MATHEMATICS

V. G. MAZ’YA, B. A. PLAMENEVSKII

ON SINGULAR EQUATIONS WITH A SYMBOL THAT VANISHES

(Presented by Academician A. D. Aleksandrov, 21 IX 1964)

By now, multidimensional integral singular equations with a symbol nowhere equal to zero have been well studied (see (1)). In this note singular equations in the plane are considered under the condition that the symbol may vanish on a finite number of rays. The equations are studied in spaces of generalized functions. The note is devoted mainly to equations of convolution type. In contrast to the case when the symbol does not degenerate, the solution of such equations is not determined uniquely. The general solution contains functions of the type of a plane wave. It is therefore natural to append to the equation additional conditions that would ensure its unique solvability. For equations of convolution type, three well-posed problems are considered. In conclusion, one problem is presented for an equation with a symbol weakly depending on the pole.

1. Let polar coordinates ((\rho,\theta)) be introduced in the plane (E_2). Denote by (\mathscr L_2^{(l)}) ((l) a positive integer) the space of functions on the circle that are square-summable together with derivatives up to order (l) inclusive. The space (\mathscr L_2^{(l)}), endowed with the norm

[
|\varphi|l^2=\int_0^{2\pi}\sum\right|^2\,d\theta,}^{l}\left|\frac{\partial^k\varphi}{\partial\theta^k
]

is a normed ring with respect to ordinary multiplication. Let (\mathscr L_2^{(-l)}) be the space conjugate to (\mathscr L_2^{(l)}) with respect to the scalar product ([f,\varphi]) in (\mathscr L_2) on the circle.

We shall consider functions defined on (E_2) as abstract functions of one variable (\rho), whose values belong to various spaces of functions on the circle. Introduce the space (H(\mathscr L_2^{(l)})) of functions of the variable (\rho) with values in (\mathscr L_2^{(l)}), square-summable on the half-axis ((0,\infty)) with weight (\rho). This space is provided with the norm

[
|u|_l^2=\int_0^\infty |u|_l^2\,\rho\,d\rho .
]

The space of functionals (H(\mathscr L_2^{(-l)})) is defined analogously.

The Fourier transform (\mathscr F u) is defined in the space (H(\mathscr L_2^{(l)})) and maps it isometrically onto itself. The space (H(\mathscr L_2^{(-l)})) is conjugate to (H(\mathscr L_2^{(l)})) with respect to the scalar product

[
(v,u)=\int_0^\infty [v,u]\rho\,d\rho .
]

Therefore the Fourier transform is also defined in (H(\mathscr L_2^{(-l)})). In order that a homogeneous function of degree zero (\Phi(z_1,z_2)=\Phi(\theta)) ((\theta) is the polar angle of the point (z)) be a multiplier in (H(\mathscr L_2^{(l)})), it is necessary and

it is sufficient that (\Phi(\theta)) belong to (\mathscr L_2^{(l)}). Consequently, in the spaces (H(\mathscr L_2^{(l)})) and (H(\mathscr L_2^{(-l)})) there is defined and bounded the singular integral operator (A=\mathcal F^{-1}\Phi\mathcal F) with symbol (\Phi(\theta)\in\mathscr L_2^{(l)}).

1. Suppose that the symbol (\Phi(\theta)) vanishes on the rays (\theta=\chi_1,\chi_2,\ldots,\chi_N); denote by (l_j) the multiplicity of the zero (\theta=\chi_j), and let
   [
   l=\max_j{l_j}.
   ]
   If (f\in L_2(E_2)), then there exists a solution of the equation (Au=f) belonging to the space (H(\mathscr L_2^{(-l)})). The general solution of the homogeneous equation has the form
   [
   \mathcal F u=\sum_{j=1}^{N}\sum_{k=0}^{l_j-1} c_{kj}(\rho)\,\delta^{(k)}(\theta-\chi_j),
   \tag{1}
   ]
   where (c_{kj}(\rho)) are arbitrary functions from (L_2[(0,\infty);\rho]).

