Bernstein singular integrals in these spaces are obtained. Estimates of simultaneous approximation by integral functions of finite degree in L^{p(.)} are proved.
In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on $L^p(\mathbb{R}^d,\,w)$, $p\in (1,\,\infty)$, which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint $p=1$, the $L\log L$ weighted weak type bound is also obtained, which has interests of its own in the theory of rough…
In this paper, we prove some BMO end-point estimates for some vector-valued multilinear operators related to certain singular integral operators.
In this paper, the authors define the weak Herz spaces and the weak Herz-type Hardy spaces with variable exponent. As applications, the authors establish the boundedness for a large class of singular integral operators including some critical cases.
In this paper, we will study the boundedness properties of multilinear Calderon--Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.
We obtain a weak type $(1,1)$ estimate for a maximal operator associated with the classical rough homogeneous singular integrals $T_Ω$. In particular, this provides a different approach to a sparse domination for $T_Ω$ obtained recently by Conde-Alonso, Culiuc, Di Plinio and Ou.
We prove sharp weighted estimates for $r$-variations of averages and truncated singular integrals on suitable spaces of homogeneous type, including homogeneous nilpotent Lie groups.
We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.
We prove that the classical one-parameter convolution singular integrals on the Heisenberg group are bounded on multiparameter flag Hardy spaces, which satisfy `intermediate' dilation between the one-parameter anisotropic dilation and the product dilation on $\mathbb{C}^{n}\times \mathbb{R}$ implicitly.
We prove optimal bounds in L^2(R^2) for the maximal oper- ator obtained by taking a singular integral along N arbitrary directions in the plane. We also give a new proof for the optimal L^2 bound for the single scale Kakeya maximal function.
We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $Δ^*Ψ$ where $Δ^*$ is Bourgain's maximal multiplier operator and $Ψ$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 < q < 2$. We also consider associated variation-norm estimates.
Singular Integrals associated with Reflection Groups on Euclidean Space
本研究针对调和分析领域的奇异积分问题，将其研究范围拓展至纳入欧氏空间上的群反射效应，并建立了对应奇异积分的T1定理，丰富了奇异积分与群作用结合的研究内容。
We show that the $L^2(w)$ operator norm of the composition $M\!\circ T_Ω$, where $M$ is the maximal operator and $T_Ω$ is a rough homogeneous singular integral with angular part $Ω\in L^{\infty}(S^{n-1})$, depends quadratically on $[w]_{A_2}$, and this dependence is sharp.
We prove a non-homogeneous T1 theorem for certain bi-parameter singular integral operators. Moreover, we discuss the related non-homogeneous Journe's lemma and product BMO theory.
We prove a dyadic representation theorem for bi-parameter singular integrals. That is, we represent certain bi-parameter operators as rapidly decaying averages of what we call bi-parameter shifts. A new version of the product space T1 theorem is established as a consequence.
Compactness of bilinear singular integral with mild kernel regularity
本文将Cao等人在2024年建立的紧致性特征扩展到具有温和核正则性的双线性奇异积分算子，所得指数与经典双线性T1定理的最佳已知充分条件一致，并引入了新的弱紧致性性质条件。
In this paper we characterize BMO in terms of the boundedness of commutators of various bilinear singular integral operators with pointwise multiplication. In particular, we study commutators of a wide class of bilinear operators of convolution type, including bilinear Calderón-Zygmund operators and the bilinear fractional integral operators.
Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting
本文通过离散Littlewood-Paley理论建立Zygmund伸缩相关奇异积分的加权估计，得到其与小bmo空间函数交换子范数上界，还证明相关奇异积分族不包含Nagel-Wainger算子且该算子具理论意义。
In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough kernel in harmonic analysis, such as Calderón commutator, higher order Calderón commutator, general Calderón commutator, Calderón commutator of Bajsanski-Coifman type and general singular integral of Muckenhoupt type, are all of weak type (1,1).
We present a framework based on modified dyadic shifts to prove multiple results of modern singular integral theory under mild kernel regularity. Using new optimized representation theorems we first revisit a result of Figiel concerning the UMD-extensions of linear Calderón-Zygmund operators with mild kernel regularity and extend our new proof to the multilinear setting improving recent UMD-valued estimates of multilinear singular integrals. Next, we develop the product space theory of the multilinear singular integrals with modified Dini-type assumptions, and use this theory to prove bi-param…
Weighted weak type $(1,1)$ estimates for oscillatory singular integrals with Dini kernels
本文研究了A1权重下的振荡奇异性积分，证明了满足Dini条件的核的加权弱类型(1,1)估计。
In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^{r}$-Hörmander condition, then $T$ can be dominated by multilinear sparse operators.
In a previous paper, we introduced a new class of Gaussian singular integrals, that we called the general alternative Gaussian singular integrals and study the boundedness of them on $L^p(γ_d)$, $ 1 < p < \infty.$ In this paper, we study the boundedness of those operators on Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot)$ following a paper by E. Dalmasso and R. Scotto.
A homage to Guido Weiss and his leadership of the Saint Louis team: Commutators of Singular Integrals and Sobolev inequalities
本文扩展了线性与非线性交换子的经典Sobolev型不等式。
A weak type $(1,1)$ estimate is established for the first order $d$-commutator introduced by Christ and Journé, in dimension $d\ge 2$.
