UDC 517.51

MATHEMATICS

A. D. DZHABRAILOV

ON AN INTEGRAL REPRESENTATION OF SMOOTH FUNCTIONS AND SOME FAMILIES OF FUNCTION SPACES

(Presented by Academician S. L. Sobolev on June 11, 1965)

1. We shall assume that \(e\) is any subset of the set of natural numbers \(e_n=\{1,\ldots,n\}\). If \(h=(h_1,\ldots,h_n)\) is a given vector, then let \(h^e=(h_1^e,\ldots,h_n^e)\), where \(h_j^e=h_j\) for \(j\in e\); \(h_j^e=0\) for \(j\in e_n\setminus e\). Let \(f(x)\) be a function defined in the Euclidean space \(E^n\) of points \(x=(x_1,\ldots,x_n)\); \(\Delta_j^{k_j}(t_j)f(x)\) is the finite difference of order \(k_j\) with respect to the variable \(x_j\) with step \(t_j\). Put
   \[ \Delta^{k^e}(t)f=\left[\prod_{j\in e}\Delta_j^{k_j}(t_j)\right]f. \]
   For each subset \(e\subseteq e_n\), \(f^{(m^e)}(x)\) is the derivative of the function \(f(x)\) of order \(m^e\) with respect to the variables \(x^e\). The order of differentiation is arbitrary; for example:
   \[ f^{(m^e)}(x)= \frac{\partial^{m_1^e}}{\partial x_1^{m_1^e}}\cdots \frac{\partial^{m_n^e}}{\partial x_n^{m_n^e}}f(x). \]
   Let
   \[ \int_0^{h^\sigma}(\cdot)\,dy^e = \int_0^{h_1^{\sigma_1}}dy_1^e\cdots \int_0^{h_n^{\sigma_n}}(\cdot)\,dy_n^e, \]
   where
   \[ \int_0^{h_j^{\sigma_j}}dy_j^e=\int_0^{h_j^{\sigma_j}}dy_j \quad\text{for } j\in e, \]
   and
   \[ \int_0^{h_j^{\sigma_j}}dy_j^e \]
   is the identity operator for \(j\in e_n\setminus e\).

Let \(\Omega\) and \(e_n^*\) be any fixed subsets of the set \(e_n\). Vectors \(h=(h_1,\ldots,h_n)\) and \(\sigma=(\sigma_1,\ldots,\sigma_n)\) with positive components are given, and a vector \(h\) is such that for \(j\in e_n\setminus\Omega\), \(h_j=h_0\). For all such vectors an integral representation has been obtained for sufficiently smooth functions \(f\):
\[ f^{(\nu)}(x)\equiv \frac{\partial^{\nu_1}}{\partial x_1^{\nu_1}}\cdots \frac{\partial^{\nu_n}}{\partial x_n^{\nu_n}}f(x) = \sum_{e\subset\Omega}\ \sum_{i\in e_n^0\setminus\Omega} (-1)^{|e|+1}I_e^{(i)}(f). \tag{1} \]
Here \(e_n^0=\{0,1,\ldots,n\}\), \(|e|\) is the number of elements of the set \(e\),
\[ I_e^{(0)}(f)\equiv I_e(f,e^*)= \]
\[ =\int_0^h dv^e\int_0^{v^\sigma}dy^e\int_0^{v^\sigma-y}dt^{e^*}\int_0^{h^\sigma} \Delta^{k^{e^*}}\!\left(\frac{t}{k}\right) f^{(m^e)}(x+y)\,R_e\,dy^{e_n\setminus e}, \]
where \(e^*=e_n^*\cap e\), \(m_i+k_i-\nu_i\ge 0\),
\[ R_e=R_e(y,t^{e^*},v^e,h^{e_n\setminus e}); \]
\[ I_e^{(i)}(f)= \begin{cases} I_{e\cup\{i\}}(f,e^*\cup\{i\}) & \text{for } i\in e_n^*,\\ I_{e\cup\{i\}}(f,e^*) & \text{for } i\in \overline{e_n^*}. \end{cases} \]

The integral kernels \(R_e\) are differentiable functions of their arguments and admit an estimate in terms of the components of the vector \(h\).

