Cauchy Singular Integral Operator Research Articles

Boundedness of Cauchy singular integral operator under Hölder norm

Boundedness of Cauchy singular integral operator under Hölder norm

Algebras of generalized Cauchy singular integral operators

Algebras of generalized Cauchy singular integral operators

On the essential norms of singular integral operators with constant coefficients and of the backward shift

Let X X be a rearrangement-invariant Banach function space on the unit circle T \mathbb {T} and let H [ X ] H[X] be the abstract Hardy space built upon X X . We prove that if the Cauchy singular integral operator ( H f ) ( t ) = 1 π i ∫ T f ( τ ) τ − t d τ (Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau )}{\tau -t}\,d\tau is bounded on the space X X , then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator a I + b H aI+bH with a , b ∈ C a,b\in \mathbb {C} , acting on the space X X , coincide. We also show that similar equalities hold for the backward shift operator ( S f ) ( t ) = ( f ( t ) − f ^ ( 0 ) ) / t (Sf)(t)=(f(t)-\widehat {f}(0))/t on the abstract Hardy space H [ X ] H[X] . Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priloz̆en. 9 (1975), pp. 73-74] for the operator a I + b H aI+bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S S .

Cauchy singular integral operator with parameters in Log-Hölder spaces

This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator $S$ on H\"older functions with parameters. To the contrary of the claim, a counter example was constructed by Tumanov which shows that $S$ with parameters fails to maintain the same H\"older regularity with respect to the parameters. In view of the example, the behavior of the Cauchy singular integral operator with parameters between a type of Log-H\"older spaces is investigated to obtain the sharp norm estimates. At the end of the paper, we discuss its application to the $\bar\partial$ problem on product domains.

On C*-algebras of singular integral operators with PQC coefficients and shifts with fixed points

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift operators , acting on the space over the unit circle . Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms with finite sets of discontinuities for their derivatives , which acts topologically freely on , where is the interior of the nonempty closed set composed by all common fixed points for all shifts , with boundary of zero Lebesgue measure. A Fredholm symbol calculus for the -algebra is constructed and a Fredholm criterion for the operators is established.

On Mellin pseudodifferential operators with quasicontinuous symbols

Let be the C∗‐algebra of quasicontinuous functions on and be the Cauchy singular integral operator, and let p ∈ (1, ∞). The paper is devoted to studying Mellin pseudodifferential operators with quasicontinuous ‐valued symbols on Lebesgue spaces , where is the Banach algebra of absolutely continuous functions of bounded total variation on the real line , and dμ(r) = dr/r. Applying obtained results on Mellin pseudodifferential operators with quasicontinuous ‐valued symbols, we study a Banach algebra of modified singular integral operators on the space , which is generated by the operators of the form where , , Rβ for is a singular integral operator with point singularities at 0 and ∞, Vα is the shift operator given by Vαf = f ∘ α, and α is a homeomorphism of [0, ∞] onto itself with . A Fredholm symbol calculus for the Banach algebra is constructed and a Fredholm criterion for the operators is established.

Singular integral operators in some variable exponent Lebesgue spaces

Abstract The paper deals with the exploration of those subclasses of the variable exponent Lebesgue space L p ⁢ ( ⋅ ) {L^{p(\,\cdot\,)}} with min ⁡ p ⁢ ( ⋅ ) = 1 {\min p(\,\cdot\,)=1} , which are invariant with respect to Cauchy singular integral operators.

$L_{p;r} $ spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework

In the present work the space $L_{p;r} $ which is continuously embedded into $L_{p} $ is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is bounded in $L_{p;r} $. The problem of basisness of the system $left{Aleft(tright)e^{{mathop{rm int}} }; Bleft(tright)e^{-{mathop{rm int}} } right}_{nin Z_{+} }, $ is also considered. It is shown that under an additional condition this system forms a basis in $L_{p;r} $ if and only if the Riemann-Hilbert problem has a unique solution in corresponding Hardy class ${ H}_{p;r}^{+} times { H}_{p;r}^{+} $.

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderon–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

Use of Legendre multiwavelets to solve Carleman type singular integral equations

• Multiscale representation of Cauchy singular integral operator and weight ω ( x ) ( = x α ( 1 − x ) β , α , β > − 1 ) is given in LMW bases. • Estimation formula of local Holder exponent helps to recast the equation with bounded solution throughout the domain. • A posteriori L 2 error of approximate solution u J M S ( x ) is given in terms of wavelet coefficients of resolutions J 2, J 1. • Numerical examples from the theory of elasticity having real as well as complex exponents of the solution are presented. • The stress intensity factor at the tip of the crack can be efficiently estimated with the aid of the approximate solution. This paper is concerned with obtaining approximate numerical solutions of Carleman type singular integral equations by using Legendre multiwavelet basis. The properties of Legendre multiwavelets are first given and the low- and high-pass filters for two-scale relations involving Legendre multiwavelets have been derived. The scheme proposed here consists of two basic steps. In the first step the multiscale representation of the given integral operator is obtained and explicit expressions for the elements of the matrix associated with the multiscale representation are derived. In the second step the given integral equation is converted into a system of linear algebraic equations. The convergence of the method is proved in L 2 space. Two illustrative examples from elasticity are given to show the efficiency of the method developed here.

