Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

On some geometric conditions of Fredholmity of one-dimensional singular integral operators

Here we consider one-dimensional singular integral operators with arbitrary bounded measurable coefficients. We study Fredholm properties of these operators inL _p spaces using geometric information on the cluster values of their coefficients.     

Two Projection Theorems and Symbol Calculus for Operators with Massive Local Spectra

In this paper we construct a symbol calculus for Banach algebras generated by two idempotents and a coefficient algebra. This, combined with local principles for ?embedding algebras”?, leads to a symbol calculus for singular integral operators on spaces with Muckenhoupt weight and for singular integral operators with measurable coefficients.     

Matrix Functions Consimilar to the Identity and Singular Integral Operators

In this paper we establish the connection between singular integral operators with conjugation and matrix functions consimilar to the identity. We show that any matrix function consimilar to the identity is factorable (in some space L _p ) if and only if it admits a special factorization, that we call antisymmetric, and that this antisymmetric factorization has a direct connection with the factorization of singular integral operators with conjugation. Submitted: April 27, 2007. Accepted: January 23, 2008.     

Gaussian upper bounds for heat kernels of second order complex elliptic operators with unbounded diffusion coefficients on arbitrary domains

In this paper we obtain Gaussian upper bounds for the integral kernel of the semigroup associated with second order elliptic differential operators with complex unbounded measurable coefficients defined in a domain Ω of ? ^N and subject to various boundary conditions. In contrast to the previous literature the diffusions coefficients are not required to be bounded or regular. A new approach based on Davies-Gaffney estimates is used. It is applied to a number of examples, including degenerate elliptic operators arising in Financial Mathematics and generalized Ornstein-Uhlenbeck operators with potentials.     

Rough oscillatory singular integral operators of nonconvolution type

In this paper, we study a classes of oscillatory singular integral operators of nonconvolution type with phases more general than polynomials. We prove that such operators are bounded on Lp provided their kernels satisfy a very weak condition. In addition, we also study the related truncated oscillatory singular integral operators. Moreover, we present a class of unbounded oscillatory singular integral operators.     

Riemann problem and singular integral equations with coefficients generated by piecewise constant functions

We study a Riemann boundary value problem with a shift into the interior of the domain. The problem has piecewise constant coefficients that take two values. We find conditions for the existence and uniqueness of a solution of the inhomogeneous problem and formulas for the number of linearly independent solutions of the homogeneous problem. We consider scalar singular integral operators with a shift and matrix characteristic operators whose coefficients are generated by piecewise constant functions and which have automorphic properties. For these operators, we find invertibility conditions.     

Weyl numbers and eigenvalues of abstract summing operators

We estimate Weyl numbers and eigenvalues of operators via studying their abstract summing norms. In particular we prove estimates of these summing norms for abstract interpolation Lorentz spaces. For this we combine factorization theorems with estimates of concavity constants. Finally we apply our general eigenvalue results to integral operators with kernels of weakly singular type. We obtain asymptotically optimal estimates which extend the well-known classical results.     

Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^p$ theory

We study estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form on Lipschitz domains subject to Dirichlet or to Neumann boundary conditions, pursuing our work [4] where we considered operators on . We obtain among other things for all if L is real symmetric and the domain bounded, which is new for . We also obtain similar results for perturbations of constant coefficients operators. Our methods rely on a singular integral representation, Calderón-Zygmund theory and quadratic estimates. A feature of this study is the use of a commutator between the resolvent of the Laplacian (Dirichlet and Neumann) and partial derivatives which carries the geometry of the boundary. Received: 12 January 2000 / Published online: 4 May 2001     

Fredholm Theory for a Class of Singular Integral Operators with Carleman Shift and Unbounded Coefficients

A criterion for the Fredholmness of singular integral operators with Carleman shift in L_P(Γ) is obtained, where Γ is either the unit circle or the real line. The approach allows to consider unbounded coefficients in a class related to that of quasicontinuous functions. Applications to Wiener-Hopf-Hankel type operators and operators with linear fractional Carleman shift on IR are included.     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

由具奇异系数的强椭圆微分算子生成的解析半群

本文提出了低阶项系数具一定奇异性的高阶强椭圆微分算子生成解析半群的条 件,其包含了低阶项系数属于Schecter位势类的情形.特别地,低阶项系数属于LP(Rn) 的情形也在其中.作为应用,得到低阶项系数属于LP(Rn)的强椭圆算子的相应半群生 成,且p还可取到一些临界值.     

带粗糙核的奇异积分交换子的有界性

本文研究了带粗糙核的奇异积分算子与BMO函数生成的交换子的有界性问题.利用原子分解的方法,获得了带粗糙核的奇异积分交换子TΩb在Lp空间、Hardy空间、弱Hardy空间上的有界性结果,推广了交换子Tb在各类函数空间上有界性的结果.     

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.     

Convolution factorability of bilinear maps and integral representations

In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called ‘zero product preserving’, and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford–Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert–Schmidt operators.     

Spectral decomposition of self-adjoint cyclically compact operators and partial integral equations

We prove a spectral decomposition theorem for self-adjoint cyclically compact operators on Hilbert–Kaplansky module over a ring of bounded measurable functions. We apply this result to partial integral equations on the space with mixed norm of measurable functions. We give a condition of solvability of partial integral equations with self-adjoint kernel.     

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.