MATHEMATICS

V. P. IL'IN and V. A. SOLONNIKOV

ON SOME PROPERTIES OF DIFFERENTIABLE FUNCTIONS OF MANY VARIABLES

(Presented by Academician V. I. Smirnov, 28 VII 1960)

In the present work we consider some properties of functions having different differentiability properties with respect to different variables. Works \((^{1-4})\) are also devoted to these questions.

1. Let \(f(x_1,\ldots,x_n)\) be a smooth function. Let \(h>0\), \(x_i\) \((i=1,2,\ldots,n)\) be arbitrary positive numbers; \(\nu_i,\ \bar l_i,\ s_i,\ k_i\) be arbitrary nonnegative integers, with \(0\leqslant \nu_i\leqslant \bar l_i\). The following identity holds:

\[ D_{x_1}^{\nu_1}D_{x_2}^{\nu_2}\cdots D_{x_n}^{\nu_n} f(x) = \frac{C}{h^r} \int_0^{h^{x_1}}\cdots\int_0^{h^{x_n}} f(x_1+y_1,\ldots,x_n+y_n)\times \]

\[ \times \prod_{j=1}^n \frac{\partial^{\bar l_j}}{\partial y_j^{\bar l_j}} \left[ \frac{y_j^{\bar l_j-\nu_j-1}}{(\bar l_j-\nu_j-1)!}\, \psi_j(y_j,h^{x_j}) \right]dy_1\cdots dy_n - \]

\[ - C\sum_{i=1}^n x_i \int_0^h \frac{dv}{v^{1+r}} \int_0^{v^{x_1}}\cdots\int_0^{v^{x_n}} \prod_{j\ne i} \frac{\partial^{\bar l_j}}{\partial y_j^{\bar l_j}} \left[ \frac{y_j^{\bar l_j-\nu_j-1}}{(\bar l_j-\nu_j-1)!}\, \psi_j(y_j,v^{x_j}) \right]dy_1\cdots dy_n \times \]

\[ \times \int_0^{v^{x_i}-y_i} \Bigl[ D_i^{\bar l_i}f(x_1+y_1,\ldots,x_i+y_i+t,\ldots,x_n+y_n) - \]

\[ - 2D_i^{\bar l_i}f(x_1+y_1,\ldots,x_i+y_i+t/2,\ldots,x_n+y_n) + \]

\[ + D_i^{\bar l_i}f(x_1+y_1,\ldots,x_i+y_i,\ldots,x_n+y_n) \Bigr] \left[ \gamma_{1i}y_i^{\bar l_i+k_i} (v^{x_i}-y_i-t)^{\bar l_i+1+s_i} + \right. \]

\[ \left. + \gamma_{2i}y_i^{\bar l_i+k_i+1} (v^{x_i}-y_i-t)^{\bar l_i+s_i} \right]dt, \]

where \(C,\ \gamma_{1i},\ \gamma_{2i}\) are certain constants,

\[ r=\sum_{i=1}^n x_i(\bar l_i+\nu_i+k_i+3), \]

\[ \psi_j(y_j),\quad v^{x_j} = (\bar l_j-\nu_j) \int_{y_j}^{v^{x_j}} (v^{x_j}-t)^{\bar l_j+s_j+2}t^{k_j+\nu_j}\,dt + \]

\[ + 2y_j\frac{\partial}{\partial y_j} \int_{y_j}^{v^{x_j}} (v^{x_j}-t)^{\bar l_j+s_j+2}t^{k_j+\nu_j}\,dt + \]

\[ + \frac{1}{\bar l_j-\nu_j-1} y_j^2 \frac{\partial^2}{\partial y_j^2} \int_{y_j}^{v^{x_j}} (v^{x_j}-t)^{\bar l_j+s_j+2}t^{k_j+\nu_j}\,dt. \]

Most of the results formulated below are obtained on the basis of this identity.

