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On the asymptotic expansion of certain plane singular integral operators
- Vladimir Vasilyev^{1}Email author
Received: 12 December 2016
Accepted: 30 July 2017
Published: 15 August 2017
Abstract
We discuss the problem of the asymptotic expansion for some operators in a general theory of pseudo-differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with the help of the Fourier transform, the Dirac mass-function and its derivatives, and the well-known distribution related to the Cauchy type integral.
Keywords
1 Introduction
In the theory of pseudo-differential equations the main difficulty is studying model operators in canonical domains according to a local principle. It shows that for a Fredholm property of a general pseudo-differential operator on a compact manifold one needs the invertibility of its local representatives in each point of a manifold [1, 2]. The author wrote many times on the nature of these local representatives, these are distinct in dependence on a point of a manifold. Each ‘singularity’ of a compact manifold (a half-space is a model situation for the smooth part of a boundary, cone for the conical point, wedge, etc.) corresponds to a certain distribution, and a convolution operator with this distribution describes a local representative of an initial pseudo-differential operator in a corresponding point of the manifold. All details can be found in [3–5]. But singularities can be of distinct dimensions and it is possible that such singularities of a low dimension can be obtained from analogous singularities of full dimension. This means we need to find distributions for limit cases when some of the parameters of the singularities tend to zero. This approach was partially realized in [6, 7], and [5] is devoted to multi-dimensional constructions. Our idea is the following. Limiting operators for thin singularities obtained in [6] may be a zero approximation for such thin singularities. It is desirable to obtain an asymptotic expansion with a small parameter for the distribution corresponding to such a singularity. We will consider here a two-dimensional case and hope these studies will help us to transfer such constructions to multi-dimensional situations [5].
The theory of differential or more generally pseudo-differential equations and boundary value problems for manifolds with non-smooth boundaries includes now a lot of interesting approaches and results.
Other approaches, technical tools and results in the theory of boundary value problems one can find in work of Mazya [8, 9], Plamenevskii [10], Schulze [11], Melrose [12–14], Taylor [15], Nistor [16], Dauge [17], Costabel [18], Mazzeo [19] and many others. We cannot enumerate all authors but in [3] a very large survey of these approaches is given.
The author wrote many times on another approach to studying solvability for pseudo-differential equations in domains with conical points and wedges, but now we would like to speak of the principal difference of our papers from other authors (Maz’ya, Plamenevski, Schulze and many others).
In all papers the conical domain is treated as a direct product of a circle and a half-axis (but in my point of view, it is a cylinder), then they apply the Mellin transform on the half-axis, and the initial problem is reduced to a problem in a domain with a smooth boundary with operator-valued symbol. It follows further it is like the generalization of well-known results on the case of an operator symbol. Of course, our approach is a generalization also, but it is a generalization on dimension space, and the principal difference is that we do not divide the cone, and it is treated as an emergent thing.
For convenience of the reader the theory of the solvability for considered pseudo-differential equations, already known in principle [3, 4], is given in the next section of this paper.
2 Solving pseudo-differential equations
2.1 Model operators and Sobolev-Slobodetskii spaces in a cone
For invertibility of such an operator with symbol \(A(\cdot,\xi)\) not depending on a spatial variable x one can apply the theory of the classical Riemann boundary value problems for upper and lower complex half-planes with a parameter \(\xi'=(\xi_{1},\ldots,\xi_{m-1})\). This step was systematically studied in [2]. But if the boundary ∂M has at least one conical point, this approach is not effective.
To study the invertibility property for the operator (1) we have introduced the concept of the wave factorization for an elliptic symbol near a singular boundary point [3, 4] and using this property we have described Fredholm properties for equation (2). We use Sobolev-Slobodetskii spaces for studying these properties.
Definition 1
If \(s=0\) then \(\tilde{H}_{0}({\mathbf{R}}^{m})=L_{2}({\mathbf{R}}^{m})\), and by virtue of Plancherel’s theorem \(H_{0}({\mathbf{R}}^{m})=F^{-1}\tilde{H}_{0}({\mathbf{R}}^{m})=L_{2}({\mathbf{R}}^{m})\).
In the case \(s=-n\), \(n>0\), n integer, the distributions from \(H_{-n}({\mathbf{R}}^{m})\) are derivatives of functions from \(L_{2}({\mathbf{R}}^{m})\) whose order is not higher than n.
2.2 Wave factorization and solvability
To describe the solvability picture for equation (2) we use the following.
Definition 2
- (1)
\(A_{\neq}(\xi)\), \(A_{=}(\xi)\) are defined for all admissible values \(\xi\in{\mathbf{R}}^{2}\), without maybe the points \(\{\xi\in {\mathbf{R}}^{2}: \vert \xi_{1} \vert ^{2}=a^{2}\xi^{2}_{2}\}\);
- (2)\(A_{\neq}(\xi)\), \(A_{=}(\xi)\) admit an analytical continuation into radial tube domains \(T(\stackrel{*}{C^{a}_{+}})\), \(T(\stackrel{*}{C^{a}_{-}})\), respectively, with estimates$$\begin{gathered} \bigl\vert A_{\neq}^{\pm1}(\xi+i \tau) \bigr\vert \leq c_{1}\bigl(1+ \vert \xi \vert + \vert \tau \vert \bigr)^{\pm\kappa}, \\ \bigl\vert A_{=}^{\pm1}(\xi-i\tau) \bigr\vert \leq c_{2}\bigl(1+ \vert \xi \vert + \vert \tau \vert \bigr)^{\pm(\alpha-\kappa)}, \forall \tau\in\stackrel{*}{C^{a}_{+}}. \end{gathered} $$
For \(\vert \kappa-s \vert <1/2\) one has the existence and uniqueness theorem [3]. For this purpose we need a certain lemma.
