ON THE BOUNDEDNESS OF A SINGULAR INTEGRAL OPERATOR IN THE SPACE $ C^{\alpha}(\overline G)$

Аниконов, Д. С.

doi:10.1070/sm1977v033n04abeh002432

Cited by 9 publications

(41 citation statements)

References 2 publications

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“…Note that some estimates in the argument below resemble the proof of necessity in [15]. Let us show first that |I − j | ≤ const.…”

Section: Auxiliary Statementsmentioning

confidence: 92%

The indicator of contact boundaries for an integral geometry problem

Аниконов

2008

Sib Math J

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We pose and study a rather particular integral geometry problem. In the two-dimensional space we consider all possible straight lines that cross some domain. The known data consist of the integrals over every line of this kind of an unknown piecewise smooth function that depends on both points of the domain and the variables characterizing the lines. The object we seek is the discontinuity curve of the integrand. This problem arose in the author's previous research in X-ray tomography. In essence, it is a generalization of one mathematical aspect of flaw detection theory, but seems of interest in its own right. The main result of this article is the construction of a special function that can be unbounded only near the required curve. Precisely for this reason we call the function the indicator of contact boundaries. A uniqueness theorem for the solution follows rather easily from the property of indicators.Consider some bounded domain D in R 2 and denote the diameter of D by d. Consider a system of domains D i , for i = 1, . . . , p, such thatIt is not difficult to see that the union of the boundaries ∂D i of D i , for i = 1, . . . , p, coincides with the boundary ∂D 0 of their union. We assume that the closed curves ∂D i are piecewise smooth of class C 2 . More exactly, we make the following assumptions: Denote the coordinates of points x, y, z, ω in R 2 in the main coordinate system by (x 1 , x 2 ), (y 1 , y 2 ), (z 1 , z 2 ), (ω 1 , ω 2 ) respectively. Define z ∈ ∂D 0 as a contact point for domains D j and D l , where 1 ≤ j and l ≤ p, if in some neighborhood V (z) of z there are no points of D i except for D j and D l , and the common part of the two boundaries V (z) ∩ ∂D j = V (z) ∩ ∂D l represents the graph of a smooth function in local coordinates. We assume that in this local coordinate system centered at z the first axis is directed along the tangent to ∂D j or ∂D l at z. The second axis is directed along the unit vector of the interior normal n j (z) to ∂D j at z so that the smallest angle of rotation from the first axis to the second goes counterclockwise, as in the main coordinate system. Using some local coordinates (ξ, η) for every point y ∈ R 2 , we assume thatwhere a positive number δ and a function ψ(ξ) may depend on z in general. Assume that all but possibly finitely many points of ∂D 0 \ ∂D are contact points.It is easy to see that |ψ(ξ)| ≤ const |ξ| 2 . Here and below the symbol const stands for some positive number.

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“…Note that some estimates in the argument below resemble the proof of necessity in [15]. Let us show first that |I − j | ≤ const.…”

Section: Auxiliary Statementsmentioning

confidence: 92%

The indicator of contact boundaries for an integral geometry problem

Аниконов

2008

Sib Math J

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“…If X and Y are finite-dimensional, then it is obvious that any N from L(X, Y ) is a Fredholm operator. Since the dimensions of the spaces X (1) and Y (1) in expansions (1.3.2) are equal in this case, it follows that ind N = dim X − dim Y .…”

Section: Fredholm Operatorsmentioning

confidence: 94%

“…Due to (3.1.5), the estimate (3.1.4) holds for any homogeneous function of power λ, satisfying the Lipschitz condition on the unit sphere Ω. Here, the norm |Q| C 0,1 (Ω) plays the role of the norm |Q| (1) .…”

Section: Homogeneous Functionsmentioning

confidence: 99%

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Singular Integral Operators and Elliptic Boundary-Value Problems. Part I

Soldatov

2020

J Math Sci

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The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here.Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted Hölder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted Hölder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved.Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

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“…The present article arises as a generalization of different approaches of the theory of inverse problems to the transport equation. For the first time, similar ideas were used by the author in [3,4]. Then this method was developed in the study of problems of X-ray tomography and integral geometry [5][6][7][8][9][10].…”

Section: Remark 32mentioning

confidence: 99%

A Method for Studying Singular Integral Equations

Аниконов

2010

Sib Math J

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We examine a singular integral equation of the first kind on a bounded open set of an n-dimensional space. Open subsets with a common (contact) (n − 1)-dimensional piecewise smooth part of boundaries are selected. The underdetermined case is treated in which the unknown part of the integrand depends on 2n independent variables whereas a given integral depends only on n variables. In this situation we pose the problem of finding the contact part of the boundaries and prove unique solvability of the problem. We use the following notation: ρ(A, B) is the distance between sets A and B in R n ; ∂A is the boundary ofConsider a function g(x, y, ω) measurable on T ×T ×Ω and bounded almost everywhere in the following sense: There exist T and Ω ( T ∈ T and Ω ∈ Ω) coinciding in measure with T and Ω and such that |g(x, y, ω)| ≤ const for (x, y, ω) ∈ T × T × Ω. Here and in what follows, the notation const is used to denote a positive constant. In order to reduce the number of denotations, the bounded functions are often written below as O(1). We put T i = T i ∩ T and T 0 = T 1 ∪ T 2 , i = 1, 2. A point of the unit sphere Ω is denoted by ω if it designates an independent variable. If this point is expressed through other variables then we employ the letter s; for example, s = (y − x)/|y − x|. Consider the integral(1.1)The functions V i (x) for x ∈ T i , i = 1, 2, are understood to be singular integrals. We assume that T i , i = 1, 2, and g(x, y, ω) satisfy the following condition: for every u ∈ T i ,Of course, this restriction means that the integrand is a summable function. For convenience, g(x, y, ω) is assumed to be extended by zero in y beyond T . Note that if we use the spherical coordinates y − u = tν, ν ∈ Ω, dy = t n−1 dtdν then (1.2) is a consequence of the condition Ω dν ∞ 0 |g(x, u + tν, ω) − g(x, u, ω)| t dt ≤ const, ω ∈ Ω, x ∈ T .

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ON THE BOUNDEDNESS OF A SINGULAR INTEGRAL OPERATOR IN THE SPACE $ C^{\alpha}(\overline G)$

Cited by 9 publications

References 2 publications

The indicator of contact boundaries for an integral geometry problem

The indicator of contact boundaries for an integral geometry problem

Singular Integral Operators and Elliptic Boundary-Value Problems. Part I

A Method for Studying Singular Integral Equations

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