§ 2. Let \(D\) be a domain of the space \(E_n\) possessing the following property: at every point \(x\in \overline D\) one can draw an \(n\)-dimensional rectangle, entirely contained in \(\overline D\), with a vertex at \(x\) and with edges parallel to the coordinate axes and having constant lengths \(\mathscr H_i\) \((i=1,2,\ldots,n)\). Moreover, suppose that if the points with coordinates \((x_1,\ldots,x_i,\ldots,x_n)\) and \((x_1,\ldots,x_i+t_i,\ldots,x_n)\) belong to \(D\), then also \((x_1,\ldots,x_i+\theta t_i,\ldots,x_n)\in D\), \(0\leq \theta\leq 1\), \(t_i\leq \mathscr H_i\). By \(\mathfrak W_{p_0,p_1\ldots p_n}^{l_1\ldots l_n}(D)\) we shall denote the space of functions that is the closure of the set of smooth functions in the norm

\[ \|f\|_{\mathfrak W_{p_0,p_1\ldots p_n}^{l_1\ldots l_n}(D)} = \|f\|_{L_{p_0}(D)} + \|f\|_{L_{p_1\ldots p_n}^{l_1\ldots l_n}(D)}, \]

where

\[ \|f\|_{L_{p_1\ldots p_n}^{l_1\ldots l_n}(D)} = \sum_{i=1}^n \left[ \int_D dx_1\cdots dx_n \int_{I_i(x)} \left| \bar D_i^{\,l_i} f(x_1,\ldots,x_i+t,\ldots,x_n) \right. \right. \]

\[ \left. \left. -2\bar D_i^{\,l_i} f\left(x_1,\ldots,x_i+\frac{t}{2},\ldots,x_n\right) +\bar D_i^{\,l_i} f(x) \right|^{p_i} \frac{dt}{t^{1+p_i\lambda_i}} \right]^{1/p_i}. \]

Here \(I_i(x)\) is the set of those values \(t\) for which the point \((x_1,\ldots,x_i+t,\ldots,x_n)\in D\), if \((x_1,\ldots,x_i,\ldots,x_n)\in D\); \(p_i>1\), \(l_i=\bar l_i+\lambda_i\), where \(\bar l_i\) is a nonnegative integer and \(0<\lambda_i\leq 1\).

It can be shown that, when \(\lambda_i<1\), in the definition of the norm \(L_{p_1\ldots p_n}^{l_1\ldots l_n}(D)\), instead of the second difference of the derivative \(\bar D_i^{\,l_i}f\) one may put the first difference; this new norm is equivalent to the one introduced above.

Theorem 1. Let the domain \(D\) be bounded and star-shaped with respect to some point. If \(f(x)\in L_{p_0}(D)\) and has, with respect to the variable \(x_i\), generalized derivatives of order \(\bar l_i\),

\[ \|f\|_{\mathfrak W_{p_0,p_1\ldots p_n}^{l_1\ldots l_n}(D)}<\infty, \]

then \(f\in \mathfrak W_{p_0,p_1\ldots p_n}^{l_1\ldots l_n}(D)\), i.e. \(f(x)\) can be approximated in the norm \(\mathfrak W_{p_0,p_1\ldots p_n}^{l_1\ldots l_n}(D)\) by smooth functions.

Theorem 2. If the domain \(D\) is a finite or infinite rectangular parallelepiped with edges parallel to the coordinate axes, then a function \(f\in \mathfrak W_{p_0,p_1\ldots p_n}^{l_1\ldots l_n}\) can be extended, with preservation of the differential properties and of the norm, to all of \(E_n\) (in the sense of equivalence of norms).

