MATHEMATICS

Ya. S. BUGROV

ON THE EXTENSION OF FUNCTIONS

(Presented by Academician S. L. Sobolev, 7 I 1963)

1. In this note we consider the question of extending functions defined on the subspace
\(R_{n-1}=\{-\infty<x_i<\infty,\ i=1,\ldots,n-1\}\) to the upper half-space
\(R_n^0=\{x_n>0,\ -\infty<x_i<\infty,\ i=1,\ldots,n-1\}\). In order to avoid cumbersome notation, we shall restrict ourselves to the case \(n=3\), although all the arguments carry over to the case of arbitrary \(n\).

Definition. Let \(r_i=\bar r_i+\alpha_i\), \(\bar r_i\ge 0\) integers, \(0<\alpha_i\le 1\) \((i=1,2,3)\). A function \(f(x,y,z)\) is said to belong to the class \(S_p^{(r_1,r_2,r_3)}H(R_3)=S_p^{(r_1,r_2,r_3)}H\) \((1\le p\le \infty)\) if: a) \(f\in L_p(R_3)\) with norm
\[ \|f\|_{L_p(R_3)}=\left(\int_{R_3}|f|^p\,dx\,dy\,dz\right)^{1/p}<\infty; \]
b) all possible generalized derivatives (in the sense of Sobolev)
\(\partial^{k_1+k_2+k_3}f/\partial x^{k_1}\partial y^{k_2}\partial z^{k_3}\) exist for \(k_i=\bar r_i\) \((i=1,2,3)\); c) for the derivatives indicated above the following relations hold:
\[ \sup_h \left\| \frac{\Delta_h^2\,\partial^{\bar r_1}f/\partial x^{\bar r_1}} {h^{\alpha_1}} \right\|_{L_p(R_3)} = M_p^{(r_1)}(f), \]
\[ \cdots \]
\[ \sup_{h,k,l} \left\| \frac{ \Delta_{h,k,l}^{2,2,2}\, \partial^{\bar r_1+\bar r_2+\bar r_3}f/ \partial x^{\bar r_1}\partial y^{\bar r_2}\partial z^{\bar r_3} } {h^{\alpha_1}k^{\alpha_2}l^{\alpha_3}} \right\|_{L_p(R_3)} = M_p^{(r_1,r_2,r_3)}(f), \]
where
\[ \Delta_h^2\psi=\psi(x+2h,y,z)-2\psi(x+h,y,z)+\psi(x,y,z), \]
\(\Delta_{h,k}^{2,2}\psi=\Delta_h^2[\Delta_k^2\psi],\ldots,\)
\[ \Delta_{h,k,l}^{2,2,2}\psi=\Delta_h^2[\Delta_k^2(\Delta_l^2\psi)] \]
(\(h\) is the increment in the variable \(x\), \(k\) in the variable \(y\), \(l\) in the variable \(z\)).

We introduce the norm in the space \(S_p^{(r_1,r_2,r_3)}H\) as follows:
\[ \|f\|_{S_p^{(r_1,r_2,r_3)}H} = \|f\|_{L_p(R_3)} + M_p^{(r_1)}(f)+\cdots+ M_p^{(r_1,r_2,r_3)}(f) <\infty. \]

It is clear from the definition that for \(n=1\) the class
\(S_p^{(r_1)}H\equiv H_p^{(r_1)}\) (for the definition of the class \(H_p^{(r_1)}\), see (1a), p. 268).

The functional classes \(S_p^{(r_1,r_2,r_3)}H\) were first introduced and studied by S. M. Nikol’skii. In the papers \((1\text{б},\text{в})\) the classes
\(S_p^{(r_1,r_2,r_3)}W\) were also considered, where \(r_i\) \((i=1,2,3)\) are integers.

One says that \(f\in S_p^{(r_1,r_2,r_3)}W\) if \(f\in L_p(R_3)\) and the generalized derivatives
\[ \frac{\partial^{r_1}f}{\partial x^{r_1}},\quad \frac{\partial^{r_2}f}{\partial y^{r_2}},\quad \frac{\partial^{r_3}f}{\partial z^{r_3}},\quad \frac{\partial^{r_1+r_2}f}{\partial x^{r_1}\partial y^{r_2}},\ldots,\quad \frac{\partial^{r_1+r_2+r_3}f}{\partial x^{r_1}\partial y^{r_2}\partial z^{r_3}} \]
are also integrable to the \(p\)-th power over \(R_3\), and
\[ \|f\|_{S_p^{(r_1,r_2,r_3)}W} = \|f\|_{L_p(R_3)} + \left\| \frac{\partial^{r_1}f}{\partial x^{r_1}} \right\|_{L_p(R_3)} +\cdots+ \left\| \frac{\partial^{r_1+r_2+r_3}f} {\partial x^{r_1}\partial y^{r_2}\partial z^{r_3}} \right\|_{L_p(R_3)} <\infty. \]

S. M. Nikol’skii proved \((1\text{г})\) that the trace of a function \(f\) from the class \(S_p^{(r_1,r_2,r_3)}H\) (for \(z=0\)) belongs to the class \(S_p^{(r_1,r_2)}H\). We shall prove the converse assertion—

namely, from the membership of the function \(\varphi(x,y)\) in the class \(S_p^{(r_1,r_2)}H\) it follows that it can be extended into the upper half-space in such a way that the extended function will belong to the class \(S_p^{(r_1,r_2,r_3)}H(R_3^0)\) for any \(r_3\).

