Abstract
Full Text
N. M. MIKHAILOVA-GUBENKO
SINGULAR INTEGRAL EQUATIONS IN LIPSCHITZ SPACES
(Presented by Academician V. I. Smirnov on 28 V 1964)
In the paper (¹) S. G. Mikhlin (see also (²)) investigated multidimensional singular integral equations in Lipschitz spaces. The main result of that paper consists in the fact that, under certain smoothness conditions imposed on the data of the equation, every solution of it satisfies a Lipschitz condition of order \(\alpha \in (0,1)\). However, this result was obtained under rather restrictive assumptions. For example, if the characteristic does not depend on the pole, it is required that it have square-summable generalized derivatives of order \(l\) on the unit sphere, where \(l \geq m/2 + 1\) for even \(m\) and \(l \geq (m+1)/2 + 2\) for odd \(m\) (\(m\) is the dimension of the space). In the case of a characteristic depending on the pole, additional restrictions arise.
The above-mentioned result of S. G. Mikhlin was obtained on the basis of his theory of singular equations in \(L_p\). The purpose of the present note is to construct a general theory of singular equations on manifolds in Lipschitz spaces. As a final result, a theorem is obtained on the membership of solutions in a Lipschitz space under assumptions weaker than in (¹).
Let \(\Gamma\) be an \(m\)-dimensional closed manifold of Lyapunov type, and let \(\varphi\) be a function on \(\Gamma\) whose support is mapped by means of a one-to-one smooth transformation \(\tau\) onto a domain \(G \subset E_m\). We shall denote by \(\varphi_\tau, \varphi^\tau\) the operators defined by the equalities
\[ \varphi_\tau u(x)=\varphi(\tau^{-1}(x))\,u(\tau^{-1}(x)), \qquad \varphi^\tau v(\xi)=\varphi(\xi)\,v(\tau(\xi)), \]
where \(\xi\) and \(x\) are arbitrary points on \(\Gamma\) and \(G\), respectively, and \(u(\xi)\) and \(v(x)\) are functions given respectively on \(\Gamma\) and \(G\).
By \(\operatorname{Lip}_\Gamma \alpha\) we shall denote the space of functions given on \(\Gamma\) and satisfying a Lipschitz condition of order \(\alpha\), \(0<\alpha<1\).
Definition 1*. An operator \(A\) is called singular on \(\Gamma\) if the following conditions are fulfilled:
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For any functions \(\varphi\) and \(\psi\) from \(\operatorname{Lip}_\Gamma \alpha\) with nonintersecting supports on \(\Gamma\), the operator \(\varphi A\psi\) is completely continuous in \(\operatorname{Lip}_\Gamma \alpha\).
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For any functions \(\varphi\) and \(\psi\) from \(\operatorname{Lip}_\Gamma \alpha\), whose supports can be mapped, by means of a single one-to-one smooth transformation \(\tau\), onto a domain of the space \(E_m\) (i.e., are situated within one coordinate system), the equality
\[ \varphi A\psi=\varphi^\tau \mathcal{A}\psi_\tau+T, \]
holds, where \(T\) is an operator completely continuous in \(\operatorname{Lip}_\Gamma \alpha\), and \(\mathcal{A}\) is a singular operator in \(E_m\), i.e.
\[ \mathcal{A}u(x)=a(x)u(x)+\lim_{\varepsilon\to 0}\int_{r>\varepsilon}\frac{f(x,\theta)}{r^m}u(y)\,dy, \]
where
\[ a(x)\in \operatorname{Lip}_{E_m}\alpha, \qquad \theta=\frac{y-x}{|y-x|}, \qquad r=|y-x|. \]
* This definition is analogous to the definition of a singular operator in \(L_p(\Gamma)\) given by Seeley (³).
The symbol \(\Phi_A\) of the operator \(A\), by definition, coincides with the symbol of the operator \(\mathcal A\).
In what follows we follow the notation adopted in the monograph of S. G. Mikhlin \((^2)\).
We formulate a theorem on the boundedness of a singular operator in \(\mathrm{Lip}_\Gamma \alpha\). Such a theorem for multidimensional singular integrals was first proved by J. Giro \((^4)\) (see also \((^2)\), § 6) under considerably more restrictive conditions.
Theorem 1. If \(f(x,\theta) \in \widehat W_1^{(1)}(S)\) and if
\[ \|f(x+h,\theta)-f(x,\theta)\|_{L(S)} \leq B |h|^\alpha, \tag{1} \]
then the operator \(A\) is bounded in \(\mathrm{Lip}_\Gamma \alpha\).
For operators defined on a two-dimensional manifold, one can obtain a more precise result.
Theorem \(1'\). Let \(m=2\). If (1) holds and \(f(x,\theta) \in \widehat L(S)\), and if
\[ \|f(x,\theta_\omega)-f(x,\theta)\|_{L(S)} \leq B\omega^\beta, \tag{2} \]
where \(\beta>\alpha\) and \(\theta_\omega\) is the rotation of the vector \(\theta\) through a constant angle \(\omega\), then the operator \(A\) is bounded in \(\mathrm{Lip}_\Gamma \alpha\).
Using a result of S. G. Mikhlin \((^5)\) (see also \((^2)\), § 31), from Theorem 1 one can obtain a criterion for the boundedness of the operator \(A\) in terms of the symbol \(\Phi_A(x,\theta)\).
