Theory Of Singular Integral Operators Research Articles

Об условиях нётеровости и индексе некоторых двумерных сингулярных интегральных операторов

The main problems in the theory of singular integral operators are the problems of boundedness, invertibility, Noethericity, and calculation of the index. The general theory of multidimensional singular integral operators over the entire space E_n was constructed by S.G. Mikhlin. It is known that in the two-dimensional case, if the symbol of an operator does not vanish, then the Fredholm theory holds. For operators over a bounded domain, the boundary of this domain significantly affects the solvability of the corresponding operator equations. In this paper, we consider two-dimensional singular integral operators with continuous coefficients over a bounded domain. Such operators are used in many problems in the theory of partial differential equations. In this regard, it is of interest to establish criteria for the considered operators to be Noetherian in the form of explicit conditions on their coefficients. The paper establishes effective necessary and sufficient conditions for two-dimensional singular integral operators to be Noetherian in Lebesgue spaces L_p (D) (considered over the field of real numbers), 1<p<∞, and formulas for calculating indices are given. The method developed by R.V. Duduchava [Duduchava R. On multidimensional singular integral operators. I: The half-space case; II: The case of compact manifolds // J. Operator Theory, 1984, v. 11, 41–76 (I); 199–214 (II)]. In this case, the study of the Noetherian properties of operators is reduced to the factorization of the corresponding matrix-functions and finding their partial indices.

Oblique derivative problems and Feller semigroups with discontinuous coefficients

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with an oblique derivative boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking (or viscosity) phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderon–Zygmund theory of singular integral operators with non-smooth kernels.

Factorization of rational matrix functions and difference equations

Abstract In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.

Local scales on curves and surfaces

In this paper, we extend a previous work on the study of local scales of a function to studying local scales of a d-dimensional surface. In the case of a function, the scale functions are computed by convolving the function with a symmetric kernel of zero mean and zero first moments of various scales. From the goodness of fit point of view, this convolution can be viewed as measuring locally the deviations from a linear function at various scales. The local scales are defined as points where deviation from a linear function reaches a local maximum. In the case of a d-dimensional surface, the analogy of the scale functions is to compute local deviations from a d-plane at various scales (this is related to Jones beta number). This analogy is realized through convolving the (surface) measure with a symmetric kernel of zero mean and zero first moments. We then apply the theory of singular integral operators on d-dimensional surfaces to show useful properties of local scales. We also show that the defined local scales are consistent in the sense that the number of local scales are invariant under dilation, and that one can relate the local scales of the original object with its dilated version via the dilating factor. In addition, with the assumption that the d-dimensional surface enjoys a certain degree of smoothness, we prove that our local scales are related to curvatures. Furthermore, this connection makes apparent that our local scales are intimately related to the change in deviation from flatness. Computational examples are also presented. In shape analysis, the local scales and the scale functions on the boundary can be used as local signatures or descriptors.

The divergence equation in weighted- and L p(·)-spaces

We study the solvability of the divergence equation in weighted spaces and Lebesgue spaces with variable exponents, where the weights are so called Muckenhoupt weights. The question of constructing divergence free test functions, which can be used for problems arising in fluid dynamics, is also addressed. The approach is based on an explicit representation formula for solutions of the divergence equation due to Bogovskiĭ and the theory of singular integral operators. The developed methods are used to prove an existence result for fluids which satisfy a p(·)-growth condition.

On the existence of Feller semigroups with discontinuous coefficients II

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with Dirichlet boundary condition, oblique derivative boundary condition and first-order Wentzell boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderón-Zygmund theory of singular integral operators with non-smooth kernels.

BMO and [formula omitted] for the Ornstein–Uhlenbeck operator

In this paper we develop a theory of singular integral operators acting on function spaces over the measured metric space ( R d , ρ , γ ) , where ρ denotes the Euclidean distance and γ the Gauss measure. Our theory plays for the Ornstein–Uhlenbeck operator the same rôle that the classical Calderòn–Zygmund theory plays for the Laplacian on ( R d , ρ , λ ) , where λ is the Lebesgue measure. Our method requires the introduction of two new function spaces: the space BMO ( γ ) of functions with “bounded mean oscillation” and its predual, the atomic Hardy space H 1 ( γ ) . We show that if p is in ( 2 , ∞ ) , then L p ( γ ) is an intermediate space between L 2 ( γ ) and BMO ( γ ) , and that an inequality of John–Nirenberg type holds for functions in BMO ( γ ) . Then we show that if M is a bounded operator on L 2 ( γ ) and the Schwartz kernels of M and of its adjoint satisfy a “local integral condition of Hörmander type,” then M extends to a bounded operator from H 1 ( γ ) to L 1 ( γ ) , from L ∞ ( γ ) to BMO ( γ ) and on L p ( γ ) for all p in ( 1 , ∞ ) . As an application, we show that certain singular integral operators related to the Ornstein–Uhlenbeck operator, which are unbounded on L 1 ( γ ) and on L ∞ ( γ ) , turn out to be bounded from H 1 ( γ ) to L 1 ( γ ) and from L ∞ ( γ ) to BMO ( γ ) .

