Zygmund Kernel Research Articles

Exotic Calderón–Zygmund Operators

We study singular integral operators with kernels that are more singular than standard Calderón–Zygmund kernels, but less singular than bi-parameter product Calderón–Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmund dilations, which is part of our motivation for studying these objects. We make the case that such kernels can, in many ways, be seen as part of the extended realm of standard kernels by proving that they satisfy both a T1 theorem and commutator estimates in a form reminiscent of the corresponding results for standard Calderón–Zygmund kernels. However, we show that one-parameter weighted estimates, in general, fail.

Oscillation Estimates for Truncated Singular Radon Operators

In this paper we prove uniform oscillation estimates on L^p, with pin (1,infty ), for truncated singular integrals of the Radon type associated with the Calderón–Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu–Wainger multiplier theorem and the Rademacher–Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calderón–Zygmund type.

Singular integrals on $$C_{w^*}^{1,\alpha }$$ regular curves in Banach duals

The modern study of singular integral operators on curves in the plane began in the 1970s. Since then, there has been a vast array of work done on the boundedness of singular integral operators defined on lower dimensional sets in Euclidean spaces. In recent years, mathematicians have attempted to push these results into a more general metric setting particularly in the case of singular integral operators defined on curves and graphs in Carnot groups. Suppose $$X = Y^*$$ for a separable Banach space Y. Any separable metric space can be isometrically embedded in such a Banach space via the Kuratowski embedding. The $$w^*$$ -derivative $$\gamma '$$ of a curve $$\gamma :[a,b] \rightarrow X$$ at $$t \in [a,b]$$ satisfies $$\tfrac{\mathrm{d}}{\mathrm{d}s} \langle \gamma (s),y \rangle |_{s=t} = \langle \gamma '(t),y\rangle$$ for any $$y \in Y$$ . Suppose $$\Gamma = \gamma ([a,b])$$ is a curve in X whose $$w^*$$ -derivative is Hölder continuous and bounded away from 0. We prove that any convolution type singular integral operator associated with a 1-dimensional Calderón–Zygmund kernel which is uniformly $$L^2$$ -bounded on lines is $$L^p$$ -bounded along $$\Gamma$$ . We also prove a version of David’s “good lambda” theorem for upper regular measures on doubling metric spaces.

Symmetrization of a Family of Cauchy-Like Kernels: Global Instability

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its \(L^2\) boundedness. In particular, a famous proof of Melnikov-Verdera (A geometric proof of the \(L^2\) boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7:325–331, 1995) relies upon an iconic symmetrization identity of Melnikov (Analytic capacity: a discrete approach and the curvature of measures. Mat Sb 186(6):57–76, 1995) linking the universal Cauchy kernel \(K_0\) to Menger curvature. Analogous identities hold for the real and the imaginary parts of \(K_0\) as well. Such connections have been immensely productive in the study of singular integral operators and in geometric measure theory. In this article, given any function \(h: {\mathbb {C}} \rightarrow {\mathbb {R}}\), we consider an inhomogeneous variant \(K_h\) of \(K_0\) which is inspired by complex function theory. While an operator with integration kernel \(K_h\) is easily seen to be \(L^2\)-bounded for all h, the symmetrization identities for each of the real and imaginary parts of \(K_h\) show a striking lack of robustness in terms of boundedness and positivity, two properties that were critical in Melnikov and Verdera (A geometric proof of the \(L^2\) boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7:325–331, 1995) and in subsequent works by many authors. Indeed here we show that for any continuous h on \({\mathbb {C}}\), the only member of \(\{K_h\}_h\) whose symmetrization has the right properties is \(K_0\)! This global instability complements our previous investigation (Lanzani and Pramanik, Symmetrization of a Cauchy-like kernel on curves. J Funct Anal (Preprint). arXiv:2001.09375) of symmetrization identities in the restricted setting of a curve, where a sub-family of \(\{K_h\}_h\) displays very different behaviour than its global counterparts considered here. Our methods of proof have some overlap with techniques in recent work of Chousionis and Prat, Some Calderòn–Zygmund kernels and their relation to Wolff capacity. Math Z 282:435–460, 2016) and Chunaev, A new family of singular integral operators whose \(L^2\) boundedness implies rectifiability. J Geom Anal 27:2725–2757, 2017).