Theorem 2. If \(\Phi_A(x,\theta) \in \widehat W_2^{(l)}(S)\), where \(l \geq (m+2)/2\), and if
\[ \|\Phi_A(x+h,\theta)-\Phi_A(x,\theta)\|_{W_2^{(l-1)}(S)} \leq B |h|^\alpha, \]
then the operator \(A\) is bounded in \(\mathrm{Lip}_\Gamma \alpha\).
The following theorem on multiplication of symbols holds.
Theorem 3. Let \(A_1\) and \(A_2\) be singular operators on \(\Gamma\). If \(\Phi_{A_i}(x,\theta) \in \widehat W_2^{(l)}(S)\), where \(l \geq (m+3)/2\), and if
\[ \|\Phi_{A_i}(x+h,\theta)-\Phi_{A_i}(x,\theta)\|_{W_2^{(l-1)}(S)} \leq B |h|^\alpha \quad (i=1,2), \]
then the operator \(A_1A_2-A_2A_1\) is completely continuous in \(\mathrm{Lip}_\Gamma \alpha\), and the symbol of the product \(A_1A_2\) is equal to the product of the symbols \(\Phi_{A_1}\Phi_{A_2}\).
In the proof of Theorem 3 the following two lemmas are used.
Lemma 1. Let \(\Phi_A(x,\theta) \in \widehat W_2^{(l)}(S)\), where \(l \geq (m+3)/2\), and
\[ \|\Phi_A(x+h,\theta)-\Phi_A(x,\theta)\|_{W_2^{(l-1)}(S)} \leq B |h|^\alpha. \]
Then the expansion of the symbol in a series in spherical functions,
\[ \Phi_A(x,\theta)= \sum_{n=0}^{\infty}\sum_{k=1}^{k_n} b_n^{(k)}(x)Y_{n,m}^{(k)}(\theta), \]
corresponds to an expansion of the operator \(A\) into a series convergent in the \(\mathrm{Lip}_\Gamma \alpha\) norm,
\[ A=A_1^{(0)}+\sum_{n=0}^{\infty}\sum_{k=1}^{k_n} A_n^{(k)}, \]
where \(A_n^{(k)}\) are singular operators on \(\Gamma\) corresponding to the operators \(\mathcal A_n^{(k)}\) in \(E_m\), where
\[ \mathcal A_1^{(0)}u(x)=b_1^{(0)}(x)u(x),\qquad \mathcal A_n^{(k)}u(x)= \frac{b_n^{(k)}(x)}{\gamma_{n,m}} \int_{E_m}\frac{Y_{n,m}^{(k)}(\theta)}{r^m}\,u(y)\,dy. \]
Lemma 2. Let \(\varphi\) and \(\psi\) be functions from the space \(\operatorname{Lip}\Gamma \alpha\) with compact supports on \(\Gamma\), and let the function \(b(x)\) satisfy a Lipschitz condition of order \(\alpha\) in \(E_m\). Then the operator \(\varphi \mathfrak B \psi\), where \(\mathfrak B\) is the integral operator with kernel
\[ \frac{b(y)-b(x)}{|y-x|^m}\,Y_{n,m}^k(\theta), \]
is completely continuous in \(\operatorname{Lip}\Gamma\alpha\).
In the following theorem conditions are given under which every bounded solution of the equation \(Au=g\), where \(g\in \operatorname{Lip}\Gamma\alpha\), belongs to the space \(\operatorname{Lip}\Gamma\alpha'\), \(\alpha'>0\).
Theorem 4. Let \(\inf |\Phi_A(x,\theta)|>0\), \(f(x,\theta)\in \widehat W_2^{(2)}(S)\cap C^{(1)}(S)\); let
\[ |f(x+h,\theta)-f(x,\theta)| \leqslant B|h|^\gamma \quad (\gamma>0), \tag{3} \]
\[ \|f(x+h,\theta)-f(x,\theta)\|_{W_2^{(1)}(S)} \leqslant B|h|^\alpha . \]
Then every bounded solution of the equation \(Au=g\) belongs to the space \(\operatorname{Lip}\Gamma\alpha'\).
For \(m=2\), using Theorem 1, this result can be strengthened.
Theorem \(4'\). Let \(\inf |\Phi_A(x,\theta)|>0\), \(f(x,\theta)\in \widehat C^{(1)}(S)\). Suppose that (3) holds and
\[ \|f(x+h,\theta)-f(x,\theta)\|_{W_2^{(\varepsilon)}(S)} \leqslant B|h|^\alpha, \tag{4} \]
where \(\varepsilon>0\). Then every bounded solution of the equation \(Au=g\) belongs to the space \(\operatorname{Lip}\Gamma\alpha'\).
Leningrad State University
named after A. A. Zhdanov
Received
20 V 1964
REFERENCES
\({}^{1}\) S. G. Mikhlin, DAN, 138, No. 3 (1961).
\({}^{2}\) S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Moscow, 1962.
\({}^{3}\) R. T. Seeley, Am. J. Math., 81, No. 3 (1959).
\({}^{4}\) G. Giraud, Ann. Sci. École Norm. Super., 51, f. 3, 4 (1934).
\({}^{5}\) S. G. Mikhlin, DAN, 126, No. 2 (1959).