UDC 517.514

MATHEMATICS

Yu. G. Reshetnyak

GENERALIZED DERIVATIVES AND DIFFERENTIABILITY ALMOST EVERYWHERE

(Presented by Academician A. D. Aleksandrov, January 17, 1966)

Let \(G\) be an arbitrary open domain of \(n\)-dimensional Euclidean space \(R^n\). By the symbol \(W_p^l(G)\), where \(l \geq 1\) is an integer, \(p \geq 1\), we shall denote the set of all real functions that are defined and locally integrable in the domain \(G\) and such that, in the sense of the theory of generalized functions, all their partial derivatives of order \(l\), in the case \(p = 1\), are completely additive set functions defined on the \(\sigma\)-ring of Borel sets contained in \(G\), while for \(p > 1\) these derivatives are ordinary functions locally integrable in \(G\) to the power \(p\).

Further, \(B\) denotes the ball \(\{x \in R^n : |x| < 1\}\); \(C\) is the set of all functions that are defined and uniformly continuous in the ball \(B\); \(L_p\) is the set of all functions integrable in \(B\) to the power \(p\), where \(p > 1\). The norm in the spaces \(C\) and \(L_p\) is defined in the usual way. By the symbol \(L_1\) we shall denote the totality of all completely additive functions defined on the \(\sigma\)-ring of Borel sets contained in the ball \(B\). We define the norm in \(L_1\) by putting, for \(\varphi \in L_1\), \(\|\varphi\|_{L_1} = |\varphi|(B)\), where \(|\varphi|(E)\) is the absolute variation of the set function \(\varphi(E)\).

Let \(\mathfrak R\) be an arbitrary Banach space whose elements are real functions defined on the ball \(B\); let \(u\) be an arbitrary measurable function defined in the domain \(G\); let \(x\) be a point of the domain; and let \(P_x(X)\) be a polynomial in the variable \(X = (X_1, X_2, \ldots, X_n)\) of degree not exceeding \(l\). We shall say that \(P_x(X)\) is the complete differential of order \(l\) of the function \(u\) at the point \(x\) in the sense of convergence in \(\mathfrak R\), if

\[ \left\| \frac{u(x+hX)-P_x(hX)}{h^l} \right\|_{\mathfrak R} \to 0 \]

as \(h \to 0\).

We introduce the following notation. Let \(\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)\), where \(\alpha_i\), \(i = 1, 2, \ldots, n\), are nonnegative integers. We put:

\[ |\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_n, \qquad \alpha! = \alpha_1! \alpha_2! \cdots \alpha_n!. \]

If \(X = (X_1, X_2, \ldots, X_n)\), then we put \(X^\alpha = X_1^{\alpha_1} X_2^{\alpha_2} \cdots X_n^{\alpha_n}\). Finally, \(D^\alpha\) denotes the differential operator

\[ D^\alpha = \frac{\partial^{|\alpha|}} {\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots \partial x_n^{\alpha_n}}. \]

Let \(u \in W_p^l(G)\). The formal differential of order \(l\) of the function \(u\) at the point \(x \in G\) is the polynomial

\[ \sum_{0 \leq |\alpha| \leq l} \frac{D^\alpha u(x)}{\alpha!} X^\alpha . \tag{1} \]

In the case \(p=1\), the derivative \(D^\alpha u\), where \(|\alpha|=l\), is a completely additive set function in \(G\). In this case \(D^\alpha u(x)\) in formula (1) should be understood as the derivative of the set function \(D^\alpha u\) with respect to Lebesgue measure in \(R^n\).

In what follows \(W_p^l\) denotes the space \(W_p^l(B)\). We introduce a norm in \(W_p^l\), setting, for \(u\in W_p^l\),

\[ \|u\|_{W_p^l}=\sum_{0\le |\alpha|\le l}\|D^\alpha u\|_{L_p}. \]

The main result of the present paper is the following theorem.

Theorem 1. For every function \(u\in W_p^l(G)\), its formal differential of order \(l\) at the point \(x\in G\) is a complete differential of order \(l\) at the point \(x\) in the sense of convergence in \(W_p^l\) for almost all \(x\in G\).

In other words, if \(u\in W_p^l(G)\), then for almost all \(x\in G\) the equality

\[ \lim_{h\to 0}\left\|\frac{1}{h^l}\left[u(x+hX)-\sum_{0\le |\alpha|\le l}\frac{D^\alpha u(x)}{\alpha!}h^{|\alpha|}X^\alpha\right]\right\|_{W_p^l}=0 \tag{2} \]

holds.