The integral representation (1) for \(e_n^*=e_n,\ \Omega=\varnothing\), i.e., when \(\Omega\) is the empty set, coincides with the integral representation obtained by V. P. Il’in \((^{13})\).

1. Let \(r=(r_1,\ldots,r_n)\) be a vector with nonnegative components. By the supporting vectors of the vector \(r\) we shall mean the smallest subset \(e\) of the set \(e_n\) such that \(r^e\equiv r\). We denote the support of this vector by \(e_r\). Suppose that \(e_r^*\) is the set of those indices \(j\) from \(e_r\) for which, when \(j\in e_r^*\), the number \(r_j\) is not an integer.

Let \(e\) be any subset of the set \(e_r\), and let \(e^*=e_r^*\cap e\). For each positive \(r_j\) put \(r_j=\bar r_j+\alpha_j\), where \(\bar r_j\) is the integer part of \(r_j\), so that \(0\leq \alpha_j<1\), and if \(r_j=0\), put \(\bar r_j=0\). Thus to each vector \(r=(r_1,\ldots,r_n)\) there corresponds a vector \(\bar r=(\bar r_1,\ldots,\bar r_n)\).

Definition. Let \(e\) be such a subset of the set \(e_r\) that \(e^*=\varnothing\), i.e., all \(r_j\) \((j\in e)\) are integers. We shall say that \(f\in L_p^{(r^e)}(E^n)\) if the function \(f(x)\) in \(E^n\) has a generalized derivative in the sense of S. L. Sobolev \((^1)\), \(f^{(r^e)}(x)\in L_p(E^n)\) \((p\geq 1)\). The norm in this space is defined as follows:

\[ \|f,L_p^{(r^e)}(E^n)\|=\|f^{(r^e)},L_p(E^n)\| =\left(\int_{E^n}|f^{(r^e)}(x)|^p\,dE^n\right)^{1/p}. \]

Now let \(e\) be such a subset of the set \(e_r\) that it intersects the set \(e_r^*\), i.e. the set \(e^*\) is nonempty. For such \(e\) we shall say that \(f\in L_p^{(r^e)}(E^n)\), if \(f\in L_p^{(\bar r^e)}(E^n)\) and the integral

\[ \mathcal{I}_e(f)= \left( \int_0^\infty\cdots\int_0^\infty \|\Delta^{\omega e^*}(t)f^{(\bar r^e)},L_p(E^n)\|^p \frac{dt^{e^*}}{\prod_{j\in e^*}t_j^{1+p\alpha_j}} \right)^{1/p} \]

is finite, where \(\omega=(1,\ldots,1)\) is the vector all of whose components are equal to one; and we define the norm in this space by

\[ \|f,L_p^{(r^e)}(E^n)\|=\|f,L_p^{(\bar r^e)}(E^n)\|+\mathcal{I}_e(f). \]

Let \(\Omega\) be any fixed subset of the set \(e_r\), and let the set \(e\) be a subset of the set \(\Omega\).

Definition. For \(i\in e_r^*\) we shall say that \(f\in L_p^{(r^e,r_i)}(E^n)\), if the generalized derivative

\[ f^{(r_i)}(x)\equiv \frac{\partial^{r_i}}{\partial x_i^{r_i}}\,f(x)\in L_p^{(r^e)}(E^n) \]

and we define the norm as follows:

\[ \|f,L_p^{(r^e,r_i)}(E^n)\|=\|f^{(r_i)},L_p^{r^e}(E^n)\|. \]