Singular Integral Operators in Generalized Morrey Spaces on Curves in the Complex Plane

We study the boundedness of the Cauchy singular integral operators on curves in complex plane in generalized Morrey spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of singular integral operators in weighted generalized Morrey spaces.

Algebraic Properties of Cauchy Singular Integral Operators on the Unit Circle

In this paper we study algebraic properties of singular integral operators with Cauchy kernel on the $L^{2}$ space of the unit circle. We give an operator equation characterization for this class of Cauchy singular integral operators. This characterization provides a direct connection between the singular integral operators and multiplication operators. We then use this characterization to study when two Cauchy singular integral operators commute. Our approach also leads to generalizations of several results on normal Cauchy singular integral operators obtained recently by Nakazi and Yamamoto.

A new efficient method with error analysis for solving the second kind Fredholm integral equation with Cauchy kernel

The main objective of this paper is to give an efficient numerical method for the solution of the second kind Fredholm integral equation with Cauchy type kernel. Although, numerical treatment of Singular integral equations (SIEs) has been considered by many researchers and many numerical methods have been proposed for the solution of this equation, strong singularity of the Cauchy singular integral operator still remains as a challenge to numerical methods. Most of the previous methods rely on specific quadrature rule or suitable base functions for capturing the singularity. Here we focus on operator transformation and graded mesh. In addition, we study error analysis. In this regard, efficiency of reproducing kernel Hilbert space (RKHS) method using smooth transformation on the graded mesh is improved and compared with some other numerical methods.

Banach algebra of the Fourier multipliers on weighted Banach function spaces

AbstractLet MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.

Cauchy integral and singular integral operator over closed Jordan curves

This paper is mostly a review paper. It contains a description of old and recent results concerning the regularity conditions on a Jordan curve in the plane that imply the boundedness of the singular integral operator as well as the boundary behavior of the Cauchy type integral. These results are of significance for boundary value problems in domains with non-smooth and non-rectifiable boundaries.

Lp -general approximations by multivariate singular integral operators

Abstract In this article, we study the Lp , 1 ≤ p < ∞ approximation properties of general multivariate singular integral operators over RN , N ≥ 1. We establish their convergence to the unit operator with rates. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly.

On the monotonicity of some singular integral operators

The paper deals with the monotonicity of singular integral operators of the form where is the Cauchy singular integral operator on the interval (0,1) of the real axis and q is a power or logarithmic function. Under suitable assumptions, such singular integral operators are proved to be monotone and maximal monotone in spaces with power weights. Moreover, two related integral equations with weakly singular kernels of logarithmic type are studied. Copyright © 2012 John Wiley & Sons, Ltd.

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Let a be a semi-almost periodic matrix function with the almost periodic representatives a l and a r at −∞ and +∞, respectively. Suppose p : R → ( 1 , ∞ ) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space L p ( ⋅ ) ( R ) . We prove that if the operator a P + Q with P = ( I + S ) / 2 and Q = ( I − S ) / 2 is Fredholm on the variable Lebesgue space L N p ( ⋅ ) ( R ) , then the operators a l P + Q and a r P + Q are invertible on standard Lebesgue spaces L N q l ( R ) and L N q r ( R ) with some exponents q l and q r lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.

Statistical Approximation by Double Poisson–Cauchy Singular Integral Operators

In this paper we show that it is possible to approximate a continuous and 2π-periodic function on the disk centered at origin with radius π by means of double Poisson–Cauchy singular integral operators which do not need to be positive in general. Our results cover not only the classical approximation but also the statistical approximation process.

Global smoothness preservation and simultaneous approximation for multivariate general singular integral operators

In this article we continue with the study of multivariate smooth general singular integral operators over R N , N ≥ 1 , regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞ , by involving multivariate higher order moduli of smoothness. Also we study their multivariate simultaneous approximation to the unit operator with rates. The multivariate Jackson type inequalities obtained are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. In the uniform case of global smoothness we prove optimality. At the end we list the multivariate Picard, Gauss–Weierstrass, Poisson–Cauchy and Trigonometric singular integral operators as applicators of our general theory.