UDC 517.9

MATHEMATICS

R. V. DUDUCHAVA

ON SINGULAR INTEGRAL OPERATORS IN A WEIGHTED HÖLDER SPACE

(Presented by Academician N. I. Muskhelishvili on 22 VII 1969)

In the present note we set forth some additions to the well-known results of N. I. Muskhelishvili \((^1)\) on singular integral equations with piecewise-Hölder coefficients. In the proofs of almost all the theorems given below, we use methods proposed by I. Ts. Gohberg and N. Ya. Krupnik in \((^{2-4})\).

1. Let \(\Gamma\) be a simple smooth closed oriented curve in the plane, enclosing the point \(z=0\). Let \(c_1,\ldots,c_n\) be some points on \(\Gamma\), and let \(\mu\) be a real number, \(0<\mu\leqslant 1\).

By \(H_\mu(\Gamma)\) we shall denote the Banach space of functions defined on \(\Gamma\), satisfying the Hölder condition with exponent \(\mu\) and with norm

\[ \|\varphi\|_H = \sup \frac{|\varphi(t_2)-\varphi(t_1)|}{|t_2-t_1|^\mu} + \max |\varphi(t)| \qquad (t,t_1,t_2\in\Gamma,\ t_1\ne t_2). \]

By \(H_\mu^0(\Gamma,c,\ldots,c_n)\) we shall denote the subspace of the space \(H_\mu(\Gamma)\) consisting of all functions \(\varphi(t)\) for which \(\varphi(c_j)=0\) \((j=1,\ldots,n)\).

Let \(\alpha_1,\ldots,\alpha_n\) be some complex numbers. Consider the function

\[ \rho(t)=\prod_{k=1}^{n}(t-c_k)^{\alpha_k}. \tag{1} \]

By \(H_\mu^0(\Gamma,\rho)\) we shall denote the set of all functions \(\varphi(t)\) satisfying the condition \(\varphi(t)\cdot \rho(t)\in H_\mu^0(\Gamma,c_1,\ldots,c_n)\). It is easy to see that the class \(H_\mu^0(\Gamma,\rho)\), endowed with the norm \(\|\varphi\|_\rho=\|\varphi\cdot\rho\|_H\), is a Banach space.

Using known methods for estimating singular operators (see \((^1)\), § 25), one can show that if the numbers \(\mu,\alpha_1,\ldots,\alpha_n\) satisfy the conditions

\[ 0<\mu<1,\qquad \mu<\operatorname{Re}\alpha_k<\mu+1 \quad (k=1,\ldots,n), \tag{2} \]

then the singular integration operator \(S\)

\[ (S\varphi)(t)=\frac{1}{\pi i}\int_{\Gamma}\frac{\varphi(\tau)}{\tau-t}\,d\tau \qquad (t\in\Gamma) \]

is a linear bounded operator in the space \(H_\mu^0(\Gamma,\rho)\).

The following lemma plays an essential role in what follows.*

Lemma. If \(a(t)\in H_\mu(\Gamma)\), and the numbers \(\mu,\alpha_1,\ldots,\alpha_n\) satisfy the relations (2), then the operator \(T=aS-SaI\) (\(I\) is the identity operator) is a completely continuous operator in the space \(H_\mu^0(\Gamma,\rho)\), where \(\rho(t)\) is defined by equality (1).

Theorem 1. Let \(c(t)\) and \(d(t)\in H_\mu(\Gamma)\); the numbers \(\mu,\alpha_1,\ldots,\alpha_n\) satisfy the relations (2), and \(\rho(t)\) is defined by equality (1).

In order that the operator

\[ A=cI+dS \qquad (A=cI+SdI) \]

be a \(\Phi_+\)- or \(\Phi_-\)-operator** in the space \(H_\mu^0(\Gamma,\rho)\), it is necessary and sufficient—

* For Hölder spaces without weight this lemma was established in \((^5)\), § 6.
** For the definition of \(\Phi_\pm\)- and \(\Phi\)-operators, see \((^6)\).

sufficient that the conditions \(c(t)+d(t)\ne0,\ c(t)-d(t)\ne0\) \((t\in\Gamma)\) be satisfied. If these conditions are satisfied and
\[ \chi=(1/2\pi)[\arg((c+d)/(c-d))]_{\Gamma}, \]
then:

1) for \(\chi>0\) the operator \(A\) is left invertible in \(H_\mu^0(\Gamma,\rho)\) and \(\dim\operatorname{coker} A=\chi\);

2) for \(\chi<0\) the operator \(A\) is right invertible in \(H_\mu^0(\Gamma,\rho)\) and \(\dim\ker A=-\chi\);

3) for \(\chi=0\) the operator \(A\) is invertible in \(H_\mu^0(\Gamma,\rho)\).