For \(i\in e_r^*\) we shall say that \(f\in L_p^{(r^e,r_i)}(E^n)\), if \(f\in L_p^{(r^e,\bar r_i)}(E^n)\) and the integral

\[ \mathcal{I}_e^{(i)}(f)= \left\{ \int_0^\infty [\mathcal{I}_e(\Delta_i(t_i)f^{(\bar r_i)})]^p t_i^{-(1+p\alpha_i)}\,dt_i \right\}^{1/p} \]

is finite, and we define the norm by

\[ \|f,L_p^{(r^e,r_i)}(E^n)\|=\|f,L_p^{(r^e,\bar r_i)}(E^n)\|+\mathcal{I}_e^{(i)}(f). \]

Basic definition. We shall say that \(f\in W_p^{(r)}(\Omega,E^n)\), if for every \(e\subset\Omega\) and every \(i\in e_r^0\setminus\Omega\), \(f\in L_p^{(r^e,r_i)}(E^n)\), where \(e_r^0=\)

\(= e_r \cup \{0\},\ r_0=0\), and define the norm in this space as follows:

\[ \|f, W_p^{(r)}(\Omega,E^n)\|= \sum_{e\subset\Omega}\sum_{i\in e_r^0\setminus\Omega} \|f,L_p^{(r^e,r_i)}(E^n)\|, \]

where

\[ L_p^{(\varnothing,0)}(E^n)\equiv L_p(E^n),\qquad L_p^{(r^e,0)}(E^n)\equiv L_p^{(r^e)}(E^n). \]

The space \(W_p^{(r)}(\Omega,E^n)\), for \(\Omega=\varnothing\) and \(e_r=e_n\), coincides with the generalized space of S. L. Sobolev \(W_p^{(r_1,\ldots,r_n)}(E^n)\) (see \((^1,{}^5,{}^7)\)), and for \(\Omega=e_r\) it coincides with the space \(S_p^{(r)}W(E^n)\), which for integral \(r_i\) \((i\in e_r)\) was defined by S. M. Nikol’skii \((^4)\).

Definition. By the space \(\mathscr W_p^{(r)}(\Omega,E^n)\) we shall mean the closure of the set of smooth finite functions in the norm \(\|f,W_p^{(r)}(\Omega,E^n)\|\).

Theorem 1. The spaces \(\mathscr W_p^{(r)}(\Omega,E^n)\) and \(W_p^{(r)}(\Omega,E^n)\) coincide for \(1<p<\infty\).

1. Let \(r=(r_1,\ldots,r_n)\) be a vector with nonnegative components \(r_i\) \((i=1,\ldots,n)\), whose support is \(e_r\). For each positive \(r_j\) put \(r_j=\bar r_j+\beta_j\), where \(\bar r_j\) is the greatest integer less than \(r_j\), so that \(0<\beta_j\le 1\), and when \(r_j=0\) put \(\bar r_j=0\); consequently, to each vector \(r=(r_1,\ldots,r_n)\) there corresponds the vector \(\bar r=(\bar r_1,\ldots,\bar r_n)\). Introduce the norms

\[ \|f,\mathscr L_p^{(r^e)}(E^n)\|= \left( \int_0^\infty\cdots\int_0^\infty \|\Delta^{2\omega^e}(t)f^{(\bar r^e)},L_p(E^n)\|^p \frac{dt^e}{\prod_{j\in e} t_j^{1+p\beta_j}} \right)^{1/p}, \]

where \(e\) is any subset of the set \(e_r\);

\[ \|f,\mathscr L_p^{(r^e,r_i)}(E^n)\|= \left( \int_0^\infty \|\Delta_i^2(t_i)f^{(\bar r_i)},\mathscr L_p^{(r^e)}(E^n)\|^p \frac{dt_i}{t_i^{1+p\beta_i}} \right)^{1/p}, \]

where \(e\) is any subset of the set \(\Omega\subset e_r\) and \(i\in e_r\setminus\Omega\);

\[ \|f,B_p^{(r)}(\Omega,E^n)\|= \sum_{e\subset\Omega}\sum_{i\in e_r^0\setminus\Omega} \|f,\mathscr L_p^{(r^e,r_i)}(E^n)\|, \]

where

\[ e_r^0=e_r\cup\{0\},\qquad \mathscr L_p^{(\varnothing,0)}(E^n)\equiv L_p(E^n),\qquad r_0=0. \]

