UDC 513.88

MATHEMATICS

V. R. PORTNOV

ON A PROJECTION OPERATOR OF S. L. SOBOLEV TYPE

(Presented by Academician S. L. Sobolev on 9 IV 1969)

In the work of S. L. Sobolev (^1), for the space \(L_p^{(m)}(\Omega)\) a projection operator \(\Pi\) was constructed whose range is the space of polynomials of degree not greater than \(m - 1\). This operator \(\Pi\) has the property that for any function \(u(x) \in L_p^{(m)}(\Omega)\) there is a representation

\[ u(x) - \Pi u(x) = \sum_{|\alpha|=m} Q_\alpha \bigl(D^\alpha u(x)\bigr), \tag{1} \]

where \(Q_\alpha\) is a certain linear bounded operator mapping \(L_p(\Omega)\) into itself. Representations of the form (1) were later generalized in various directions (see (^2–^4)). In (^5), applications of such representations were given to the proof of solvability of the first boundary-value problem for one class of degenerate equations.

In the present note a projection operator \(\Pi\) is constructed, and a representation of the form (1) is obtained for a certain collection \(\mathscr L_1, \mathscr L_2, \ldots, \mathscr L_R\) of partial differential operators that are products of ordinary differential operators with respect to different variables. The corresponding result of (^5) is contained here as a special case.

Let us first introduce a number of notations: \(n\) and \(R\) are a fixed pair of natural numbers; the letters \(i\) and \(k\) below will denote natural numbers not exceeding \(n\) and \(R\), respectively; \(\bar E\) is the extended real line; \(E_n\) is the \(n\)-dimensional Euclidean space of points \(x = (x_1, \ldots, x_n)\). The remaining notations will be introduced in the course of the exposition.

Suppose that for each \(i\) an interval \((a_i, c_i)\) is given on the axis \(Ox_i\) of the space \(E_n\), where \(a_i\) and \(c_i \in \bar E\). This means that \(-\infty \le a_i < c_i \le \infty\). Put \(D = (a_1, c_1) \times \cdots \times (a_n, c_n)\).

Let, further, to each \(k\) there correspond a set \((m_{1,k}, m_{2,k}, \ldots, m_{n,k})\) of \(n\) nonnegative integers and an operator \(\mathscr L_k = \mathscr L_{1,k}\mathscr L_{2,k}\ldots \mathscr L_{n,k}\), whose domain is the space \(C^{(\infty)}(D)\), and for each \(i\)

\[ \mathscr L_{i,k} = \begin{cases} \dfrac{\partial^{m_{i,k}}}{\partial x_i^{m_{i,k}}} + \displaystyle\sum_{j=1}^{m_{i,k}} b_{i,k}^{(j)}(x_i) \dfrac{\partial^{j-1}}{\partial x_i^{j-1}}, & \text{if } m_{i,k} > 0,\\[1.2em] I \ (I \text{ is the identity operator}), & \text{if } m_{i,k} = 0, \end{cases} \]

where \(b_{i,k}^{(j)}(x_i) \in L_1^{\mathrm{loc}}(D)\) for all \(j = 1, \ldots, m_{i,k}\).

If \(m_{i,k} > 0\), then the operator \(\mathscr L_{i,k}\) may be regarded as an ordinary differential operator on the interval \((a_i, c_i)\). Let \(\{\varepsilon_{i,k}^{(j)}(x_i)\}\) \((j = 1, \ldots, m_{i,k})\) be its fundamental system of solutions on the indicated interval. We denote the linear span of this system by \(S_{i,k}\). If \(m_{i,k} = 0\), then put \(S_{i,k} = \{0\}\), where \(\{0\}\) denotes the set consisting of the single function identically equal to zero on \((a_i, c_i)\). This notation will be used below.