For the case of the Hölder space without weight, Theorem 1 was established in (7).

We note that, in order to establish the theorem formulated above, a new method was needed for proving the necessity of its conditions.

From Theorem 1, in particular, it follows that if the functions \(c(t)\) and \(d(t)\in H_\mu(\Gamma)\), then the spectrum of the operator \(A\) does not depend on the choice of the numbers \(\alpha_1,\ldots,\alpha_n\) satisfying relations (2). This assertion ceases to be true if the coefficients \(c(t)\) and \(d(t)\) have discontinuities of the first kind at certain fixed points on \(\Gamma\) (see Theorem 2).

1. Let \(x\) and \(\xi\) be two points of the complex plane; \(\mu\) real, and \(\alpha\) a complex number, with \(0<\mu<1,\ \mu<\operatorname{Re}\alpha<\mu+1\). By \(\nu_{\mu,\alpha}(z,\xi)\) we denote an arc of a circle joining the points \(z\) and \(\xi\) and possessing the following properties: (I) in the case \(\operatorname{Re}\alpha-\mu<1/2\), from the interior points of the arc \(\nu_{\mu,\alpha}(z,\xi)\) the segment joining the points \(z\) and \(\xi\) is seen under the angle \(\theta=(\operatorname{Re}\alpha-\mu)2\pi\), and the direction from \(z\) to \(\xi\) along the arc \(\nu_{\mu,\alpha}(z,\xi)\) is counterclockwise; (II) in the case \(\operatorname{Re}\alpha-\mu>1/2\), put
   \[ \nu_{\mu,\alpha}(z,\xi)=\nu_{\mu,\beta}(\xi,z), \]
   where \(\beta=1+2\mu-\alpha\); (III) in the case \(\operatorname{Re}\alpha-\mu=1/2\), by \(\nu_{\mu,\alpha}(z,\xi)\) we denote the segment joining the points \(z\) and \(\xi\).

By \(H_\mu(\Gamma,c_1,\ldots,c_n)\) we denote the set of functions continuous from the left on \(\Gamma\), satisfying everywhere the Hölder condition with exponent \(\mu\), except, possibly, at the points \(c_1,\ldots,c_n\), at which they may have discontinuities of the first kind.

Let \(a(t)\in H_\mu(\Gamma,c_1,\ldots,c_n)\), and let \(\omega=(\mu,\alpha_1,\ldots,\alpha_n)\) denote a vector with coordinates satisfying relations (2). To the function \(a(t)\) and the vector \(\omega\) we associate a continuous, closed, naturally oriented curve \(V_\omega(a)\), obtained by adding \(n\) arcs \(\nu_{\mu,\alpha_k}(a(c_k),a(c_k+0))\) to the set of values of the function \(a(t)\). The function \(a(t)\) will be called \(\omega\)-nonexceptional if \(0\notin V_\omega(a)\). The index (or \(\omega\)-index*) of an \(\omega\)-nonexceptional function is the number of turns of the curve \(V_\omega(a)\) about the point \(z=0\):
\[ \operatorname{ind}_{\omega} a=\frac{1}{2\pi}[\arg V_\omega(a)]_{\Gamma}. \]

Theorem 2. Let \(c(t)\) and \(d(t)\in H_\mu(\Gamma,c_1,\ldots,c_n)\); let the vector
\[ \omega=(\mu,\alpha_1,\ldots,\alpha_n) \]
satisfy relations (2), and let
\[ \rho(t)=\prod_{k=1}^n (t-c_k)^{\alpha_k}. \]
In order that the operator
\[ A=cI+dS \quad (A=cI+SdI) \tag{3} \]
be a \(\Phi_+\)- or \(\Phi_-\)-operator in the space \(H_\mu^0(\Gamma,\rho)\), it is necessary and sufficient that the following two conditions be satisfied:

I. \(\inf |b(t)|>0\) \((t\in\Gamma)\).

II. The function \(a(t)/b(t)\) is \(\omega\)-nonexceptional, where \(a(t)=c(t)+d(t)\) and \(b(t)=c(t)-d(t)\).

If conditions I and II are satisfied and \(\chi=\operatorname{ind}_{\omega}[a/b]\), then:

1) for \(\chi>0\) the operator \(A\) is left invertible in \(H_\mu^0(\Gamma,\rho)\) and \(\dim\operatorname{coker} A=\chi\);

2) for \(\chi<0\) the operator \(A\) is right invertible in \(H_\mu^0(\Gamma,\rho)\) and \(\dim\ker A=-\chi\);

3) for \(\chi=0\) the operator \(A\) is invertible in \(H_\mu^0(\Gamma,\rho)\).

* The orientation of the curve \(V_\omega(a)\) is chosen so that, on the intervals of continuity of the function \(a(t)\), the motion along the curve \(V_\omega(a)\) is determined by the motion of the variable \(t\) on \(\Gamma\) in the positive direction, and along the arcs \(\nu_{\mu,\alpha_k}(a(c_k),a(c_k+0))\)—from the point \(a(c_k)\) to the point \(a(c_k+0)\).

It is not difficult to verify that conditions I and II are equivalent to the following:

I′. \(\inf |a(t)|>0\) and \(\inf |b(t)|>0\) \((t\in\Gamma)\).

II′. \(\beta_k\ne \operatorname{Re}\alpha_k-\mu\), where
\[ \beta_k=(1/2\pi)\arg\,[a(c_k)b(c_k+0)/a(c_k+0)b(c_k)]. \]

Let conditions I and II be fulfilled. Consider the function
\[ \psi(t)=\prod_{k=1}^{n} t^{\gamma_k}, \]
where the numbers
\[ \gamma_k=(1/2\pi)\ln [a(c_k)b(c_k+0)/a(c_k+0)b(c_k)] \]
are chosen so that
\[ \operatorname{Re}\alpha_k-\mu>\operatorname{Re}\gamma_k>\operatorname{Re}\alpha_k-\mu-1. \]
The latter is possible by virtue of condition II′. We also agree that the point of discontinuity for the function \(t^{\gamma_k}\) (if \(\gamma_k\) is not an integer) is the point \(c_k\). It is easy to see that the function
\[ g(t)=a(t)/b(t)\psi(t) \]
belongs to the class \(H_\mu(\Gamma)\) and \(g(t)\ne 0\) \((t\in\Gamma)\). If
\[ g_\pm(t)=\exp\,[\,\tfrac12((I\pm S)t^{-\varkappa}g)(t)\,], \]
where
\[ \varkappa=(1/2\pi)[\arg g(t)]_\Gamma, \]
then one of the inverse operators to the operator \(A=cI+dS\) (the left inverse for \(\varkappa\ge 0\), the right inverse for \(\varkappa\le 0\)) can be written in the form
\[ A^{-1}=g_+^{-1}(t)\left[\frac{1+g(t)}{2}I+\frac{1-g(t)}{2}S\right]\times \]
\[ \times \psi_+^{-1}(t)\left[\frac{t^{-\varkappa}+\psi(t)}{2}I+\frac{t^{-\varkappa}-\psi(t)}{2}S\right] b^{-1}(t)g_-^{-1}(t)\psi_-^{-1}(t)I, \]
where
\[ \psi_+(t)=\prod_{k=1}^{n}(t-c_k)^{\alpha_k},\qquad \psi_-(t)=\prod_{k=1}^{n}\left(\frac{t}{t-c_k}\right)^{\alpha_k}. \]

Theorem 2, in its sufficient part, is a refinement of certain results of N. I. Muskhelishvili (see \((^1)\), § 97). From Theorem 2 the following theorem on the spectrum of the operator \(A\), defined by equality (3), is easily derived.