Definition. By the space \(B_p^{(r)}(\Omega,E^n)\) we shall mean the closure of the set of sufficiently smooth finite functions in the norm \(\|f,B_p^{(r)}(\Omega,E^n)\|\). The space \(B_p^{(r)}(\Omega,E^n)\), for \(\Omega=\varnothing\) and \(e_r=e_n\), coincides with the known space \(B_p^{(r_1,\ldots,r_n)}(E^n)\), defined and studied by O. V. Besov \((^6)\) (see also \((^7,{}^8,{}^{10})\), etc.), and for \(\Omega=e_r\) it coincides with the known space \(S_p^{(r)}B(E^n)\), defined by the author \((^{12})\).

1. Let, for any subset \(e\) of the set \(e_r\),

\[ M^{(r^e)}(f)=\sup_{t^e}\left\| \prod_{j\in e} t_j^{-\beta_j}\Delta^{2\omega^e}(t)f^{(\bar r^e)},\, L_p(E^n) \right\|. \]

For any subset \(e\) of the set \(\Omega\subset e_r\) and any \(i\in e_r\setminus\Omega\) put

\[ M^{(r^e,r_i)}(f)=\sup_{t_i} t_i^{-\beta_i}M^{(r^e)}\bigl(\Delta_i^2(t_i)f^{(\bar r_i)}\bigr). \]

Definition. By the space \(H_p^{(r)}(\Omega,E^n)\) we shall mean the closure of the set of sufficiently smooth finite functions in the norm

\[ \|f,H_p^{(r)}(\Omega,E^n)\|= \sum_{e\subset\Omega}\sum_{i\in e_r^0\setminus\Omega} M^{(r^e,r_i)}(f), \]

where

\[ e_r^0=e_r\cup\{0\},\qquad M^{(\varnothing,0)}(f)=\|f,L_p(E^n)\|. \]

An analogous remark is also made for \(H_p^{(r)}(\Omega,E^n)\) (see \((^{2,3,7,11})\).

1. Let the vector \(r=(r_1,\ldots,r_n)\) be such that \(e_r=e_n\), i.e., the components of this vector are positive numbers. Further, let \(1<p\leq q\leq\infty\), let \(\Omega\) be any fixed subset of the set \(e_n=\{1,\ldots,n\}\), let \(m\) be any natural number \(\leq n\) and \(e_m=\{1,\ldots,m\}\), and let \(\nu=(\nu_1,\ldots,\nu_n)\) be a vector with nonnegative integer components. Put

\[ \varepsilon_j=1-\left(\frac1p-\frac1q\right)\frac1{r_j}-\frac{\nu_j}{r_j}>0 \quad (j\in\Omega\cap e_m\equiv\Omega^*),\qquad \varepsilon_i=1-\frac1{pr_i}-\frac{\nu_i}{r_i}>0 \]

\[ (i\in\Omega\setminus\Omega^*),\qquad \varepsilon_\Omega=1-\frac1p\sum_{j\in e_n\setminus\Omega}\frac1{r_j} -\sum_{j\in e_n\setminus\Omega}\frac{\nu_j}{r_j} +\frac1q\sum_{j\in e_m\setminus\Omega^*}\frac1{r_j}>0. \]

Theorem 2. Let \(f\in W_p^{(r)}(\Omega,E^n)\), and let the vector \(\rho=(\rho_1,\ldots,\rho_m)\) be such that its components satisfy the conditions:

a) for \(j\in\Omega^*\), \(0<\rho_j<\varepsilon_j r_j\), if at least one of \(\rho_j\) and \(r_j\) is an integer; \(0<\rho_j\leq \varepsilon_j r_j\), if \(\rho_j\) and \(r_j\) are simultaneously nonintegers;

b) for \(j\in e_m\setminus\Omega^*\), \(0<\rho_j<\varepsilon_\Omega r_j\), if at least one of \(\rho_j\) and \(r_i\) \((i\in e_n\setminus\Omega)\) is an integer; \(0<\rho_j\leq \varepsilon_\Omega r_j\), if \(\rho_j\) and \(r_i\) \((i\in e_n\setminus\Omega)\) are simultaneously nonintegers.