For each pair \((i,k)\) let us consider the set \(\Psi_{i,k}\) of all possible tuples \(\tau=(k_1,\ldots,k_l)\) \((1\leq l\leq R\) and \(l\), generally speaking, depends on \(\tau)\) of natural numbers satisfying the following conditions: 1) \(k<k_1<\cdots<k_l\leq R\); 2) there is a tuple \((i_1,i_2,\ldots,i_R)\) of \(R\) natural numbers not exceeding \(n\) such that \(m_{i_1,1}>0,\ldots,m_{i_R,R}>0\), and, moreover,
\[ i_k=i_{k_1}=\cdots=i_{k_l}=i, \]
while \(i_j\ne i\) for \(j>k\) and \(j\ne k_1,\ldots,j\ne k_l\).

For some pairs \((i,k)\) it may happen that \(\Psi_{i,k}=\varnothing\). For example, \(\Psi_{i,R}=\varnothing\) for all \(i\), and \(\Psi_{i,k}=\varnothing\) when \(m_{i,k}=0\).

If \(\Psi_{i,k}\ne\varnothing\), then for any tuple \(\tau=(k_1,\ldots,k_l)\in\Psi_{i,k}\) we introduce the sets \(S^{(\tau)}_{i,k}\) and \(\Gamma^{(\tau)}_{i,k}\) of functions defined on the interval \((a_i,c_i)\):
\[ S^{(\tau)}_{i,k}=\bigcap_{1\leq j\leq l} S_{i,k_j};\qquad \Gamma^{(\tau)}_{i,k}=S_{i,k}\cap S^{(\tau)}_{i,k}. \]

Now, for each pair \((i,k)\), we formulate

Condition A. One of the following assertions is true:
1) \(\Psi_{i,k}=\varnothing\);
2) \(\Psi_{i,k}\ne\varnothing\), and for every \(\tau\in\Psi_{i,k}\) there exists a subspace
\[ \nabla^{(\tau)}_{i,k}\subset S^{(\tau)}_{i,k} \]
such that
\[ S^{(\tau)}_{i,k}=\Gamma^{(\tau)}_{i,k}\oplus \nabla^{(\tau)}_{i,k}, \]
and
\[ S_{i,k}\cap \nabla_{i,k}=\{0\}, \]
where \(\nabla_{i,k}\) is the space spanned by
\[ \bigcup_{\tau\in\Psi_{i,k}}\nabla^{(\tau)}_{i,k}. \]
(For the notation and terminology, see \((^6)\), pp. 30 and 44, 46.)

Everywhere in what follows it is assumed that for every pair \((i,k)\) condition A is fulfilled and, in the case when \(\Psi_{i,k}\ne\varnothing\), the family of spaces
\[ \{\nabla^{(\tau)}_{i,k}\}\quad(\tau\in\Psi_{i,k}) \]
is fixed.

Lemma. Let \(m_{i,k}>0\). Then there exists a system of functions
\[ \{f^{(j)}_{i,k}(x_i)\}\quad (j=1,\ldots,m_{i,k}), \]
defined on the interval \((a_i,c_i)\), finite and infinitely differentiable, which has the property that
\[ \int_{a_i}^{c_i} f^{(j)}_{i,k}(x_i)e^{(s)}_{i,k}(x_i)\,dx_i=\delta_{js}, \]
where \(\delta_{js}\) is the Kronecker symbol, and, in the case when \(\Psi_{i,k}\ne\varnothing\),
\[ \int_{a_i}^{c_i} f^{(j)}_{i,k}(x_i)v(x_i)\,dx_i=0 \]
for any function \(v(x_i)\in\nabla_{i,k}\) and for all \(j=1,\ldots,m_{i,k}\).

If \(m_{i,k}>0\), then the functions \(\{e^{(j)}_{i,k}(x_i)\}\) and the functions \(\{f^{(j)}_{i,k}(x_i)\}\) \((j=1,\ldots,m_{i,k})\) appearing in the lemma will be regarded as fixed and defined on the whole domain \(D\).

