Compositions of singular integral operators

The composition of two Calderón-Zygmund singular integral operators is given explicitly in terms of the kernels of the operators. For φ?L¹(Rⁿ) and ε = 0 or 1 and ∝ φ = 0 if ε = 0, let Ker(φ) be the unique function on R^{n + 1} homogeneous of degree ?n ? 1 of parity ε that equals φ on the hypersurface x₀ = 1. Let Sing(φ, ε) denote the singular integral operator , which exists under suitable growth conditions on ? and φ. Then Sing(φ, ε₁) Sing(ψ, ε₂)f = ?2π²(∝ φ)(∝ ψ)f + Sing(A, ε₁, + ε₂)f, where

(with notation $\text{¦t¦}{}_0{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{and}\text{¦t¦}{}_1{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{sgn}\text{t}$). This result is used to show that the mapping ψ → A is a classical pseudo-differential operator of order zero if φ is smooth, with top-order symbol

$\text{ω}{}_0\text{(x,?)=?πiθ(?)∫θ(x?y)}\text{sgn}\text{y·?dy}\text{if}\text{?}{}_1\text{=1}$

,

$\text{=?2θ(?)∫θ(x?y)}\text{log}\text{|y·?|dy}\text{if}\text{?}{}_1\text{=0}$

where θ(ξ) is a cut-off function. These results are generalized to singular integrals with mixed homogeneity.     

Certain singular integral operators revisited

This note presents an alternate approach to the analysis of the composition of some (classical) singular integral oprators that arise in a number of applications. It is based on the facts that certain naturally paired operators have the same range and are injective. In the course of this analysis the proofs of some classical identities are unified     

Nonisotropic strongly singular integral operators

We consider a class of strongly singular integral operators which include those studied by Wainger, and Fefferman and Stein, and extend the results concerning the boundedness of these operators to the nonisotropic setting. We also describe a geometric property of the underlying space which helps us show that our results are sharp.

     

Maximal operators and singular integral operators along submanifolds

In this paper we prove, for certain values of p, the L^p boundedness of the maximal operator

     

A note on singular integral operators

In this paper, we study the L^p boundedness for the singular integral operators of R. Fefferman when the kernel satisfies certain size condition. We also consider the corresponding maximal singular integral operators.     

Two-weight estimates for singular and strongly singular integral operators

We give a Pontryagin-Thom type construction for Stein factorizations of fold maps of 3-manifolds into the plane. As an application, we compute the cobordism group of Stein factorizations of fold maps of oriented 3-manifolds into the plane and the oriented cobordism group of fold maps of 3-manifolds into the plane. It turns out that these two groups are isomorphic to Z ₂ ⊕ Z ₂. We have the analogous results about bordism groups as well.      

Essential norms of some singular integral operators

Let a\alpha and b\beta be bounded measurable functions on the unit circle T. The singular integral operator S_a, bS_{\alpha ,\,\beta } is defined by S_a, b f = aPf + bQf(f ? L² (T))S_{\alpha ,\,\beta } f = \alpha Pf + \beta Qf(f \in L^2 (T)) where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of S_a, bS_{\alpha ,\,\beta } was calculated in general, using a,b\alpha ,\beta and a[`(b)] + H<sup>￥</sup>\alpha \bar {\beta } + H^\infty where H<sup>￥</sup>H^\infty is a Hardy space in L<sup>￥</sup> (T).L^\infty (T). In this paper, the essential norm ||S<sub>a, b</sub> ||<sub>e</sub>\Vert S_{\alpha ,\,\beta } \Vert _e of S<sub>a, b</sub>S_{\alpha ,\,\beta } is calculated in general, using a[`(b)] + H^￥ + C\alpha \bar {\beta } + H^\infty + C where C is a set of all continuous functions on T. Hence if a[`(b)]\alpha \bar {\beta } is in H^￥ + CH^\infty + C then ||S_a, b ||_e = max(||a||_￥ , ||b||_￥ ).\Vert S_{\alpha ,\,\beta } \Vert _e = \max (\Vert \alpha \Vert _\infty , \Vert \beta \Vert _\infty ). This gives a known result when a, b\alpha , \beta are in C.     