Assume that for all (\lambda) the function (\Phi(\theta)-\lambda) vanishes only on a finite number of rays, and that the multiplicities of the zeros do not exceed a fixed number (l). Then the spectrum of the operator (A) in the space (H(\mathscr L_2^{(-l)})) coincides with the set of values of the symbol (\Phi(\theta)), and every point of the spectrum is an eigenvalue of infinite multiplicity. The form of the eigenfunctions follows directly from (1).

1. We pose the following problem:
   [
   Au=f,\qquad [A_k u,1]=\psi_k(\rho)\quad
   \left(k=1,\ldots,s=\sum_{j=1}^{N} l_j\right),
   \tag{2}
   ]
   where (A_k) are singular operators with symbols (\Phi_k(\theta)\in\mathscr L_2^{(l)}), and (\psi_k(\rho)) are known functions from (L_2[(0,\infty);\rho]).

Let (B) be a square matrix of order (s) of the form (B=(B_1,\ldots,B_N)), where
[
B_j=
\begin{pmatrix}
\Phi_1(\chi_j)\ldots \Phi_1^{(l_j-1)}(\chi_j)\
\Phi_2(\chi_j)\ldots \Phi_2^{(l_j-1)}(\chi_j)\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\
\Phi_s(\chi_j)\ldots \Phi_s^{(l_j-1)}(\chi_j)
\end{pmatrix}.
]
For the existence and uniqueness of a solution of problem (2) for arbitrary (\psi_k(\rho)), it is necessary and sufficient that the determinant (\det B) be nonzero. If this condition is fulfilled, the solution depends continuously on (f) and (\psi_k).

Considerations analogous to those carried out in items 2, 3 are also possible in the space of generalized functions over a countably normed space in which the topology is given by a countable system of norms (|u|_l) ((l=0,1,\ldots)).

1. Suppose that the multiplicity of each zero (\theta=\chi_j) ((j=1,\ldots,N)) of the symbol (\Phi) is equal to one. Choose Cartesian coordinates so that none of the rays (\theta=\chi_j) lies on the axis (x_1=0), and let the rays (\theta=\chi_j) ((j=1,\ldots,M)) project onto the positive half-axis (x_2=0), while the remaining rays project onto the negative one.

In posing the second and third problems we shall use the fact that ((-\Delta+E)^{-2}u\in\mathscr L_2^{(1)}) for almost all (\rho) for any (u\in H(\mathscr L_2^{(-1)})), and therefore the function ((-\Delta+E)^{-2}u) is defined almost everywhere on every straight line.

We shall seek a solution of the equation
[
Au=0,
\tag{3}
]
satisfying the conditions
[
(-\Delta+E)^{-2}A_k\operatorname{Re}u\big|_{x=0}=\varphi_k(x_1)
\quad (k=1,\ldots,N),
\tag{4}
]
where (A_k) are real singular operators with symbols (\Phi_k(\theta)). By (\varphi_k(x_1)) are denoted prescribed real functions from (L_2(-\infty,+\infty)),

subject to the condition

[
\int_{-\infty}^{+\infty} |\widetilde{\varphi}_k|^2 (1+t^2)^2 \frac{dt}{|t|} < \infty,
\tag{5}
]

where (\widetilde{\varphi}_k) is the Fourier transform of the function (\varphi_k).

For the existence and uniqueness of the solution of problem (3)—(4) it is necessary and sufficient that the determinant

[
\det
\begin{pmatrix}
\Phi_1(\theta_1) & \ldots & \Phi_1(\theta_M) & \Phi_1(\theta_{M+1}+\pi) & \ldots & \Phi_1(\theta_N+\pi)\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots\
\Phi_N(\theta_1) & \ldots & \Phi_N(\theta_M) & \Phi_N(\theta_{M+1}+\pi) & \ldots & \Phi_N(\theta_N+\pi)
\end{pmatrix}
]

not be equal to zero. Under this condition the solution depends continuously on (\varphi_k).