The proof of the theorem is based on the following considerations. Denote the expression under the norm sign in equality (2) by \(R_x(h,X)\). For \(|\alpha|=l\), obviously, we have

\[ D_X^\alpha R_x(h,X)=D^\alpha u(x+hX)-D^\alpha u(x). \tag{3} \]

If \(v\) is a function locally integrable in \(G\) to the power \(p\ge 1\), then, as is known (see, for example, (2)), for almost all \(x\in G\)

\[ \|v(x+hX)-v(x)\|_{L_p}\to 0 \]

as \(h\to 0\). Hence, by virtue of equality (3), it follows that for almost all \(x\in G\)

\[ \sum_{|\alpha|=l}\|D_X^\alpha R_x(h,X)\|_{L_p}\to 0 \]

as \(h\to 0\).

Let \(\varphi_\alpha(X)\), where \(|\alpha|\le l-1\), be finite in \(B\) infinitely differentiable functions such that

\[ \int_B \varphi_\alpha(X)X^\beta\,dX=0 \quad \text{for } \alpha\ne\beta; \qquad \int_B \varphi_\alpha(X)X^\beta\,dX=1 \quad \text{for } \alpha=\beta. \]

By virtue of the well-known theorems of S. L. Sobolev on equivalent norms in \(W_p^l\), in order to complete the proof of the theorem it is sufficient to show that, for almost all \(x\in G\),

\[ \int_B \varphi_\alpha(X)R_x(h,X)\,dX\to 0 \]

as \(h\to 0\). The validity of this assertion follows from the property of functions locally integrable to the power \(p\ge 1\) indicated above, and also from the equality

\[ \int_B \varphi_\alpha(X)R_x(h,X)\,dX = \frac{1}{h^l}\int_0^h t^{\,l-1} \left[ \int_B \sum_{|\beta|=l} \frac{D^\beta u(x+tX)-D^\beta u(x)}{\beta!} X^\beta \varphi_\alpha(X)\,dX \right]dt, \]

valid for almost all \(x\in G\).

As one of the consequences of Theorem 1, we note the following result.

Theorem 2. Let \(u \in W_p^l(G)\), where \(lp>n\). Then for almost all \(x \in G\) the equality

\[ u(x+X)-\sum_{0\le |\alpha|\le l}\frac{D^\alpha u(x)}{\alpha!}X^\alpha = o(|X|^l). \tag{4} \]

holds.

Proof. By Theorem 1, for almost all \(x \in G\),

\[ \left\| \frac{1}{h^l} \left[ u(x+hX)-\sum_{0\le |\alpha|\le l} \frac{D^\alpha u(x)}{\alpha!}h^{|\alpha|}X^\alpha \right] \right\|_{W_p^l} \to 0 \]

as \(h\to 0\). By the embedding theorem for the class \(W_p^l\) in \(C\) when \(lp>n\) \((^1)\), it follows that, for almost all \(x \in G\),

\[ \left\| \frac{1}{h^l} \left[ u(x+hX)-\sum_{0\le |\alpha|\le l} \frac{D^\alpha u(x)}{\alpha!}h^{|\alpha|}X^\alpha \right] \right\|_{C} \to 0 \]

as \(h\to 0\), which, as is not difficult to see, is equivalent to equality (4). For the particular case \(l=1\), the theorem was proved earlier by A. Calderón \((^3)\) (see also \((^4)\)).

In conclusion we note that, in an analogous way, by combining Theorem 1 with embedding theorems, one can obtain a number of known results on differentiability almost everywhere for certain classes of functions, for example the theorem on the existence almost everywhere of the second differential of a convex function \((^5)\), the theorem on differentiability of a monotone function of the class \(W_p^1\), where \(p>h-1\) \((^6)\), and others.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
27 XII 1965

REFERENCES

\(^1\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^2\) A. Zigmund, Trigonometric Series, 1, 1965.
\(^3\) A. Calderón, Riv. Mat. Univ. Parma, 2, 203 (1951).
\(^4\) J. Serrin, Arch. Rat. Mech. and Analysis, 7, 359 (1961).
\(^5\) A. D. Aleksandrov, Uch. zap. Leningrad. gos. univ., ser. matem., 6, 3 (1939).
\(^6\) G. Väisälä, Ann. Acad. Sci. Fenn., Ser. A. 1, No. 362 (1965).