UDC 517.946

GENERALIZATION OF THE LEIBNIZ–HÖRMANDER FORMULA

(Multiplication of differential operators with variable coefficients)

V. R. FRIDLENDER

In L. Hörmander’s work [1], p. 32, the following generalization of Leibniz’s formula is given:

\[ P(D)(uv)=P(D_u+D_v)uv=\sum_\gamma \frac{D^\gamma v}{\gamma!}\,P^{(\gamma)}(D)u, \tag{1} \]

where \(P(D)=\sum_\alpha a_\alpha D^\alpha\) is a differential operator with constant coefficients; \(\alpha=(\alpha_1,\ldots,\alpha_n)\); \(\gamma=(\gamma_1,\ldots,\gamma_n)\); \(\gamma!=\gamma_1!\cdots\gamma_n!\); \(u=u(x)\); \(v=v(x)\); \(x=(x_1,\ldots,x_n)\); \(|\gamma|=\gamma_1+\cdots+\gamma_n\);

\[ D^\alpha=D_1^{\alpha_1}\cdots D_n^{\alpha_n},\qquad D_j \equiv \frac{\partial}{\partial x_j};\qquad P^{(\gamma)}(D)\equiv \frac{\partial^{|\gamma|}P(D)}{\partial D_1^{\gamma_1}\cdots \partial D_n^{\gamma_n}} . \]

(The notation used here differs inessentially from Hörmander’s notation.)

Hörmander’s formula (1) may be regarded as a formula for multiplying the differential operator \(P(D)\) and the operator \(Q\) of multiplication by the function \(u(x)\). Therefore the question naturally arises of constructing an analogous formula for multiplying arbitrary differential operators, including those with variable coefficients. Such a formula proves useful in a number of questions in the general theory of partial differential equations.

Introduce the following notation:

\[ P\equiv P(x,D)\equiv \sum_\alpha a_\alpha(x)D^\alpha, \]

\[ Q\equiv Q(x,D)\equiv \sum_\beta b_\beta(x)D^\beta, \]

\(PQ\) is the operator product of \(P\) and \(Q\), i.e. \(PQu=P(Qu)\); \(P\times Q\) is the algebraic product of \(P\) and \(Q\), regarded as polynomials in \(D\); \(P_D^{(\gamma)}\) is the partial derivative of the polynomial \(P(x,D)\) of order \(\gamma=(\gamma_1,\ldots,\gamma_n)\) with respect to \(D=(D_1,\ldots,D_n)\), i.e.

\[ P_D^{(\gamma)} = \frac{\partial^{|\gamma|}P(x,D)} {\partial D_1^{\gamma_1}\cdots \partial D_n^{\gamma_n}} = \sum_{\alpha\ge \gamma} a_\alpha(x)\, \frac{\alpha!}{(\alpha-\gamma)!}\, D^{\alpha-\gamma}, \]

\[ Q_x^{(\gamma)}=\sum_{\beta} D^\gamma b_\beta(x)D^\beta \]
is the operator obtained from \(Q\) by differentiating the coefficients.

Theorem. The product of the operators \(P(x,D)\) and \(Q(x,D)\) is computed by the formula

\[ P(x,D)Q(x,D)=\sum_{\gamma}\frac{1}{\gamma!}\,P_D^{(\gamma)}\times Q_x^{(\gamma)} . \tag{2} \]

Proof. \(1^\circ\). Let \(P=D^\alpha\). Then, by Leibniz’s formula, we have

\[ PQ=D^\alpha\left(\sum_{\beta} b_\beta(x)D^\beta\right)= \]

\[ =\sum_{\beta}\sum_{\gamma\leq \alpha} \frac{\alpha!}{\gamma!(\alpha-\gamma)!}\, D^\gamma[b_\beta(x)]D^{\beta+\gamma-\gamma}= \]

\[ =\sum_{\gamma}\frac{1}{\gamma!}\sum_{\beta} \frac{\alpha!}{(\alpha-\gamma)!}\, D^{\alpha-\gamma}\times D^\gamma[b_\beta(x)]D^\beta= \]

\[ =\sum_{\gamma}\frac{1}{\gamma!}\sum_{\beta} P_D^{(\gamma)}\times D^\gamma[b_\beta(x)]D^\beta = \sum_{\gamma}\frac{1}{\gamma!}P_D^{(\gamma)}\times Q_x^{(\gamma)}, \]

as was required.

\(2^\circ\). The general case: \(P=\sum_{\alpha}a_\alpha(x)D^\alpha\),

\[ PQ=\sum_{\alpha}a_\alpha(x)D^\alpha Q = \sum_{\alpha}a_\alpha(x)\sum_{\gamma}\frac{1}{\gamma!} (D^\alpha)_D^{(\gamma)}\times Q_x^{(\gamma)} = \]

\[ = \sum_{\gamma}\frac{1}{\gamma!}\sum_{\alpha} a_\alpha(x)(D^\alpha)_D^{(\gamma)}\times Q_x^{(\gamma)} = \sum_{\gamma}\frac{1}{\gamma!}P_D^{(\gamma)}\times Q_x^{(\gamma)}, \]

which completes the proof.

Remarks and examples. \(1^\circ\). Formula (2) remains meaningful also in the case when the operators \(P\) and \(Q\) are matrix operators. The passage to this case is trivial.

\(2^\circ\). For \(n=1\), \(P=\dfrac{d^k}{dx^k}\), \(Qv=u(x)v\) (the multiplication operator), we obtain the usual Leibniz formula

\[ (uv)^{(k)} = \sum_{\gamma=0}^{k}\frac{1}{\gamma!}\, \frac{k!}{(k-\gamma)!}\, u^{(\gamma)}(x)v^{(k-\gamma)}(x). \]

\(3^\circ\). Hörmander’s formula (1) is obtained from (2) for arbitrary \(P\) and \(Qv=u(x)v\).

Literature

1. Hörmander L. On the theory of general differential operators in partial derivatives. IL, Moscow, 1959.

Received by the editors
September 16, 1965.

Kazan State Pedagogical Institute