A metric approach to sparse domination

We present a general approach to sparse domination based on single-scale \(L^p\)-improving as a key assumption. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ–Hytönen–Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderón–Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of \({\mathbb {R}}^n\).

Discrete analogues of maximally modulated singular integrals of Stein–Wainger type

Consider the maximal operator \mathscr{C} f(x) = \sup_{\lambda\in\mathbb{R}}\,\Bigl|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}} f(x-y) e(\lambda |y|^{2d}) K(y)\Bigr|\quad\ (x\in\mathbb{Z}^n), where d is a positive integer, K a Calderón–Zygmund kernel and n\ge 1 . This is a discrete analogue of a real-variable operator studied by Stein and Wainger. The nonlinearity of the phase introduces a variety of new difficulties that are not present in the real-variable setting. We prove \ell^2(\mathbb{Z}^n) -bounds for \mathscr{C} , answering a question posed by Lillian Pierce.

On Approximate Solution of Certain Equations

In this paper, we consider problems of discrete approximation of special integral operators with the Calderon–Zygmund kernel. We introduce discrete spaces and bounded discrete operators acting in these spaces; then we use these operators for the search for approximate solutions of the corresponding equations. We state theorems on the solvability of equations with discrete operators, compare integral operators with their discrete analogs, and obtain estimates of errors of approximate solutions.

Maximal polynomial modulations of singular integrals

Let K be a standard Hölder continuous Calderón–Zygmund kernel on Rd whose truncations define L2 bounded operators. We show that the maximal operator obtained by modulating K by polynomial phases of a fixed degree is bounded on Lp(Rd) for 1<p<∞. This extends Sjölin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.

Weighted boundedness for Toeplitz type operator associated to singular integral operator with variable Calderón-Zygmund kernel

In this paper, we establish the weighted sharp maximal function inequalities for the Toeplitz type operator associated to the singular integral operator with variable Calderón- Zygmund kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.

Oscillatory Singular Integral Operators with Hölder Class Kernels

We establish the boundedness on $$L^p({\mathbb {R}}^n)$$ of oscillatory singular integral operators whose kernels are the products of an oscillatory factor with bilinear phase and a Calderon–Zygmund kernel K(x, y) satisfying a Holder condition. Our results also hold on weighted $$L^p$$ spaces with $$A_p$$ weights.

The Traveling Salesman Theorem in Carnot groups

Let $${\mathbb {G}}$$ be any Carnot group. We prove that, if a subset of $${\mathbb {G}}$$ is contained in a rectifiable curve, then it satisfies Peter Jones’ geometric lemma with some natural modifications. We thus prove one direction of the Traveling Salesman Theorem in $${\mathbb {G}}$$ . Our proof depends on new Alexandrov-type curvature inequalities for the Hebisch–Sikora metrics. We also apply the geometric lemma to prove that, in every Carnot group, there exist $$-\,1$$ -homogeneous Calderon–Zygmund kernels such that, if a set $$E \subset {\mathbb {G}}$$ is contained in a 1-regular curve, then the corresponding singular integral operators are bounded in $$L^2(E)$$ . In contrast to the Euclidean setting, these kernels are nonnegative and symmetric.

Boundedness of Toeplitz operators related to singular integral operators

We establish that Toeplitz-type operators related to singular integral operators with variable Calderón– Zygmund kernels are bounded on weighted Morrey spaces. To do this, we prove weighted inequalities for the sharp maximal functions of such operators.

Multilinear Commutators of Singular Integral Operators in Variable Exponent Herz-Type Spaces

In this paper, we study the boundedness of multilinear commutators of Hardy–Littlewood maximal operators in variable exponent Herz and Herz–Morrey spaces, which in turn are used to obtain the boundedness for a large class of the multilinear commutators related to sublinear operators. Moreover, based on the atomic decomposition and on generalization of the BMO norm, we study the boundedness of multilinear commutators of singular integral operators with Calderon–Zygmund kernels in variable exponent Herz-type Hardy spaces.

