V. G. KHRIPTUN

FUNCTIONS ANALYTIC WITH RESPECT TO A HYPERBOLIC OPERATOR WITH TWO INDEPENDENT VARIABLES

(Presented by Academician V. I. Smirnov on 28 IV 1958)

Mathematics

§ 1. The subject of this work is the extension of certain results of the theory of operator-analytic functions, obtained by M. K. Fage \((^1)\) for functions of one variable, to functions of two variables that are analytic with respect to a hyperbolic operator of second order.

Let the hyperbolic operator

\[ H=\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2} +p(x,y)\frac{\partial}{\partial x}+q(x,y)\frac{\partial}{\partial y} +r(x,y)I \tag{1} \]

be given in the square \(D=(|x|+|y|<1)\), whose sides are characteristics; moreover \(p(x,y)\) and \(q(x,y)\) have continuous partial derivatives of first order in \(D\), and \(r(x,y)\) is continuous. For \(|x_0|<a\leq 1\), denote by \(D(x_0,a)\) the square \(|x-x_0|+|y|<a-|x_0|\) lying in \(D\); by \(D(x_0,a,\varepsilon)\) denote the intersection of \(D(x_0,a)\) with the strip \(|x-x_0|<\varepsilon\).

Definition 1. A function \(f(x,y)\), given in the domain \(D\), will be called infinitely \(H\)-differentiable in \(D\) if, for every \(k=0,1,2,3,\ldots\), the functions \(H^k f(x,y)\) have continuous partial derivatives up to the second order inclusive.

Definition 2. An infinitely \(H\)-differentiable function \(f(x,y)\) in the domain \(D\) will be called \((H,x)\)-analytic in \(D\) if, for every closed domain \(R\subset D\), there exists a constant \(C>0\) (depending on \(f,H\), and \(R\)) such that in \(R\) the inequalities

\[ \left| \frac{\partial^s}{\partial x^s}H^k f(x,y) \right| \leq C^{2k+s}(2k+s)! \tag{2} \]

hold for all \(s=0,1;\ k=0,1,2,\ldots\), with the possible exception of \(k=s=0\) (in the exponent \(2k+s\), the number two is the order of the operator \(H\) with respect to the variable \(x\)).

By an operator inverse to the operator \(H\) we shall mean an integral operator \(K_{x_0}\) that transforms a function \(\varphi(x,y)\), given and continuous in the domain \(D(x_0,1)\), into the solution of the equation \(Hu=\varphi(x,y)\) under zero initial conditions on the segment \(x=x_0\). By \(C^{(m)}(a,b)\) we denote the class of functions continuously differentiable \(m\) times on the interval \((a,b)\). For \(s=0,1,\ |x_0|<1\), introduce the operator \(K^{(s)}_{x_0}\), which transforms functions \(\varphi(y)\in C^{(2-s)}(|x_0|-1,\,1-|x_0|)\) into solutions of the equation \(Hu=0\) under the following initial conditions:

\[ \text{if } s=0,\qquad u(x_0,y)=\varphi(y),\qquad u'_x(x_0,y)=0; \]

\[ \text{if } s=1,\qquad u(x_0,y)=0,\qquad u'_x(x_0,y)=\varphi(y). \]

To study the structure of an \((H,x)\)-analytic function \(f(x,y)\), given in the domain \(D\), for each \(x_0\in(-1,+1)\) we construct a system of functions \(\varphi_{s,k}(y)\)

by the formulas

\[ \varphi_{s,k}(y)=\left.\frac{\partial^s}{\partial x^s}H^k f(x,y)\right|_{x=x_0},\qquad s=0,1,\quad k=0,1,2,\ldots^* . \tag{3} \]

Let us note here that \(\varphi_{s,k}(y)\in C^{(2-s)}(|x_0|-1,\;1-|x_0|)\), and on every segment
\([-a,a]\subset (|x_0|-1,\;1-|x_0|)\) the inequalities

\[ |\varphi_{s,k}(y)|\leq C^{2k+s}(2k+s)! \tag{4} \]

hold.

