MATHEMATICS

Yu. K. SOLNTSEV

ON AN ESTIMATE OF THE MIXED DERIVATIVE IN \(L_p(G)\)

(Presented by Academician P. S. Novikov, 3 VI 1961)

1. Let \(f(x_1,x_2)\) in a domain \(G\) have generalized unmixed derivatives \(\partial^2 f/\partial x_1^2,\ \partial^2 f/\partial x_2^2\). We shall say that \(f \in W_p^{(2,2)}(G)\) if

\[ \|f\|_{W_p^{(2,2)}}= \left\{ \|f\|_{L_p(G)}^p +\sum_1^2 \left\|\frac{\partial f}{\partial x_i}\right\|_{L_p(G)}^p +\sum_1^2 \left\|\frac{\partial^2 f}{\partial x_i^2}\right\|_{L_p(G)}^p \right\}^{1/p} <\infty . \tag{1} \]

From the results of S. M. Nikol’skii \((^{1-3})\) on the extension of functions and of S. G. Mikhlin \((^{8,9})\) it follows that the function \(f \in W_p^{(2,2)}(G)\) has a generalized mixed derivative \(\partial^2 f/\partial x_1\partial x_2\) in \(G\), and moreover, for \(1<p<\infty\),

\[ \left\| \frac{\partial^2 f}{\partial x_1\,\partial x_2} \right\|_{L_p(G_\varepsilon)} \le C\|f\|_{W_p^{(2,2)}(G)}, \tag{2} \]

where \(G_\varepsilon\) is the domain internal to \(G\), consisting of those points of \(G\) whose distances to the boundary of \(G\) are greater than \(\varepsilon>0\), and the constant \(C\) does not depend on \(f\), but depends on the domain and on \(\varepsilon\).

1. In the following particular cases it is known that in inequality (2) one may replace \(G_\varepsilon\) by \(G\):

\[ \left\| \frac{\partial^2 f}{\partial x_1\,\partial x_2} \right\|_{L_p(G)} \le C\|f\|_{W_p^{(2,2)}(G)} . \tag{3} \]

1) The case of a periodic function; \(p=2\) (S. N. Bernstein \((^4)\), p. 97); for \(1<p<\infty\), (3) follows from the theorem of I. Marcinkiewicz \((^5)\) on multipliers of Fourier series.

2) The domain \(G\) is a rectangle, finite or infinite, with sides parallel to the coordinate axes (L. N. Slobodetskii \((^{6,7})\); for the case of the plane \(G \equiv R_2\) it follows from the works \((^{8,9})\) of S. G. Mikhlin).

3) The function \(f\) is equal to zero on the boundary of the domain \(G\), bounded by a curve of class \(C^3\); \(p=2\) (S. N. Bernstein \((^{10})\), S. G. Mikhlin \((^{12})\)); \(1<p<\infty\) (A. I. Koshelev \((^{13})\)).

4) The boundary values of the function and of its normal derivative belong to the classes:
\[ f|_\Gamma \in W_p^{(2-1/p)}(\Gamma), \qquad \left.\frac{\partial f}{\partial n}\right|_\Gamma \in W_p^{(1-1/p)}(\Gamma), \qquad 1<p<\infty \]
(L. N. Slobodetskii \((^{10})\)).

In the case \(p=\infty\) (the metric \(C\)) estimate (3) does not hold (B. S. Mityagin \((^{11})\)).

1. We have established that inequality (3) is valid for an arbitrary domain \(G\) with twice continuously differentiable boundary \(\Gamma\), without any assumptions about boundary values. Namely:

Theorem 1. Let the domain \(G\) be bounded by a finite number of closed or unbounded curves \(\Gamma_i\) of class \(C^2\),

\[ \min_{i\ne k}\rho(\Gamma_i,\Gamma_k)>0,\qquad 1<p<\infty;\qquad f\in W_p^{(2,2)}(G). \]

Then the mixed derivative is summable in the \(p\)-th power in the domain \(G\) and the inequality holds

\[ \left\|\frac{\partial^2 f}{\partial x_1 \partial x_2}\right\|_{L_p(G)} \leq C \|f\|_{W_p^{(2,2)}(G)}, \tag{4} \]

where the constant \(C\) does not depend on \(f\).

The proof is based on lemmas, proved by us, concerning the properties of boundary values of the function \(f\) and its derivatives.

Lemma 1. Let the domain \(G\) contain the strip \(a < x_1 < b\), \(\alpha(x_1) < x_2 < \alpha(x_1)+d\), where the curve \(x_2=\alpha(x_1)\) is a part \(\gamma\) of the boundary \(\Gamma\) of the domain \(G\), and

\[ \int_a^b |\alpha''(t)|^p\,dt < \infty,\qquad |\alpha'(x_1)| \leq k < 1. \]

Let \(f(x_1,x_2)\in W_p^{(2,2)}(G)\), \(1<p<\infty\). Let

\[ \varphi(x_1)=f\big|_\gamma,\qquad \mu(x_1)=\frac{\partial f}{\partial x_2}\bigg|_\gamma . \]

Then

\[ \varphi(x_1)\in W_p^{(2-1/p)}(a_1,b_1),\qquad \mu(x_1)\in W_p^{(1-1/p)}(a_1,b_1), \]

\[ \|\varphi\|_{W_p^{(2-1/p)}(a_1,b_1)} +\left\|\frac{d\varphi}{dx_1}\right\|_{W_p^{(1-1/p)}(a_1,b_1)} +\|\mu\|_{W_p^{(1-1/p)}(a_1,b_1)} \leq C\|f\|_{W_p^{(2,2)}(G)}, \tag{5} \]

where \((a_1,b_1)\) is an arbitrary interval lying inside \((a,b)\).

