Tuomas Hytönen, Henri Martikainen
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.
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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 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.
Let $(X,d,μ)$ be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling measure conditions. In this paper, the boundedness of multilinear fractional integral operator in this setting is proved. Via a sharp maximal operator, the boundedness of commutators generated by multilinear fractional integral operator with $RBMO(μ)$ function on non-homogeneous metric measure spaces in Lebesgue spaces is obtained.
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.
We obtain sharp $L^p$ bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition which is an important notion introduced by Greenleaf, Pramanik and Tang. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint $L^p$ estimates.
Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces
本文研究了极大算子在三维欧几里得空间中的有界性问题，其中超曲面为$c+f$的图，其中$f$是离原点光滑的混合齐次函数，$c$是常数.我们的结果推广了A. Iosevich和E.索伊尔
Bounds for singular fractional integrals and related Fourier integral operators
在非零旋转曲率的假设下，证明了奇异分数次积分算子及其相关的傅里叶积分算子的L^p-L^q端点有界.
We consider a generalization of the notion of spaces of homogeneous type, inspired by recent work of Street [21] on the multi-parameter Carnot-Caratheodory geometry, which imbues such spaces with differentiability structure. The setting allows one to formulate estimates for scalar oscillatory integrals on these spaces which are uniform and respect the underlying geometry of both the space and the phase function. As a corollary we obtain a generalization of a theorem of Bruna, Nagel, and Wainger [1] on the asymptotic behavior of scalar oscillatory integrals with smooth, convex phase of finite t…
Calderón-Zygmund decompositions of functions have been used to prove weak-type (1,1) boundedness of singular integral operators. In many examples, the decomposition is done with respect to a family of balls that corresponds to some family of dilations. We study singular integral operators $T$ that require more particular families of balls, providing new spaces of homogeneous type. Rotations play a decisive role in the construction of these balls. Boundedness of $T$ can then be shown via Calderón-Zygmund decompositions with respect to this space of homogeneous type. We prove weak-type (1,1) and…
Estimates for singular integrals along surfaces of revolution
我们证明了某些$L^p$估计（$1<p<\infty$）的非各向同性奇异积分沿表面的革命。作为应用，我们得到了奇异积分在核的一个尖锐尺寸条件下的$L^p$有界性.
We consider a version of M. Riesz fractional integral operator on a space of homogeneous type and show an analogue of the well-known Hardy--Littlewood--Sobolev theorem in this context. In our main result, we investigate the dependence of the operator norm on weighted spaces on the weight constant, and find the relationship between these two quantities. It it shown that the estimate obtained is sharp in any given space of homogeneous type with infinitely many points. Our result generalizes the recent Euclidean result by Lacey, Moen, P{é}rez and Torres.
For any $0<α<n$, the homogeneous fractional integral operator $T_{Ω,α}$ is defined by \begin{equation*} T_{Ω,α}f(x)=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy. \end{equation*} In this paper, we prove that if $Ω$ satisfies certain Dini smoothness conditions on $\mathbf{S}^{n-1}$, then $T_{Ω,α}$ is bounded from $L^{p,κ}(w^p,w^q)$ (weighted Morrey space) to $\mathrm{BMO}(\mathbb R^n)$.
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 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).
Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_Ω$ and commutators of $BMO$ symbols and $T_Ω$ are obtained. In particular the following estimates are proved: % \[ \|T_Ω\|_{L^p(w)}\le c_{n,p}\|Ω\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)} \] % and % \[ \| [b,T_Ω]f\|_{L^{p}(w)}\leq c_{n,p}\|b\|_{BMO}\|Ω\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)}, \] % for $1<p<\infty$ and $1/p+1/p'=1$.
Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.
Estimates for singular integrals and extrapolation
根据奇异Radon变换的核的大小条件，证明了奇异Radon变换的一个精确Lp估计，这对于外推是有用的。
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.
Commutators of singular integrals on generalized $L^p$ spaces with variable exponent
将Coifman，Rochberg和Weiss关于奇异积分的积分算子的一个经典定理推广到变指数广义$L^p$空间的情形.
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 describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with fe…
Multilinear singular integrals with homogeneous kernels near $L^1$
本文研究了具有近 $L^1$ 齐次核的多线性奇异积分算子，在核函数 $Ω$ 属于 $L^q$ 且积分消失的条件下，得到了其从乘积 Lebesgue 空间到 $L^p$ 空间有界性的最优开区间。
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.