Lemma 1
Then the multiplication operator by the function \(B_{\ne}(\xi)\) boundedly acts from space \(\tilde{H}_{s}(C_{+}^{a})\) into \(\tilde{H}_{s-\alpha}(C_{+}^{a})\), and the multiplication operator by the function \(B_{=}(\xi)\) from space \(\tilde{H}_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) into space \(\tilde{H}_{s-\alpha}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\).
Proof
The fact that multiplication operators by functions \(B_{\ne}(\xi)\), \(B_{=}(\xi)\) boundedly act from spaces \(\tilde{H}_{s}(C_{+}^{a})\), \(\tilde{H}_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) into space \(\tilde{H}_{s-\alpha}({\mathbf{R}}^{2})\) is well known [2]. For clarity we denote \(u\in\tilde{H}_{s}(C_{+}^{a})\) by \(u_{+}\) and \(u\in\tilde{H}_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) by \(u_{-}\). Let us show that \(B_{\ne}(\xi)\tilde{u}_{+}(\xi)\in\tilde{H}_{s-\alpha}(C_{+}^{a})\) for any \(\tilde{u}_{+}\in\tilde{H}_{s}(C_{+}^{a})\).
Transfer to the general case \(u_{-}\in H_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\) is realized by virtue of the density of class \(C_{0}^{\infty}({\mathbf{R}}^{2}\setminus \overline{C_{+}^{a}})\) in space \(H_{s}({\mathbf{R}}^{2}\setminus\overline{C_{+}^{a}})\).
So, it was shown that \(B_{=}(\xi)\tilde{u}_{-}(\xi)\in\tilde{H}_{s-\alpha }({\mathbf{R}}^{2})\) and \(\operatorname{supp} F^{-1}(B_{=}\tilde{u}_{-})\subset ({\mathbf{R}}^{2}\setminus C_{+}^{a})\). Hence, \(B_{=}(\xi)\tilde{u}_{-}(\xi)\in\tilde{H}_{s-\alpha}({\mathbf{R}}^{2}\setminus \overline{C_{+}^{a}})\). □
As above we use the notation \(u_{+}\) for the function \(u\in H_{s}(C^{a}_{+})\).
Theorem 1
Proof
We give here this proof to explain the appearance of the operator \(K_{a}\) because it plays a crucial role in our studies.
3 An initial approximation
We will consider two spaces of basic functions for distributions. If \(D({\mathbf{R}}^{2})\) denotes a space of infinitely differentiable functions with a compact support then \(D'({\mathbf{R}}^{2})\) is the corresponding space of distributions over the space \(D({\mathbf {R}}^{2})\); analogously if \(S({\mathbf{R}}^{2})\) is the Schwartz space of functions infinitely differentiable rapidly decreasing at infinity, then \(S'({\mathbf{R}}^{2})\) is a corresponding space of distributions over \(S({\mathbf{R}}^{2})\).
4 A decomposition formula for distributions
Remark 2
In [6] the author has considered the two cases \(a\to\infty\) and \(a\to0\); the first case corresponds to a zero angle but the second one corresponds to a half-space; the last was done for a comparison with [2]. Since the half-space case is studied in [2] in detail we do not stop in this here.
4.1 A rough decomposition
4.2 A sharp decomposition
Here we consider \(\varphi\in S({\mathbf{R}}^{2})\).
First \(T_{k,N}(\xi_{1})\equiv0\), \(\forall k=2n-1\), \(n\in{\mathbf{N}}\). So the non-trivial case is \(k=2n\), \(n\in{\mathbf{N}}\). Let us recall \(T_{0,\infty}(\xi_{1})=\pi i2^{-1}\xi_{1}^{-1}\) [6, 7]. For other cases we can calculate this integral so we have the following:
5 A local behavior of a boundary operator
Lemma 2
Proof
Theorem 2
Proof
Returning to formula (8) and using calculations \(T_{k,N}(\xi_{1})\) and Lemma 2 we obtain the required assertion. □
Remark 3
One can easily reconstruct the coefficients \(c_{m,n}(a)\) starting from the above calculations.
6 Towards a pseudo-differential equation
Below we denote \(lv\equiv V\).
Theorem 3
Proof
We need to apply Theorem 2 and to recall correlations between distributions and pseudo-differential operators. It proves the theorem. □
Remark 4
The reader can easily write an analog of Theorem 3 corresponding to a rough decomposition.
7 Conclusion
It was shown that the solution of equation (2) for a smooth enough right-hand side v can be represented in the form (10). It shows that in this series the first summand belongs to the space \(H_{s}(C^{a}_{+})\) only. Secondary summands can be useful for certain special situations related to some additional properties of the right-hand side v.
Declarations
Acknowledgements
The author thanks the anonymous referee for making several helpful corrections and suggestions.
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Authors’ Affiliations
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