Suppose that

\[ \varkappa_i\equiv \frac{1}{l_i}\left(1-\sum_{j=1}^n \frac{1}{p_jl_j}+\frac{1}{p_i}\sum_{j=1}^n \frac{1}{l_j}\right)>0. \]

Then the following theorems are valid:

Theorem 3. Let \(f\in \mathfrak W_{p_0,p_1\ldots p_n}^{l_1\ldots l_n}(D)\). Then:

1) If

\[ \varepsilon_0=1-\sum_{i=1}^n \frac{1}{p_il_i}-\sum_{i=1}^n \varkappa_i\nu_i>0, \]

then \(f(x)\) is equivalent to a continuous function differentiable in \(\overline D\), and

\[ \left|D_{x_1}^{\nu_1}D_{x_2}^{\nu_2}\cdots D_{x_n}^{\nu_n}f\right| \leq C\left( \|f\|_{L_{p_0}(D)} h^{-\frac{1}{p_0}\sum_{i=1}^n \varkappa_i-\sum_{i=1}^n \varkappa_i\nu_i} + \|f\|_{L_{p_1\ldots p_n}^{l_1\ldots l_n}(D)} h^{\varepsilon_0} \right). \]

2) If

\[ \varepsilon_s = 1-\sum_{j=1}^n \frac{1}{p_jl_j} -\sum_{j=1}^n \varkappa_j\nu_j +\frac{1}{q}\sum_{j=1}^s \varkappa_j >0, \qquad q\geq p_i>1 \]

\[ (i=0,1,\ldots,n), \]

then

\[ \left\|D_{x_1}^{\nu_1}D_{x_2}^{\nu_2}\cdots D_{x_n}^{\nu_n}f\right\|_{L_q(D_s)} \le C\left(\|f\|_{L_{p_0}(D)}\,h^{ \frac1q\sum_{j=1}^s x_j-\frac1{p_0}\sum_{j=1}^n x_j-\sum_{j=1}^n x_j\nu_j} + \|f\|_{L_{p_1\ldots p_n}^{\,l_1\ldots l_n}(D)}\,h^{\varepsilon_s}\right). \]

3) If \(q>p_i>1\) \((i=0,1,\ldots,n)\),
\[ 1-\sum_{j=1}^n\frac1{p_jl_j}-\sum_{j=1}^n x_j\nu_j+\frac1q\sum_{j=1}^s x_j=0, \]
then
\[ \left\|D_{x_1}^{\nu_1}D_{x_2}^{\nu_2}\cdots D_{x_n}^{\nu_n}f\right\|_{L_q(D_s)} \le C\left(\|f\|_{L_{p_0}(D)}\,h^{ \frac1q\sum_{j=1}^s x_j-\frac1{p_0}\sum_{j=1}^n x_j-\sum_{j=1}^n x_j\nu_j} + \|f\|_{L_{p_1\ldots p_n}^{\,l_1\ldots l_n}(D)}\right). \]

Here \(h>0\) is an arbitrary number satisfying the condition
\[ h\le \mathcal H\equiv \min_{i=1,\ldots,n}\mathcal H_i^{1/x_i}, \]
and \(D_s\) is the section of \(D\) by the hyperplane \(x_{s+1}=\mathrm{const},\ldots,x_n=\mathrm{const}\).

Theorem 4. Let \(f\in \mathfrak W_{p_0,p_1\ldots p_n}^{\,l_1\ldots l_n}(D)\). Then:

1) If
\[ \beta_k=\frac1{x_k}\left(1-\sum_{j=1}^n\frac1{p_jl_j}-\sum_{j=1}^n x_j\nu_j\right)>0,\quad 0\le \alpha_k\le \beta_k,\quad \alpha_k<m, \]
then
\[ \left|\Delta_{m,k}^{h}D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f\right| \le CH^{\alpha_k}\left( \|f\|_{L_{p_0}(D)}\,h^{ -\frac1{p_0}\sum_{j=1}^n x_j-\sum_{j=1}^n x_j\nu_j-\alpha_kx_k} + \|f\|_{L_{p_1\ldots p_n}^{\,l_1\ldots l_n}(D)}\,h^{(\beta_k-\alpha_k)x_k} \right). \]

