Algebra of singular integral operators with a Carleman backward slowly oscillating shift

In this paper we study the Banach algebra of singular integral operators with piecewise continuous coefficients and a Carleman orientation-reversing slowly oscillating shift on the Lebesgue space with a power weight on the unit circle. The slow oscillation of the shift derivative, in contrast to the classic assumption on its piecewise continuity, leads to the appearance of massive local spectra for the considered operators. Applying localization techniques and the theory of Mellin pseudodifferential and associated limit operators, we construct a symbol calculus for the above-mentioned operator algebra and find a Fredholm criterion and an index formula for the operators in this algebra in terms of their symbols.Partially supported by CONaCYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.Partially supported by F. C. T. grant Praxis XXI/2/2.1/MAT/441/94, Portugal.     

Defect numbers of singular integral operators with Carleman shift and almost periodic coefficients

This paper is devoted to the study of properties of the kernel and the cokernel of singular integral operators with almost periodic coefficients and a Carleman shift. In particular, the dimensions of their kernels and cokernels are obtained. This is done by considering appropriate properties of the related almost periodic elements and, in special, the partial indices of some of their relevant factorizations.     

Factorization of singular integral operators with a Carleman backward shift: The case of bounded measurable coefficients

In this paper, we generalize our recent results concerning scalar singular integral operators with a Carleman backward shift, allowing more general coefficients, bounded measurable functions on the unit circle. Our aim is to obtain an operator factorization for singular integral operators with a backward shift and bounded measurable coefficients, from which such Fredholm characteristics as the kernel and the cokernel can be described. The main tool is the factorization of matrix functions. In the course of the analysis performed, we obtain several useful representations, which allow us to characterize completely the set of invertible operators in that class, thus providing explicit examples of such operators Dedicated to Professor A. Ferreira dos Santos on the occasion of his seventieth birthday     

A normalization problem for a class of singular integral operators with carleman shift and unbounded coefficients

The solution to a normalization problem for singular integral operators with Carleman shift and degenerate and unbounded coefficients inL _p () is obtained, where is either the unit circle or the real line. The approach followed consists mainly in two steps: the reduction to a singular integral operator with bounded coefficients and the use of the solution to an abstract normalization problem.This research was supported by JNICT under the grant PBIC/C/CEN/1040/92.     

Explicit solutions of Cauchy singular integral equations with weighted Carleman shift

Existence and uniqueness of solutions, as well as their explicit representations, are obtained for singular integral equations with weighted Carleman shift which cannot be reduced to binomial boundary value problems.     

Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip

It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener-Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More generally, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which and Fredholm operators can be expressed in terms of the partial indices of an associated Wiener-Hopf factorization problem.In this paper we establish corresponding results for Toeplitz plus Hankel operators and singular integral operators with flip under the assumption that the generating functions are sufficiently smooth (e.g., Hölder continuous). We are led to a slightly different factorization problem, in which pairs , instead of the partial indices appear. These pairs provide the relevant information about the dimension of the kernel and cokernel and thus answer the invertibility problem.     

A contribution to the theory of singular integral equations with carleman shift

Let the operator N be defined by . It is shown that in the spaces L^P(R_ü;h) (h(x) = x^o|x+i|; -1<_oo+相似文献   

On algebras of singular integral operators with shift in Hölder weighted spaces

We consider algebras of singular integral operators with shift and piecewise Hölder coefficients in a Hölder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Hölder coefficients in a Hölder space with “canonical” weight on the circle. We construct the symbol calculus, formulate necessary and sufficient conditions for the Fredholm property, and give the formula for the index of Fredholm operators.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

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.     

Class of singular integral operators

A class of singular integral operators is studied from the point of view of the boundedness of their action from some symmetric spaces into other spaces.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 105–110, January, 1991.     

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.