Asymptotic Moore-Penrose invertibility of singular integral operators

The topic of the paper is the asymptotic Moore-Penrose inversion of singular integral operators by means of the trigonometric collocation and the finite section method. Two versions (a weak and a strong one) of these methods are considered. Necessary and sufficient conditions for the applicability of the weak method, and sufficient conditions for the strong method, are derived. These conditions involve a certain asymptotic resp. exact kernel structure of the approximation matrices.Partially supported by the HCM project ROLLS under contract CHRX-CT 93-0416 (both authors) and by a DFG Heisenberg grant (S. Roch)     

Composition of variable co-efficient singular integral operators

In this paper we give a formula for composition of two singular integral operators with variable co-efficients by explicitly calculating the lower order terms. Also we discuss the boundedness of the lower order terms inL ^p-spaces.     

Continuity and compactness of singular integral operators

In the space of square integrable functions we establish effective sufficient continuity and compactness conditions for singular integral operators with Cauchy kernels on a segment of the real axis.     

Rough singular integral operators on Hardy-Sobolev spaces

The authors study the singular integral operator TΩ,αf(x)=p.v.∫R^nb(|y|Ω(y′)|y|^-n-αf(x-y)dy, defined on all test functions f, where b is a bounded function, α＞0, Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions. It is proved that, for n/(n α)＜p＜∞,TΩ,α is a bounded operator from the Hardy-Sobolev space H^pα to the Hardy space H^p. The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden‘s paper(1969). The proof is based on a new atomic decomposition of the space H^pα by Han, Paluszynski and Weiss(1995). By using the same proof,the singluar integral operators with variable kernels are also studied.     

Vector-valued singular integral operators with rough kernels

In this paper, we establish a weak-type (1,1) boundedness criterion for vector-valued singular integral operators with rough kernels. As applications, we obtain weak-type (1,1) bounds for the convolution singular integral operator taking value in the Banach space Y with a rough kernel, the maximal operator taking vector value in Y with a rough kernel and several square functions with rough kernels. Here, $Y={[H,X]}_{\theta }$ is a complex interpolation space between a Hilbert space H and a UMD space X.     

Estimates for oscillatory strongly singular integral operators

In this paper we consider oscillatory integral operators with strong singularities on . We obtain sharp decay estimates for L² operator norm. We also obtain H^p estimates for difference of oscillatory strongly singular integral operators and strongly singular integral operators, which gives L^p mapping properties of oscillatory strongly singular integral operators.     

Exponential decay estimates for singular integral operators

The following subexponential estimate for commutators is proved $$\begin{aligned} |\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\le c\,e^{-\sqrt{\alpha \, t\Vert b\Vert _{BMO}}}\, |Q|, \qquad t>0. \end{aligned}$$ where $c$ and $\alpha $ are absolute constants, $T$ is a Calderón–Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q\subset \mathbb{R }^n$ . We also obtain that $$\begin{aligned} |\{x\in Q: |f(x)-m_f(Q)|>tM_{\lambda _n;Q}^\#(f)(x) \}|\le c\, e^{-\alpha \,t}|Q|,\qquad t>0, \end{aligned}$$ where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{\lambda _n;Q}^\#$ is Strömberg’s local sharp maximal function with $\lambda _n=2^{-n-2}$ . As a consequence we derive Karagulyan’s estimate: $$\begin{aligned} |\{x\in Q: |Tf(x)|> tMf(x)\}|\le c\, e^{-c\, t}\,|Q|\qquad t>0, \end{aligned}$$ from [21] improving Buckley’s theorem [3]. A completely different approach is used based on a combination of “Lerner’s formula” with some special weighted estimates of Coifman–Fefferman type obtained via Rubio de Francia’s algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón–Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón–Zygmund operators. In each case, $M$ will be replaced by a suitable maximal operator.     

Rank one perturbations and singular integral operators

We consider rank one perturbations Aα=A+α(⋅,φ)φ of a self-adjoint operator A with cyclic vector φ∈H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aα is given by a singular integral operator of special form. Such operators exhibit what we call ‘rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms Tε are uniformly (in ε) bounded operators from L²(μ) to L²(μα), where μ and μα are the spectral measures of A and Aα, respectively. As an application, a sufficient condition for Aα to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of A with respect to φ. Some examples, like Jacobi matrices and Schrödinger operators with L² potentials are considered.     

Some remarks on weakly singular integral operators

The eigenvalues _n of weakly singular integral operators, the order of singularity of the kernel of which is one half of the dimension of the domain, are shown to be of order O((1n(n+1)/n^1/2). There are convolution operators which demonstrate that this order cannot be improved in general.Research supported by the SFB 72 at the University of Bonn