Transferring the scheme for constructing the theory of \(L\)-Taylor series for functions of one variable (1), we establish the validity of the following theorems.

Theorem 1. Let the function \(f(x,y)\) be \((H,x)\)-analytic in \(D\). Then, for any \(x_0\in(-1,+1)\) and \(a\in(0,1)\), there exists an \(\varepsilon>0\) \((\varepsilon<1-|x_0|)\) such that in \(D(x_0,a,\varepsilon)\) the function \(f(x,y)\) has an expansion into an “\((H,x)\)-series”:

\[ f(x,y)=\sum_{k=0}^{\infty}K_{x_0}^k\sum_{s=0}^{1}K_{x_0}^{(s)}\varphi_{s,k}(y). \tag{5} \]

Theorem 2. The functions \(\varphi_{s,k}(y)\) of formula (5) are determined by the function \(f(x,y)\) according to formulas (3), i.e. every \((H,x)\)-series is an “\((H,x)\)-Taylor series.”

Theorem 3. An \((H,x)\)-series admits termwise \(H\)-differentiation any number of times.

Theorem 4. If on the diagonal \(\Delta(x=x_0)\) of the square \(D(x_0,1)\) there is given a system of functions \(\varphi_{s,k}(y)\in C^{(2-s)}(|x_0|-1,\;1-|x_0|)\), satisfying conditions (4) on every closed segment \([-a,a]\subset \Delta\), then the function \(f(x,y)\), constructed by formula (5), is \((H,x)\)-analytic in the domain \(D(x_0,a,\varepsilon)\).

Here \(\varepsilon\) \((0<\varepsilon<1-|x_0|)\) is determined in an appropriate way for each \(a\in(0,1)\).

Definition 3. A system of functions \(\varphi_{s,k}(y)\in C^{(2-s)}(|x_0|-1,\;1-|x_0|)\), satisfying conditions (4), will be called a defining system for the \((H,x)\)-analytic function \(f(x,y)\) constructed by formula (5).

§ 2. Let \(x_0\in(-1,+1)\); denote by \(A_{H,x_0}\) the set of functions each of which is defined and \((H,x)\)-analytic in some (its own) domain \(D(x_0,a,\varepsilon)\). The set of functions \(A_{H,x_0}\) is a linear vector space. We introduce a topology in \(A_{H,x_0}\) as follows. A sequence of functions \(f_m(x,y)\in A_{H,x_0}\) \((m=0,1,2,\ldots)\) will be called regularly convergent if the conditions are satisfied: a) there is a common domain \(D(x_0,a,\varepsilon)\) in which all \(f_m(x,y)\) are defined and satisfy inequalities of the form (2); b) each \(\frac{\partial^s}{\partial x^s}H^k\)-derivative \(\frac{\partial^s}{\partial x^s}H^k f_m(x,y)\) tends in this domain uniformly to some limit.

Along with the operator \(H\), consider another operator \(\overline H\) of the form (1); for simplicity of exposition we assume that both operators are given in the domain \(D\). We construct the topological spaces \(A_{H,x_0}\) and \(A_{\overline H,x_0}\). They consist respectively of sums

\[ f(x,y)=\sum_{k=0}^{\infty}K_{x_0}^k\sum_{s=0}^{1}K_{x_0}^{(s)}\varphi_{s,k}(y),\qquad g(x,y)=\sum_{k=0}^{\infty}\overline K_{x_0}^{\,k}\sum_{s=0}^{1}\overline K_{x_0}^{(s)}\varphi_{s,k}(y), \]

where \(\varphi_{s,k}(y)\) and \(\overline\varphi_{s,k}(y)\) are functions of the defining systems, while \(\overline K_{x_0}\) and \(\overline K_{x_0}^{(s)}\) are operators constructed for \(\overline H\) in the same way as \(K_{x_0}\) and \(K_{x_0}^{(s)}\) for \(H\).