If the number of zeros of the symbol is even and the multiplicity of each zero is equal to one, then one can pose one more correct problem. Choose Cartesian coordinates so that (M) is equal to (N/2) (this is always possible). We shall seek a solution of the equation (Au=0) such that

[
(-\Delta+E)^{-2}A_k u\big|_{x_2=0}=\varphi_k(x_1)
\qquad (k=1),\ldots,N/2),
\tag{6}
]

where (A_k) are singular operators (not necessarily real) with symbols (\Phi_k). By (\varphi_k(x_1)) are denoted prescribed complex functions satisfying condition (5).

For the existence and uniqueness of the solution of problem (3)—(6) for all (\varphi_k), it is necessary and sufficient that the determinants

[
\det |\Phi_k(\chi_j)|,\qquad
\det |\Phi_k(\chi_{N/2+j})|
\qquad (k,j=1,\ldots,N/2)
]

not be equal to zero.

1. Let (f(x,z)) and (\Phi(x,z)) be functions homogeneous of degree zero with respect to (z). Consider the singular operator

[
Au=a(x)u(x)+\int_{E_2}\frac{f(x,y-x)}{r^2}\,u(y)\,dy
]

with an infinitely differentiable symbol (\Phi(x,z)\equiv\Phi(x,\theta)). Suppose that, for sufficiently large (|x|), the symbol depends only on (z). The operator (A) is bounded in the spaces (H(\mathscr L_2^{(1)})) and (H(\mathscr L_2^{(-1)})).

A bounded operator (T), acting from (H(\mathscr L_2^{(-1)})) into (L_2(E_2)), will be called smoothing; the symbol of (T), by definition, is equal to zero. A general singular operator will mean an operator of the form (A+T). For such operators the rule of multiplication of symbols is valid.

The curve (F(x)=\mathrm{const}) will be called characteristic for the operator (A+T) if

[
\Phi(x,\operatorname{grad} F)=0.
\tag{7}
]

Equation (7) remains invariant under a change of coordinates. Suppose that the symbol (\Phi(x,\theta)) of the operator (A) has the form

[
\Phi(x,\theta)=\bigl(e^{i\chi(x)}-e^{i\theta}\bigr)\Psi(x,\theta)
\qquad (\operatorname{Im}\chi\equiv0),
]

where (|\Psi(x,\theta)|>0), and that the characteristic curves are given by the equation
(x_1\cos\omega(x)+x_2\sin\omega(x)=\mathrm{const}). The functions (\Psi(x,\theta)-\Psi(\infty,\theta)) and (\omega(x)-\omega(\infty)) are identically equal to zero for large (|x|), and, together with their derivatives with respect to (x) and (\theta) of sufficiently high order, do not exceed in absolute value a prescribed small positive number.

Under these assumptions, the problem

[
Au=f,\qquad [u,1]=\psi(\rho),
]

where (f\in L_2(E_2)), (\psi(\rho)\in L_2[(0,\infty);\rho]), is uniquely solvable in the space (H(\mathcal L_2^{(-1)})).

For the proof, by means of a special change of variables preserving the space (H(\mathcal L_2^{(-1)})), we straighten the characteristic curves of the operator (A). Then, multiplying (A) by a certain operator with a nondegenerate symbol, we obtain a singular operator whose symbol does not depend on the pole. Finally, using the results of §§ 2, 3 and the method of successive approximations, we prove the existence and uniqueness of the solution of the problem.

The authors express their sincere gratitude to Prof. S. G. Mikhlin for his constant attention to, and discussion of, the work.

Leningrad State University
named after A. A. Zhdanov

Received
17 IX 1964

References

1. S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Moscow, 1962.