2) If
\[ \beta_k=\frac1{x_k}\left(1-\sum_{j=1}^n\frac1{p_jl_j}-\sum_{j=1}^n x_j\nu_j+\frac1q\sum_{j=1}^s x_j\right)>0 \quad (k=1,2,\ldots,s), \]
\[ q\ge p_i>1\quad (i=0,1,\ldots,n),\quad 0<\alpha_k\le \beta_k,\quad \alpha_k<m,\quad 1\le s\le n, \]
then
\[ \left[ \iint_{D_s}\cdots\int dx_1\ldots dx_s \int_{I_k(x)} \frac{\left|\Delta_{m,k}^{t/m}D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f(x)\right|^q\,dt} {|t|^{1+q\alpha_k}} \right]^{1/q} \le \]
\[ \le C\left[ \|f\|_{L_{p_0}(D)}\,h^{ -\frac1{p_0}\sum_{j=1}^n x_j-\sum_{j=1}^n x_j\nu_j-\alpha_kx_k+\frac1q\sum_{j=1}^s x_j} + \|f\|_{L_{p_1\ldots p_n}^{\,l_1\ldots l_n}(D)}\,h^{(\beta_k-\alpha_k)x_k} \right]. \]

Here
\[ \Delta_{m,k}^{t}\varphi(x_1,\ldots,x_n) = \sum_{i=0}^m(-1)^{m-i}C_m^i \varphi(x_1,\ldots,x_k+it,\ldots,x_n), \]
and \(D_s\) is the section of \(D\) by the hyperplane
\(x_{s+1}=\mathrm{const},\ldots,x_n=\mathrm{const}\).

Theorems 3 and 4 generalize the embedding theorems proved in \((4\text{–}6)\). Analogous theorems for the case of the spaces of S. L. Sobolev \(W_p^l\) are proved in \((7)\).

Theorem 5. Let the domain \(D\) be finite and let the set of functions \(\{f\}\) be bounded in the norm
\(\mathfrak W_{p_0,p_1\ldots p_n}^{\,l_1\ldots l_n}(D)\). Then:

1) If condition 1) or 2) of Theorem 3 is satisfied, then the set
\[ \{D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f\} \]
is compact, respectively, in \(C\) or \(L_q(D_s)\).

2) If condition 1) or 2) of Theorem 4 is satisfied, with \(\alpha_k<\beta_k\), then the set
\[ \{D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f\} \]
is compact, respectively, in the spaces
\[ \mathrm{Zig}_{x_1\ldots x_n}^{\alpha_1\ldots\alpha_n}\,* \quad\text{and}\quad \mathfrak W_q^{\alpha_1\ldots\alpha_s}(D_s). \]

\[ \text{*} \]
By \(\mathrm{Zig}_{x_1\ldots x_n}^{\alpha_1\ldots\alpha_n}\) is meant the set of functions having continuous derivatives \(D_{x_i}^{\bar\alpha_i}u\) satisfying the condition
\[ \left|\Delta_{2,i}^{h}D_{x_i}^{\bar\alpha_i}u\right|\le Ch^{\varepsilon_i}, \]
where \(\alpha_i=\bar\alpha_i+\varepsilon_i,\ 0<\varepsilon_i\le1\). This condition, for \(\varepsilon_i<1\), is known to be equivalent to the Hölder condition.

1. Let \(D=E_n\); then one can define the space \(L_{p_1\ldots p_n}^{l_1\ldots l_n}\) as the closure of the set of smooth finite functions in the norm

\[ \|f\|_{L_{p_1\ldots p_n}^{l_1\ldots l_n}(E_n)} = \sum_{i=1}^{n} \left[ \int_{E_n} dx \int_{0}^{\infty} \left| \Delta_{2,i}^{h} D_i^{\bar l_i} f(x) \right|^{p_i} \frac{dh}{h^{1+p_i\lambda_i}} \right]^{1/p_i}. \]