Sobolev spaces associated to singular and fractional Radon transforms

The purpose of this paper is to study the smoothing properties (in L^p Sobolev spaces) of operators of the form f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt , where \gamma_t(x) is a C^\infty function defined on a neighborhood of the origin in (t,x)\in \mathbb{R}^N\times \mathbb{R}^n , satisfying \gamma_0(x)\equiv x , \psi is a C^\infty cut-off function supported on a small neighborhood of 0\in \mathbb{R}^n , and K is a "multi-parameter fractional kernel" supported on a small neighborhood of 0\in \mathbb{R}^N . When K is a Calderón–Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when K is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on \gamma were given under which the above operator is bounded on L^p ( 1 &lt; p &lt; \infty ). Under these same conditions, we introduce non-isotropic L^p Sobolev spaces associated to \gamma . Furthermore, when K is a fractional kernel which is smoothing of an order which is close to 0 (i.e., very close to a singular kernel) we prove mapping properties of the above operators on these non-isotropic Sobolev spaces. As a corollary, under the conditions introduced on \gamma by Christ, Nagel, Stein, and Wainger, we prove optimal smoothing properties in isotropic L^p Sobolev spaces for the above operator when K is a fractional kernel which is smoothing of very low order.

On finite discrete operators and equations

AbstractFor discrete operators generated by Calderon–Zygmund kernels we consider certain finite‐dimensional approximations. We compare an initial discrete operator with its finite‐dimensional analogue and obtain invertibility conditions and approximation rate. The error estimate for approximate finite‐dimensional solution of corresponding equation is given also for a special right‐hand side. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Weighted boundedness of multilinear operator associated to singular integral operator with variable Calderón–Zygmund Kernel

In this paper, we establish the weighted sharp maximal function inequalities for the multilinear operator associated to the singular integral operator with variable Calderon–Zygmund kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.

Some Calderón–Zygmund kernels and their relations to Wolff capacities and rectifiability

We consider the Calderon–Zygmund kernels \(K_ {\alpha ,n}(x)=(x_i^{2n-1}/|x|^{2n-1+\alpha })_{i=1}^d\) in \({\mathbb R}^d\) for \(0<\alpha \le 1\) and \(n\in \mathbb {N}\). We show that, on the plane, for \(0<\alpha <1\), the capacity associated to the kernels \(K_{\alpha ,n}\) is comparable to the Riesz capacity \(C_{\frac{2}{3}(2-\alpha ),\frac{3}{2}}\) of non-linear potential theory. As consequences we deduce the semiadditivity and bilipschitz invariance of this capacity. Furthermore we show that for any Borel set \(E\subset {\mathbb R}^d\) with finite length the \(L^2(\mathcal {H}^1 \lfloor E)\)-boundedness of the singular integral associated to \(K_{1,n}\) implies the rectifiability of the set E. We thus extend to any ambient dimension, results previously known only in the plane.

On the solvability of certain discrete equations and related estimates of discrete operators

The solvability of discrete equations with Calderon–Zygmund kernels is studied, the limit passage from the discrete to the continuous case is substantiated, and estimates of the errors of discrete solutions are obtained.

On multilinear square function and its applications to multilinear Littlewood–Paley operators with non-convolution type kernels

Let m≥2, n≥1 and x∈Rn, define the multilinear square function T by T(f→)(x)=(∫0∞|∫(Rn)mKt(x,y1,…,ym)∏j=1mfj(yj)dy1⋯dym|2dtt)1/2, where the kernel K satisfies a class of integral smooth conditions which is much weaker than the standard Calderón–Zygmund kernel conditions. In this paper, we first established the Lp1(w1)×⋯×Lpm(wm)→Lp(νω→) estimate of T when each pi>1 and weak type Lp1(w1)×⋯×Lpm(wm)→Lp,∞(νω→) estimate of T when there is a pi=1, where νω→=∏i=1mωip/pi and each wi is a nonnegative function on Rn. As applications of the above results, we obtained the boundedness of multilinear Littlewood–Paley operators with non-convolution type kernels, including multilinear g-function, Marcinkiewicz integral and gλ⁎-function.

Mean oscillation and boundedness of Toeplitz-type operator related to singular integral operator with a variable Calderón–Zygmund kernel

In this paper, the boundedness from Lebesgue space to Orlicz space of certain Toeplitz-type operator related to the singular integral operator with a variable Calderón–Zygmund kernel is obtained.