Theorem 3. Let \(c(t)\) and \(d(t)\in H_\mu(\Gamma,c_1,\ldots,c_n)\), and let the vector
\[ \omega=(\mu,\alpha_1,\ldots,\alpha_n) \]
satisfy relations (2), and
\[ \rho(t)=\prod_{k=1}^{n}(t-c_k)^{\alpha_k}. \]

Then the complement \(C\Phi_A\) to the \(\Phi\)-set* of the operator \(A\) in the space \(H_\mu^0(\Gamma,\rho)\) consists of the union of the set of values of the functions \(a(t)\) and \(b(t)\) and the set of complex numbers \(\lambda\) satisfying, for at least one pair of numbers \(k\) and \(x\) \((k=1,\ldots,n;\ 0\le x\le 1)\), the equation
\[ [a(c_k)-\lambda]\frac{\sin\theta_k(1-x)}{\sin\theta_k}e^{ix\theta_k} + [a(c_k+0)-\lambda]\frac{\sin\theta_k x}{\sin\theta_k}e^{i\theta_k(x-1)} =0, \]
where
\[ \theta_k=\pi-2\pi(\operatorname{Re}\alpha_k-\mu). \]

The spectrum of the operator \(A\) in \(H_\mu^0(\Gamma,\rho)\) consists of all points of \(C\Phi_A\) and the points
\[ \lambda\in\Phi_A, \]
for which
\[ \operatorname{ind}_\omega[(a-\lambda)/(b-\lambda)]\ne 0. \]

1. By \(H_\mu^m(\Gamma,\rho)\) we denote the space of vector-functions
   \[ \Psi=(\psi_1,\ldots,\psi_n) \]
   with components \(\psi_j\in H_\mu^0(\Gamma,\rho)\). Consider in the space \(H_\mu^m(\Gamma,\rho)\) the matrix singular integral operators
   \[ (K\Psi)(t)=\mathscr{G}(t)\Psi(t)+\mathscr{D}(t)(S\Psi)(t), \]
   \[ (L\Psi)(t)=\mathscr{G}(t)\Psi(t)+(S\mathscr{D}\Psi)(t), \]
   where \(\mathscr{G}(t)\) and \(\mathscr{D}(t)\) are matrix-functions of order \(m\), each element of which belongs to the set \(H_\mu^m(\Gamma,c_1,\ldots,c_n)\) (i.e. \(\mathscr{G}(t)\) and \(\mathscr{D}(t)\in H_\mu^m(\Gamma,c_1,\ldots,c_n)\)), and \(S\) is the matrix operator of singular integration.

If
\[ \mathscr{G}(t)=\|g_{ij}\|_{i,j=1}^{m}, \]
then we associate with it the discontinuous matrix-function
\[ V_\omega(\mathscr{G})=\|V_\omega(g_{ij})\|_{i,j=1}^{m}, \]
where \(\omega=(\mu,\alpha_1,\ldots,\alpha_n)\) is a vector whose coordinates satisfy relations (2).

* The \(\Phi\)-set of the operator \(A\) is the set of those points \(\lambda\) of the complex plane for which the operator \(A-\lambda I\) is a \(\Phi\)-operator.

We shall call a matrix-function \(\mathcal G(t)\in H_\mu^m(\Gamma,c_1,\ldots,c_n)\) \(\omega\)-nonsingular if the function \(\det V_\omega(\mathcal G)\) does not vanish.

Theorem 4. Let \(\mathcal G(t)\) and \(\mathcal D(t)\in H_\mu^m(\Gamma,c_1,\ldots,c_n)\). In order that the operator \(K=\mathcal GI+\mathcal DS\) \((L=\mathcal GI+S\mathcal DI)\) be a \(\Phi\)-operator in the space \(H_\mu^m(\Gamma,\rho)\), it is necessary and sufficient that the following two conditions be fulfilled:

I. \(\inf |\det \mathcal B(t)|>0\) \((t\in\Gamma)\).

II. The matrix \(\mathcal B^{-1}\mathcal A\) \((\mathcal A\mathcal B^{-1})\) is \(\omega\)-nonsingular, where \(\mathcal A(t)=\mathcal G(t)+\mathcal D(t)\) and \(\mathcal B(t)=\mathcal G(t)-\mathcal D(t)\).

If conditions I and II are fulfilled, then

\[ \operatorname{ind}K=\operatorname{ind}\det V_\omega(\mathcal B^{-1}\mathcal A), \qquad \bigl(\operatorname{ind}L=\operatorname{ind}\det V_\omega(\mathcal A\mathcal B^{-1})\bigr)^*. \]

It is easily verified that conditions I and II are equivalent to the following:

I′. \(\inf |\det \mathcal A(t)|>0,\ \inf |\det \mathcal B(t)|>0\) \((t\in\Gamma)\).