Theorem 6. Let \(f\in L_{p_1\ldots p_n}^{l_1\ldots l_n}(E_n)\),

\[ \varkappa_n= \frac{1}{l_k} \left( 1-\sum_{j=1}^{n}\frac{1}{p_jl_j} -\frac{1}{p_k}\sum_{i=1}^{n}\frac{1}{l_i} \right)>0. \]

Then:

1) If

\[ \beta_k= \frac{1}{\varkappa_k} \left( 1-\sum_{j=1}^{n}\frac{1}{p_jl_j} -\sum_{j=1}^{n}\varkappa_j\nu_j \right)>0,\quad m>\beta_k, \]

then

\[ \left|\Delta_{m,R}^{H}D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n} f\right| \le CH^{\beta_k}\|f\|_{L_{p_1\ldots p_n}^{l_1\ldots l_n}(E_n)}. \]

2) If

\[ \beta_k= \frac{1}{\varkappa_k} \left( 1-\sum_{j=1}^{n}\frac{1}{p_jl_j} -\sum_{j=1}^{n}\varkappa_j\nu_j +\frac{1}{q}\sum_{j=1}^{s}\varkappa_j \right)>0, \]

then

\[ D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f\in L_q^{\beta_1\ldots\beta_s}(E_s),\quad \left\|D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f\right\|_{L_q^{\beta_1\ldots\beta_s}(E_s)} \le C\|f\|_{L_{p_1\ldots p_n}^{l_1\ldots l_n}(E_n)}. \]

3) If

\[ 1-\sum_{j=1}^{n}\frac{1}{p_jl_j} -\sum_{j=1}^{n}\varkappa_j\nu_j +\frac{1}{q}\sum_{j=1}^{s}\varkappa_j =0,\quad q>p_i>1, \]

then

\[ D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f\in L_q(E_s),\quad \left\|D_{x_1}^{\nu_1}\cdots D_{x_n}^{\nu_n}f\right\|_{L_q(E_s)} \le C\|f\|_{L_{p_1\ldots p_n}^{l_1\ldots l_n}(E_n)}. \]

This theorem is a special case of Theorem 4 and item 3) of Theorem 3, from which it is obtained for \(h=\infty\). Theorems 3, 4, and 6 are customarily called “direct embedding theorems.” If in item 2) of Theorem 6 \(p_1=\cdots=p_n=p\) and \(q=p\), then this part of the theorem can be inverted. Namely:

Theorem 7. Suppose numbers \(p>1\), \(l_k>0\) \((k=1,2,\ldots,n)\), integers \(s,n\) such that \(0<s<n\), and \(N\) distinct sequences of nonnegative integers \(\nu_j^{(i)}\) \((j=s+1,\ldots,n;\ i=1,\ldots,N)\) are given. Suppose the conditions

\[ \mu_i= 1-\sum_{j=s+1}^{n}\frac{\nu_j^{(i)}}{l_j} -\frac{1}{p}\sum_{j=s+1}^{n}\frac{1}{l_j}>0 \]

hold, and on the hyperplane \(E_s\) of the space \(E_n\) there are given \(N\) functions

\[ \varphi^{(i)}(x_1\ldots x_s)\in L_p^{\frac{l_1}{\mu_i},\ldots,\frac{l_s}{\mu_i}}(E_s) \quad (i=1,2,\ldots,N). \]

Then there exists a function \(\varphi(x_1\ldots x_n)\in L_p^{l_1\ldots l_n}(E_n)\) such that

\[ D_{x_{s+1}}^{\nu_{s+1}^{(i)}}\cdots D_{x_n}^{\nu_n^{(i)}}\varphi\big|_{x\in E_s} = \varphi^{(i)},\quad \|\varphi\|_{L_p^{l_1\ldots l_n}(E_n)} \le C\sum_{i=1}^{n} \left\|\varphi^{(i)}\right\|_{L_p^{\frac{l_1}{\mu_i},\ldots,\frac{l_s}{\mu_i}}(E_s)}. \]

We note that this theorem reduces to its special case \(s=n-1\).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR

Received
21 VII 1960

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