Let $L=-Δ+V$ with non-negative potential $V$ satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to $L$ such as Riesz transforms and fractional integrals with the new BMO functions introduced in \cite{BHS1} on the weighted spaces $L^p(w)$ where $w$ belongs to the new classes of weights introduced by \cite{BHS2}.
Let $r>\frac{4}{3}$ and let $Ω\in L^{r}(\mathbb{S}^{2n-1})$ have vanishing integral. We show that the bilinear rough singular integral $$T_Ω(f,g)(x)= \textrm{p.v.} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{Ω((y,z)/|(y,z)|)}{|(y,z)|^{2n}}f(x-y)g(x-z)\,dydz,$$ satisfies a sparse bound by $(p,p,p)$-averages, where $p$ is bigger than a certain number explicitly related to $r$ and $n$. As a consequence we deduce certain quantitative weighted estimates for bilinear homogeneous singular integrals associated with rough homogeneous kernels.
We prove sufficient and necessary conditions for compactness of Calderón-Zygmund operators on the endpoint from $L^{\infty }(\mathbb R)$ into ${\rm CMO}(\mathbb R)$. We use this result to prove compactness on $L^{p}(\mathbb R)$ with $1<p<\infty $ of certain perturbations of the Cauchy integral on curves with normal derivatives satisfying a ${\rm CMO}$-condition.
We prove that in any metric space $(X,d)$ the singular integral operators {equation*} T^k_{μ,\ve}(f)(x)=\int_{X\setminus B(x,\varepsilon)}k(x,y)f(y)dμ(y).{equation*} converge weakly in some dense subspaces of $L^2(μ)$ under minimal regularity assumptions for the measures and the kernels.
In this paper, we consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calderón-Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $μ$. For any given homogeneous polynomial phases, except monomial phases, of degreee $n$, we give the range of $p$ for which the sharp decay rate $-\frac{1-μ}{n}$ on $L^2$ spaces can be preserved on $L^p$ spaces.
This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-Hörmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such…
We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation. Extrapolation gives powerful applications -- for example, a free access to mixed-norm estimates in the full range of exponents.
This paper studies dyadic singular integral forms associated with $r$-partite $r$-uniform hypergraphs such that all their connected components are complete. We characterize their $L^p$ boundedness by T(1)-type conditions in two different ways. We also dominate these forms by positive sparse forms and prove weighted estimates with multilinear Muckenhoupt weights.
We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the continuous dependence of the solution with respect to the parameter and the corresponding explicit solution to the considered problem.
We prove the full range of estimates for a five-linear singular integral of Brascamp-Lieb type. The study is methodology-oriented with the goal to develop a sufficiently general technique to estimate singular integral variants of Brascamp-Lieb inequalities that do not obey Hölder scaling. The invented methodology constructs localized analysis on the entire space from local information on its subspaces of lower dimensions and combines such tensor-type arguments with the generic localized analysis. A direct consequence of the boundedness of the five-linear singular integral is a Leibniz rule whi…
An Approach To Endpoint Problems in Oscillatory Singular Integrals
本文用初等鸽笼原理和驻相法，证明了振荡奇异积分极大截断从 L^1 到弱 L^1 的有界性。
In this paper it is shown that for $Ω\in L\log L(\mathbb{S}^{d-1})$, the rough maximal singular integral operator $T_Ω^*$ is of weak type $L\log\log L(\mathbb{R}^d)$. Furthermore, for $w\in A_1$ and $Ω\in L^\infty(\mathbb{S}^{d-1})$, it is shown that $T_Ω^*$ is of weak type $L\log\log L(w)$ with weight dependence $[w]_{A_1}[w]_{A_{\infty}}\log([w]_{A_{\infty}}+1),$ which is same as the best known constant for the singular integral $T_Ω$.
Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension
本文研究了Cantor集上奇异积分主值的存在性，证明了在测度线性密度消失时主值存在点集的Hausdorff维数等于集合维数，并推广至高维Riesz奇异积分，同时引入鞅辅助证明。
In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of $L^2 \mapsto L^2$ type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in $\mathbb{R}^3$.
The complete boundedness of singular integrals on weighted flag and product Hardy space
本文旨在证明奇异积分在加权flag和乘积Hardy空间上的完全有界性，包括单参数奇异积分和flag奇异积分的有界性。
In this paper, the weighted Lp boundedness of multilinear commutators and iterated commutators of multilinear singular integral operators with generalized kernels is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels.
The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals above cannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $w\in A_{p}^{+}$, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted $L^{p}$-boundednesss of the oscillatory singular integral is given.
We prove weighted $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators in higher dimensions. The vector-valued extensions of these inequalities are also given.
Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces
该研究证明奇异积分的精确加权BMO估计，且给出将这类估计外推到Banach函数空间的方法，丰富了调和分析相关理论。
In this paper the jump formulas for the double layer potential and other singular integrals are proved for arbitrary rectifiable sets, by defining suitable non-tangential limits. The arguments are quite straightforward and only require some Calderón-Zygmund techniques.
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