We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound that is quadratic in the $A_2$ constant $[w]_{A_2}$. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on…
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…
Some BMO estimates for vector-valued multilinear singular integral operators
本文证明了与某些奇异积分算子相关的向量值多重线性算子的BMO端点估计。
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 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.
Oscillatory integral operators with homogeneous polynomial phases in several variables
对于相函数为m次齐次多项式且满足各种一般性假设的振荡积分算子，我们得到了$λ$中的$L^2$衰减估计.得到的衰减率是最佳的情况下（2 + 2）-维的任何m，而在更高的维度，结果是尖锐的m足够大。大$m$的证明主要是从代数上考虑的。对于（2 + 2）维的三次方程，证明涉及到使用Phong和Stein [1994]的一般方法的详细说明，在相位函数的Hessian行列式的圆锥零点簇附近分解算子。
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.
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 article we study a homogeneous transient diffusion process $X$. We combine the theories of differential equations and of stochastic processes to obtain new results for homogeneous diffusion processes, generalizing the results of Salminen and Yor. The distribution of local time of $X$ is found in a closed form. To this end, a second order differential equation corresponding to the generator of $X$ is considered, and properties of its monotone solutions as functions of a parameter are established using their probabilistic representations. We also provide expressions and upper bounds for …
Let $({\mathcal X},d,μ)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional integrals over $({\mathcal X},d,μ)$. The authors also prove that multilinear commutators of fractional integrals with ${\mathop\mathrm{\,RBMO(μ)}}$ functions are bounded on Orlicz spaces over $({\mathcal X},d,μ)$, which include Lebesgue spaces as special cases. The weak type endpoint estimates for multilinear commutators of fractional integrals with functions in t…
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.
We consider singular integrals associated to homogeneous kernels on self similar sets. Using ideas from ergodic theory we prove, among other things, that in Euclidean spaces the principal values of singular integrals associated to real analytic, homogeneous kernels fail to exist a.e. on self similar sets satisfying some separation conditions. Furthermore in general metric groups, using similar techniques, we generalize a criterion of $L^2$-unboundedness for singular integrals on self similar sets.
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.
Algebras of singular integral operators on Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points
在具有对数涡点的Carleson曲线上的Nakano空间（变指数的广义Lebesgue空间）上，建立了具有Khvedelidze权的分段连续系数的奇异积分算子的Banach代数中任意算子的Fredholm判别准则.
Boundedness on inhomogeneous Lipschitz spaces of fractional integrals, singular integrals and hypersingular integrals associated to non-doubling measures
在测度满足增长条件的有限测度度量空间中，证明了非齐次Lipschitz空间上分数次积分、奇异积分和超奇异积分有界的“T1”型充要条件.我们还指出了如何将结果推广到无限测度的情形。最后，我们展示了实分析和复分析的应用。
In this note, we generalize the Fresnel integrals using oscillatory integral, and then we obtain an extention of the stationary phase method.
In this note, by using the result in one variable, we obtain asymptotic expansions of oscillatory integrals for certain multivariable phase functions with {\bf degenerate} singular points. Moreover by using this result, we have asymptotic expansions of oscillatory integrals with phase function of type $A_{k}$, $E_6$, $E_8$-function germs.
Landau singularities and singularities of holonomic integrals of the Ising class
我们认为家庭的多重和简单的积分的"伊辛类“和线性常微分方程的多项式系数，他们的解决方案。我们比较完整的一组奇点给出的根的头多项式的这些线性常微分方程的和子集的奇点发生在积分，从朗道条件得到的奇点。对于这些伊辛类积分，我们表明，朗道条件可以制定出，要么给相应的线性微分方程的奇异性或奇异性发生在积分。这些积分的奇异行为是在自对偶变量$w= s/2/(1+s^2)$中得到的，其中$s= \sinh(2K)$，其中$K=J/kT$是通常的伊辛模型耦合常数。切换到变量$s$，我们表明，这些积分的级数展开的解析延拓的奇异性实际上打破了Kramers-Wannier对偶。我们重新审视（J.Phys.A { \bf 38}（2005）9439-9474）中对Ising模型$χ^{(3)}$在点$1+3w+4w^2= 0$处的磁化率的第三贡献，并且示出了$χ^{(3)}(s)$在单位圆$| s |=1$内的对应点处不是奇异的，而其在变量$s$中的解析延拓实际上在单位圆（$| s | > 1$）外部的对应点$ 2+s+s^2=0$处是奇异的。
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