Lemma 2. Let the domain \(G\) contain the strip \(a < x_1 < b\), \(\alpha(x_1) < x_2 < \alpha(x_1)+d\), where the curve \(x_2=\alpha(x_1)\) is a part \(\gamma\) of the boundary \(\Gamma\) of the domain \(G\), and

\[ \int_a^b |\alpha''(t)|^p\,dt < \infty,\qquad 0<k_1\leq \alpha'(x_1)\leq k_2<\infty. \]

Let \(f\in W_p^{(2,2)}(G)\), \(1<p<\infty\),

\[ \varphi(x_1)=f\big|_\gamma,\qquad \lambda(x_1)=\frac{\partial f}{\partial x_1}\bigg|_\gamma,\qquad \mu(x_1)=\frac{\partial f}{\partial x_2}\bigg|_\gamma . \]

Then

\[ \varphi(x_1)\in W_p^{(2-1/p)}(a_1,b_1),\qquad \lambda(x_1),\mu(x_1)\in W_p^{(1-1/p)}(a_1,b_1), \]

\[ \|\varphi\|_{W_p^{(2-1/p)}(a_1,b_1)} +\left\|\frac{d\varphi}{dx_1}\right\|_{W_p^{(1-1/p)}(a_2,b_1)} +\|\lambda\|_{W_p^{(1-1/p)}(a_1,b_1)} + \]

\[ +\|\mu\|_{W_p^{(1-1/p)}(a_1,b_1)} \leq C\|f\|_{W_p^{(2,2)}(G)}, \tag{6} \]

where \((a_1,b_1)\) is an arbitrary interval lying inside \((a,b)\).

Lemma 3. Almost everywhere on the set \(E\) of those points of the boundary \(\Gamma\) of the domain \(G\) at which the tangent to \(\Gamma\) is not parallel to either of the coordinate axes, the formula holds

\[ \varphi'(x_1)=\lambda(x_1)+\mu(x_1)\alpha'(x_1), \]

where \(x_2=\alpha(x_1)\) is the equation of an arc of the boundary.

Analogous circumstances hold for \(n\) variables.

Theorem 2. Let

\[ \|f\|_{W_p^{(2,\ldots,2)}(G)} = \left\{ \|f\|_{L_p(G)}^p +\sum_1^n \left\|\frac{\partial f}{\partial x_i}\right\|_{L_p(G)}^p +\sum_1^n \left\|\frac{\partial^2 f}{\partial x_i^2}\right\|_{L_p(G)}^p \right\}^{1/p} <\infty \tag{7} \]

and let the domain \(G\) be bounded by a finite number of surfaces \(\Gamma_i\) of class \(C^2\),

\[ \min_{i\ne k}\rho(\Gamma_i,\Gamma_k)>0. \]

Then the mixed derivatives are summable to the \(p\)-th power on the domain \(G\), and

\[ \left\| \frac{\partial^{2} f}{\partial x_i\, \partial x_k} \right\|_{L_p(G)} \leq C \, \|f\|_{W_p^{(2,\ldots,2)}(G)} . \tag{8} \]

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
25 V 1961

References

\(^{1}\) S. M. Nikol’skii, Izv. AN SSSR, Ser. Mat., 22, 321 (1958).
\(^{2}\) S. M. Nikol’skii, Matem. sborn., 32 (74), 2 (1953).
\(^{3}\) S. M. Nikol’skii, Matem. sborn., 40 (82), 2 (1956).
\(^{4}\) S. N. Bernstein, Collected Works, 1, 1952.
\(^{5}\) J. Marcinkiewicz, Stud. Math., 8, 78 (1939).
\(^{6}\) L. N. Slobodetskii, DAN, 120, No. 3 (1958).
\(^{7}\) L. N. Slobodetskii, Uch. zap. fiz.-matem. fak. Leningrad State Pedagogical Institute named after A. I. Herzen, 197, 54 (1958).
\(^{8}\) S. G. Mikhlin, DAN, 109, No. 4 (1956).
\(^{9}\) S. G. Mikhlin, Vestn. LGU, No. 7, 143 (1957).
\(^{10}\) L. N. Slobodetskii, DAN, 123, No. 4 (1958).
\(^{11}\) B. S. Mityagin, DAN, 123, No. 4 (1958).
\(^{12}\) S. G. Mikhlin, DAN, 78, No. 3 (1951).
\(^{13}\) A. I. Koshelev, Matem. sborn., 32 (74), 3 (1953).