Spectral Theory in Banach Spaces and Harmonic Analysis

The conference was motivated by the recent solutions of two longstanding questions. The first one is Kato's square root problem, i.e. whether for an elliptic operator in divergence form on L_2( \mathbb R^n) we have \|L^{1/2}u\|\leq C \|\nabla u\| + C\|u\| for u\in W^1_2( \mathbb R^n) . After a long development over 40 years this was shown in a joint effort by P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and Ph. Tchamitchian. For a new approach to this important result via Dirac operators and extensions of it, see the abstracts of A. McIntosh and S. Keith. The second problem, attributed to Brézis in the eighties, asks whether the Cauchy problem for every generator of an analytic semigroup A in an L_q(\Omega) -space with 1<q<\infty has maximal L_p -regularity, i.e. does the solution y of the Cauchy problem y'=Ay+f , y(0)=0 , satisfy \|y\|_{W^1_p( \mathbb R_+,L_q)} \leq C \|f\|_{L_p( \mathbb R_+,L_q)} . Recently G. Lancien and N .J. Kalton gave counterexamples to this questions while L. Weis gave a characterization of maximal L_p -regularity in terms of R-boundedness. This criterium has since been shown to be useful in establishing maximal regularity for large classes of Cauchy problems for elliptic differential operators with rather general coefficients and boundary values, e.g. Schrödinger operators with singular potentials and Stokes operators (see the abstracts of M. Hieber, P. Kunstmann, J. Prüss and R. Schnaubelt). Applications of maximal regularity to non-linear differential equations were presented in the talks of H. Amann and G. Simonett. The two problems of Kato and Brézis share a common background in the mathematical methods employed in their solution, e.g. the H^\infty -functional calculus for sectorial operators in Banach spaces, boundedness of singular integral operators and square function estimates. Obviously, further progress will depend on an indepth study of these methods and their interrelations. The workshop contributions featured some of the most recent progress in these directions. N.J. Kalton used deep results from Banach space theory to crystallize the difficult perturbation theory of the H^\infty -functional calculus. Some of the randomization techniques he used can be seen as an extension of the classical square function estimates from L_p -spaces to general Banach spaces. This method also underlies boundedness results for Fourier multiplier operators, Calderón–Zygmund operators and wavelet transforms for functions with values in a Banach space with the UMD-property (see the abstracts of W. Arendt, O. Blasco, T. Hytonen and C. Kaiser) and applications of those to spectral theory (A. Gillespie). Also in the scalar-valued case several extensions of basic methods in the theory of singular integral operators were presented, e.g. new spaces of BMO- and H^1 -type associated to a given operator (X. Duong), Calderón-Zygmund operators associated to non-doubling measures (J. Garcia-Cuerva), bilinear pseudo-differential operators (A. Nahmod) and optimal domains for convolution operators (W. Ricker). As so often, new tools in harmonic analysis lead to new results for the H^\infty -functional calculus of partial differential operators (M. Cowling, X. Duong). Some of the talks reached beyond the circle of questions raised by the problems of Kato and Brézis. They were closely connected to the main topics by a shared methological background in spectral theory and harmonic analysis. Ch. Thiele discussed an approach to non-linear differential equations via the non-linear Fourier transform. B. Jefferies and J. van Neerven described the connection between spectral theory, square function estimates and stochastic integration on infinite dimensional Banach spaces. S. Grivaux and Y. Latushkin gave applications of spectral theory to instability and stability theorems for evolutionary systems. The talks of of J. Eschmeier, M. Haase and F. Sukochev were focused on the functional calculus and its applications in operator theory. The varied background of the participants lead to a number of new collaborations started during the workshop. Last but not least, the unique setting of Oberwolfach and a week of beautiful sunshine contributed to a memorable and successful meeting.