We now define the transformation \(T=T_{\overline H,x_0;\,H,x_0}\) of the space \(A_{\overline H,x_0}\) onto \(A_{H,x_0}\) by the formula \(Tg(x,y)=f(x,y)\), if \(f(x,y)\) and \(g(x,y)\) are constructed according to one and the same defining system of functions \(\varphi_{s,k}(y)\). We obtain mutually—

\[ {}^* \text{The dependence of these functions on } x_0 \text{ is not explicitly indicated.} \]

a one-to-one and continuous mapping of the linear topological space \(A_{\bar H,x_0}\) onto \(A_{H,x_0}\). Since

\[ Hf(x,y)=\sum_{k=1}^{\infty} K_{x_0}^{\,k-1}\sum_{s=0}^{1}K_{x_0}^{(s)}\varphi_{s,k}(y),\qquad \bar H g(x,y)=\sum_{k=1}^{\infty}\bar K_{x_0}^{\,k-1}\sum_{s=0}^{1}\bar K_{x_0}^{(s)}\varphi_{s,k}(y), \]

these functions correspond to one another:

\[ Hf(x,y)=T\bar H g(x,y). \]

Substituting here \(g(x,y)=T^{-1}f(x,y)\), we obtain \(H=T\bar H T^{-1}\).

We have arrived at the theorem:

Theorem 5. All hyperbolic operators of the second order are locally equivalent.

§ 3. Comparing the results formulated above with the results of M. K. Fage \({}^{1}\) for functions analytic with respect to an ordinary linear differential operator

\[ L=\frac{d^n}{dt^n}+p_{n-1}(t)\frac{d^{\,n-1}}{dt^{\,n-1}}+\cdots+p_0(t)I,\qquad a_1<t<b_1, \tag{6} \]

with continuous coefficients, we come to the conclusions:

a) the role of the sequence of coefficients \(a_0,a_1,a_2,\ldots\) of the \(L\)-series

\[ \sum_{k=0}^{\infty} a_k f_k(t,t_0) \]

passes to the defining system of functions (3);

b) each term \(a_k f_k(t,t_0)\) of the \(L\)-series is replaced by a term of the \((H,x)\)-series (5), but from this latter term it is impossible to isolate functions of any “\(H\)-basis.”

§ 4. Functions analytic with respect to operators can be used in solving the Cauchy problem (in \(t\)) for equations of the form

\[ L^p u(x,y,t)=H^m u(x,y,t) \tag{7} \]

(where \(L\) is the operator (6) of order \(n\); \(H\) is the operator (1); \(p,m\) are natural numbers) with initial values

\[ \frac{\partial^s}{\partial t^s}L^q u(x,y,t_0)=\varphi_{s,q}(x,y) \tag{8} \]

\[ (s=0,1,2,\ldots,n-1;\ q=0,1,2,\ldots,p-1;\ a_1<t_0<b_1), \]

which are \((H,x)\)-analytic functions in the domain \(D\).

Theorem 6. If \(np\ge 2m\), then for every domain \(D(x_0,a)\subset D\) one can indicate an \(\varepsilon>0\) such that, for \(t_0\in(a_1,b_1)\), in the domain \((|x-x_0|+|y|<a-|x_0|,\ t_0-\varepsilon<t<t_0+\varepsilon)\): 1) the solution of problem (7)—(8) is given by the formula

\[ u(x,y,t)=\sum_{k=0}^{\infty}\sum_{s=0}^{n-1}\sum_{q=0}^{p-1} f_{s+nq+kpn}(t,t_0)\cdot H^{km}\varphi_{s,q}(x,y) \]

(here \(f_{s+nq+kpn}(t,t_0)\) are functions of the \(L\)-basis \({}^{1}\)); 2) this solution is an \((H,x)\)-analytic function in \(x,y\) and an \(L\)-analytic function in \(t\); 3) the solution is unique in the class of such functions.

In an analogous way one can construct functions analytic in \(x\) with respect to the parabolic operator

\[ \frac{\partial}{\partial x}-\frac{\partial^2}{\partial y^2}+p(x,y)I. \]

In conclusion, I express my sincere gratitude to the supervisor of the present work, M. K. Fage, for posing the problem and for valuable guidance in the course of its execution.

Chernivtsi
State University

Received
26 IV 1958

CITED LITERATURE

1. M. K. Fage, DAN, 112, No. 6 (1957).