Then, for any fixed \(x_{m+1},\ldots,x_n\), \(f^{(\nu)}\in W_q^{(\rho)}(\Omega^*,E^m)\), and the inequality holds

\[ \bigl\|f^{(\nu)},\, W_q^{(\rho)}(\Omega^*,E^m)\bigr\| \leq c\bigl\|f,\, W_p^{(r)}(\Omega,E^n)\bigr\|, \]

where \(c\) is a constant independent of \(f\).

For brevity we shall denote the assertion of this theorem as follows:

\[ W_p^{(r)}(\Omega,E^n)\to W_q^{(\rho)}(\Omega^*,E^m). \]

Theorem 3.

\[ B_p^{(r)}(\Omega,E^n)\to B_q^{(\rho)}(\Omega^*,E^m), \]

where the components of the vector \(\rho=(\rho_1,\ldots,\rho_m)\) satisfy the conditions:
\(0<\rho_j\leq \varepsilon_j r_j\) for \(j\in\Omega^*\); \(0<\rho_j\leq \varepsilon_\Omega r_j\) for \(j\in e_m\setminus\Omega^*\).

Theorem 4. Under the conditions of Theorem 3,

\[ H_p^{(r)}(\Omega,E^n)\to H_q^{(\rho)}(\Omega^*,E^m). \]

Theorem 5.

\[ W_p^{(r)}(\Omega,E^n)\to B_q^{(\rho)}(\Omega^*,E^m), \]

where the components of the vector \(\rho=(\rho_1,\ldots,\rho_m)\) satisfy the conditions:

a) for \(j\in\Omega^*\), \(0<\rho_j<\varepsilon_j r_j\), if \(r_j\) is an integer; \(0<\rho_j\leq \varepsilon_j r_j\), if \(r_j\) is noninteger;

b) for \(j\in e_m\setminus\Omega^*\), \(0<\rho_j<\varepsilon_\Omega r_j\), if at least one of the \(r_i\) \((i\in e_n\setminus\Omega)\) is an integer; \(0<\rho_j\leq \varepsilon_\Omega r_j\), if all \(r_i\) \((i\in e_n\setminus\Omega)\) are nonintegers.

Theorem 6. Under the conditions of Theorem 3,

\[ B_p^{(r)}(\Omega,E^n)\to H_q^{(\rho)}(\Omega^*,E^m). \]

I express my sincere gratitude to Academician S. L. Sobolev and Professor L. D. Kudryavtsev for their attention to this work.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
9 IV 1965

CITED LITERATURE

\(^{1}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^{2}\) S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 38 (1951).
\(^{3}\) L. D. Kudryavtsev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 55 (1959).
\(^{4}\) S. M. Nikol’skii, Matem. sborn., 61 (103), 2 (1963).
\(^{5}\) L. N. Slobodetskii, Uch. zap. LGU, fiz.-matem. fak., 197 (1958).
\(^{6}\) O. V. Besov, Tr. Matem. inst. V. A. Steklova AN SSSR, 60 (1961).
\(^{7}\) V. P. Il’in, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 66 (1962).
\(^{8}\) S. V. Uspenskii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 60 (1961).
\(^{9}\) S. M. Nikol’skii, Sibirsk. matem. zhurn., 4, No. 6 (1963).
\(^{10}\) P. I. Lizorkin, Matem. sborn., 60 (102), 3 (1963).
\(^{11}\) Ya. S. Burenkov, Izv. AN SSSR, ser. matem., 22 (1958).
\(^{12}\) A. D. Dzhabrailov, DAN, 159, No. 2 (1964).
\(^{13}\) V. P. Il’in, V. A. Solonnikov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 66 (1962).