II′. For each eigenvalue \(\lambda_j^{(k)}\) of the matrix

\[ \mathcal A^{-1}(c_k+0)\mathcal B(c_k+0)\mathcal B^{-1}(c_k)\mathcal A(c_k) \]

\[ \bigl(\mathcal B(c_k+0)\mathcal A^{-1}(c_k+0)\mathcal A(c_k)\mathcal B^{-1}(c_k)\bigr) \]

the relation

\[ \operatorname{Re}\alpha_k\ne \mu+\beta_j^{(k)} \qquad (j=1,\ldots,m;\ k=1,\ldots,n), \]

holds, where \(\beta_j^{(k)}=(1/2\pi)\arg \lambda_j^{(k)},\ 0\le \beta_j^{(k)}<1\).

1. All the results stated admit a generalization to the case when \(\Gamma\) consists of a finite number of closed smooth contours, and to the case when \(\Gamma\) consists of a finite number of smooth simple closed and open arcs. In doing so, the method described in \((^2)\) is used.

Let us illustrate the results of the second point for the case when \(A=S\) (i.e. \(c(t)=0\) and \(d(t)=1\)) and \(\Gamma\) is a contour consisting of \(p\) simple smooth oriented open arcs. Denote by \(c_1,\ldots,c_{2p}\) the ends of these arcs, and by \(\rho(t)\) the weight defined by the equality

\[ \rho(t)=\prod_{k=1}^{2p}(t-c_k)^{\alpha_k}, \]

where \(0<\mu<1,\ \mu<\operatorname{Re}\alpha_k<\mu+1\) \((k=1,\ldots,2p)\).

Theorem 5. In order that the operator \(S\) be a \(\Phi_+\)- or \(\Phi_-\)-operator in the space \(H_\mu^0(\Gamma,\rho)\), it is necessary and sufficient that \(\operatorname{Re}\alpha_k-\mu\ne \tfrac12\) for all \(k=1,\ldots,2p\).

Let \(\operatorname{Re}\alpha_k-\mu\ne \tfrac12\) for all \(k\), and let \(q\) be the number of points \(c_k\) at which \(\operatorname{Re}\alpha_k-\mu>\tfrac12\).

Then:

1) if \(p>q\), the operator \(S\) is right invertible in \(H_\mu^0(\Gamma,\rho)\) and \(\dim\ker S=p-q\);

2) if \(p<q\), the operator \(S\) is left invertible in \(H_\mu^0(\Gamma,\rho)\) and \(\dim\operatorname{coker}S=q-p\);

3) if \(p=q\), the operator \(S\) is invertible in \(H_\mu^0(\Gamma,\rho)\).

The complement \(C\Phi_A\) to the \(\Phi\)-set of the operator \(S\) is the union of \(2p\) arcs of \(Z\)-circumferences, each of which joins the points \(-1\) and \(1\) and passes through one of the corresponding points \(i\operatorname{ctg}[\pi(\operatorname{Re}\alpha_k-\mu)]\) or \(-i\operatorname{ctg}[\pi(\operatorname{Re}\alpha_k-\mu)]\), depending on whether \(c_k\) is the end or the beginning of the corresponding arc.

Institute of Mathematics with Computing Center
Academy of Sciences of the Moldavian SSR
Kishinev

Tbilisi Mathematical Institute named after A. M. Razmadze
Academy of Sciences of the Georgian SSR

Received
4 VII 1969

References

1. N. I. Muskhelishvili, Singular Integral Equations. Moscow, 1968.
2. I. Ts. Gokhberg, N. Ya. Krupnik, DAN, 185, No. 4 (1969).
3. I. Gokhberg, N. Krupnik, Studia Math., 31, 347 (1968).
4. I. Ts. Gokhberg, N. Ya. Krupnik, DAN, 186, No. 5 (1969).
5. M. S. Budyanu, I. Ts. Gokhberg, Mathematical Research, Kishinev, 3, issue 3 (1968).
6. I. Ts. Gokhberg, M. G. Krein, UMN, 12, issue 2 (74) (1957).
7. I. Ts. Gokhberg, M. K. Zambitskii, Ukrainian Mathematical Journal, 18, issue 1 (1966).

\[ {}^*\operatorname{ind}K=\dim\ker K-\dim\operatorname{coker}K. \]