Logistic Dirichlet problems with discontinuous coefficients

This paper is devoted to the study of the existence of positive solutions of semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singular integral operators. Moreover, we make use of an L p variant of an estimate for the Green operator of the Dirichlet problem introduced in the study of Feller semigroups.

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderon-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szego projections and the corresponding Lp-decompositions. Our approach relies on an extension of the classical Calderon-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.

Mellin pseudodifferential operators with slowly varying symbols and singular integrals on Carleson curves with Muckenhoupt weights

We show that by using Mellin pseudodifferential operators whose double symbol depends analytically on the co-variable we can rather quickly arrive at descriptions of the local spectra of the Cauchy singular integral operator over a large class of Carleson curves with Muckenhoupt weights. The approach of this paper extends some recent results of the spectral theory of singular integral operators to the case of piecewise slowly varying coefficients and also yields new and surprising interpretations of these results.

Noether theory of unbounded singular integral operators with a Carleman shift

In [i] we have obtained a Noethericity criterion and an index formula for singular integral operators with a Carleman shift and with continuous coefficients in the spaces Lp (i < p < ~). In a series of publications of other authors (see, e.g., [2]), these results have been extended to the case of coefficients having discontinuities of the first kind. The purpose of this note is the construction of the Noether theory of singular integral operators with a Carleman shift and with coefficients having discontinuities of the second kind.

Pseudo-differential operators with nonregular symbols

where a power of 27~ is ignored and G(tr ,..., 5,) is the Fourier transform of U(Xi ,...) x,). We shall call xi the space variables and fi the dual variables. The notion of pseudo-differential operators has grown in recent years out of attempt to obtain sharp a priori estimates for the solutions of the partial differential equations. Since its discovery it has been found to be one of the most powerful tools in attacking various problems in partial differential equations such as the existence and uniqueness of the boundary value problems [I], regularity of the solutions of the partial differential equations [2], solvability of a general partial differential operator [3], etc. The basic calculus formulas for the pseudo-differential operators are due to J. Kohn and L. Nirenberg [4] as a development of the theory of Singular Integral Operators of Calderon and Zygmund. They constructed an algebra of pseudo-differential operators with symbols having bounded derivatives in the space variables and proved the boundedness of such operators. However their algebra is too restrictive for many purposes. One of the

Analytic Singular Integral Operators

The following paper extends to real analytic manifolds the general theory of singular integral operators as described in [lO] and [13]. The definition of an analytic singular integral operator is made in terms of the kernel of the operator. The symbol of the operator is discussed and in the case of an elliptic operator, a regularity theorem is proved. I t should be pointed out, however, that the regularity theorem is a purely local one. The question of obtaining a global inverse to an elliptic operator or more generally an operator with a prescribed symbol is still open.

Singular integrals on Hilbert space

exists as a bounded operator on L(H) as o tends to zero and p tends to infinity. A theorem of this type extends the Calderon-Zygmund theory of singular integral operators on En to infinite dimensions. For if fc(#)||x||- is a Calderon-Zygmund kernel and if £ is a bounded Borel set which is disjoint from a neighborhood of the origin then v{E) =fEk(x)\\x\\~ dx satisfies v(tE) = v(E) for £>0; if g(x) is an integrable radial function on En with support in a bounded annulus disjoint from a neighborhood of the origin, then fEng(x)k(x)\\x\\~ dx = 0. When fx satisfies a smoothness condition and n(H) = 0 , the set function v(E) —fâ[i{E/i)dt/t has these properties. The results in this paper are taken from the author's Cornell doctoral dissertation. The author wishes to express his most hearty thanks to his thesis advisor Professor Leonard Gross for his interest, advice, and encouragement during the preparation of the thesis.

Analytic singular integral operators

The following paper extends to real analytic manifolds the general theory of singular integral operators as described in [lO] and [13]. The definition of an analytic singular integral operator is made in terms of the kernel of the operator. The symbol of the operator is discussed and in the case of an elliptic operator, a regularity theorem is proved. I t should be pointed out, however, that the regularity theorem is a purely local one. The question of obtaining a global inverse to an elliptic operator or more generally an operator with a prescribed symbol is still open.

On the spectral theory of singular integral operators

A complete spectral representation is given for a class of singular integral operators. The influence of spectral multiplicity is partially taken into account. (T.F.H.)

On the spectral theory of singular integral operators

On the spectral theory of singular integral operators