We construct the projection operator \(\Pi_{i,k}\), defining it on functions \(u(x)\) from \(L^{\mathrm{loc}}_1(D)\), in the case \(m_{i,k}>0\), as follows:
\[ \Pi_{i,k}u(x)= \sum_{j=1}^{m_{i,k}} \left( \int_{a_i}^{c_i} f^{(j)}_{i,k}(\xi_i)\, u(x_1,\ldots,x_{i-1},\xi_i,x_{i+1},\ldots,x_n)\,d\xi_i \right) e^{(j)}_{i,k}(x_i). \]

If \(m_{i,k}=0\), then \(\Pi_{i,k}\) will be regarded as the zero operator on the space \(L^{\mathrm{loc}}_1(D)\).

We now define the operators \(\Pi_k\) and \(\Pi\), mapping the space \(L^{\mathrm{loc}}_1(D)\) into itself. Put
\[ \Pi_k=I-(I-\Pi_{1,k})(I-\Pi_{2,k})\cdots(I-\Pi_{n,k}) \qquad (k=1,\ldots,R), \]
\[ \Pi=\Pi_1\Pi_2\cdots\Pi_R \qquad (I\text{ is the identity operator}). \]

Theorem 1. The following equalities hold:

\[ \Pi_k^2=\Pi_k,\qquad \Pi_k\Pi=\Pi\quad (k=1,\ldots,R);\qquad \Pi^2=\Pi. \]

Definition. An additive and homogeneous operator \(Q\) mapping the space \(L_1^{\mathrm{loc}}(D)\) into itself will be called strongly continuous if for every compact set \(K^{(1)}\subset D\) there exists a compact set \(K^{(2)}\subset D\), depending, generally speaking, on \(K^{(1)}\), such that for all functions \(w(x)\in L_1^{\mathrm{loc}}(D)\) the inequality

\[ \|Qw(x)\|_{L_1(K^{(1)})}\le C\|w(x)\|_{L_1(K^{(2)})}, \]

holds, where \(C\) is a real number independent of \(w(x)\).

Theorem 2. For any function \(u(x)\in C^{(\infty)}(D)\) and any \(k\), the equality

\[ u(x)-\Pi_k u(x)=T_k(\mathscr{L}_k u(x)), \tag{2} \]

holds, where \(T_k\) is a strongly continuous operator in the space \(L_1^{\mathrm{loc}}(D)\), possessing the property that \(T_k w(x)=0\) implies \(w(x)=0\).

Theorem 3. For any function \(u(x)\in C^{(\infty)}(D)\) and any collection \(\beta=(\beta_1,\ldots,\beta_n)\) of nonnegative integers such that \(\beta_i\le \min_{1\le k\le R} m_{i,k}\) \((i=1,\ldots,n)\), the equality

\[ \mathscr{D}^{\beta}(u(x)-\Pi u(x)) = \sum_{k=1}^{R} Q_k^{(\beta)}(\mathscr{L}_k u(x)), \tag{3} \]

holds, where \(Q_k^{(\beta)}\) \((k=1,\ldots,R)\) are strongly continuous operators in the space \(L_1^{\mathrm{loc}}(D)\).

Remark 1. The operators \(\Pi_{i,k}\), \(\Pi_k\), and \(\Pi\) are, obviously, strongly continuous in the space \(L_1^{\mathrm{loc}}(D)\).

Remark 2. Equalities (2) and (3) are in fact valid, of course, for a class of functions broader than \(C^{(\infty)}(D)\). This class will not be defined here; however, let us note that, for example, equality (3) with \(\beta=(0,\ldots,0)\), and in the case when all the operators \(\mathscr{L}_{i,k}\) with \(m_{i,k}>0\) have constant coefficients, is valid for all functions for which in the domain \(D\) there exist generalized, in the sense of S. L. Sobolev, operators \(\mathscr{L}_1,\mathscr{L}_2,\ldots,\mathscr{L}_R\), as is easily verified by means of passage to the limit in equality (3).

Received
24 III 1969

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