2. Theorem 1. Let a system of functions \(\varphi_0(x,y),\ldots,\varphi_{s-1}(x,y)\) be given, belonging to the class \(S_p^{(r_1,r_2)}H(R_2)\) \((1\le p\le\infty,\ r_1>0,\ r_2>0)\). Then in the half-space \(R_3^0\) one can construct a function \(f(x,y,z)\) having the following properties:

a) \(f\in S_p^{(r_1,r_2,r_3)}H(R_3^0)\) for any \(r_3>0\), and the norm of \(f\) is estimated in terms of the norms of the functions \(\varphi_k\) \((k=0,1,\ldots,s-1)\);

b) \(\partial^k f(x,y,0)/\partial z^k=\varphi_k(x,y)\) \((k=0,1,\ldots,s-1)\). \(\tag{2,1}\)

Proof. We shall seek the function \(f(x,y,z)\) in the form

\[ f(x,y,z)=\exp(-z)\sum_{j=0}^{s_1-1}\left[\psi_j(x,y)\cos\beta_j z+\eta_j(x,y)\sin\beta_j z\right], \tag{2,2} \]

where \(\beta_i\ne\beta_j\) \((i\ne j)\) and \(\beta_j>0\) \((j=0,1,\ldots,s_1-1)\) for \(s=2s_1\); \(\beta_j>0\) \((j=1,2,\ldots,s_1-1)\), \(\beta_0=0\) for \(s=2s_1-1\).

In view of the fact that \(\beta_0=0\) when \(s=2s_1-1\), the function \(\eta_0(x,y)\) may always be taken to be identically zero. Thus, for \(s=2s_1-1\) we have \(2s_1-1\) unknown functions \(\psi_j,\eta_j\). From formula (2,2), taking account of the equalities (2,1), in order to determine the functions \(\psi_j,\eta_j\) we obtain the system of equations:

\[ \sum_{j=0}^{s_1-1}\psi_j=\varphi_0, \]

\[ \sum_{j=0}^{s_1-1}\left[-\psi_j+\beta_j\eta_j\right]=\varphi_1, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ \sum_{j=0}^{s_1-1} \left[ \left\{\binom{k}{0}-\binom{k}{2}\beta_j^2+\binom{k}{4}\beta_j^4-\cdots\right\}\psi_j + \left\{\binom{k}{1}\beta_j^{k-2}-\binom{k}{3}\beta_j^{k-4}+\cdots\right\}(-1)^{k_1}\beta_j\eta_j \right] = \varphi_k \quad (k=2k_1), \tag{2,3} \]

\[ \sum_{j=0}^{s_1-1} \left[ \left\{-\binom{k}{0}+\binom{k}{2}\beta_j^2-\cdots\right\}\psi_j + \left\{\binom{k}{k}\beta_j^{k-1}-\binom{k}{k-2}\beta_j^{k-3}+\cdots\right\}(-1)^{k_1}\beta_j\eta_j \right] = \varphi_k \quad (k=2k_1+1) \]

\[ (k=0,1,\ldots,s-1). \]

Denote by \(\Delta\) the determinant of the system (2,3), the \(j\)-th columns of which can be written as

\[ 1,\ -1,\ (1-\beta_j^2),\ldots \]

\[ \ldots \left\{ \begin{array}{ll} \left[\binom{k}{0}-\binom{k}{2}\beta_j^2+\binom{k}{4}\beta_j^4-\cdots\right] & \text{for } k=2k_1,\\[6pt] \left[-\binom{k}{0}+\binom{k}{2}\beta_j^2-\cdots\right] & \text{for } k=2k_1+1 \end{array} \right\} \ldots \]

for \(j\le s_1-1\);

\[ 0,\ \beta_{j-s_1},\ -2\beta_{j-s_1},\ldots \]

\[ \ldots \left\{ \begin{array}{ll} (-1)^{k_1}\beta_{j-s_1} \left[\binom{k}{1}\beta_{j-s_1}^{k-2}-\binom{k}{3}\beta_{j-s_1}^{k-4}+\cdots\right] & \text{for } k=2k_1,\\[6pt] (-1)^{k_1}\beta_{j-s_1} \left[\binom{k}{k}\beta_{j-s_1}^{k-1}-\binom{k}{k-2}\beta_{j-s_1}^{k-3}+\cdots\right] & \text{for } k=2k_1+1 \end{array} \right\} \ldots \]

for \(s_1\le j\le 2s_1-1\).

In order that \(\Delta \ne 0\), it is necessary that all \(\beta_i\) be distinct. Choosing the numbers \(\beta_i\) so that \(\Delta \ne 0\) \((\beta_i \ne \beta_j,\ i \ne j)\), we obtain a solution of the posed problem.

In the general case it is obvious that

\[ f(x,y,z)=\exp(-z)\sum_{i=0}^{s-1}\psi_i(z)\varphi_i(x,y), \tag{2.4} \]

where \(\psi_i(z)\) are bounded functions and are a linear combination of the trigonometric functions \(\cos \beta_i z,\ \sin \beta_i z\) with coefficients that ensure the fulfillment of conditions (2.1).

Let us give the solution of system (2.3) in the simplest cases:

1) \(s=1,\ f(x,y,z)=\exp(-z)\varphi_0(x,y)\);

2) \(s=2,\ f(x,y,z)=\exp(-z)[(\cos z+\sin z)\varphi_0(x,y)+\varphi_1(x,y)\sin z]\). \((2.5)\)

Here we have put \(\beta_0=1\).

3) \(s=3\) \((\beta_0=0,\ \beta_1=1)\),

\[ f(x,y,z)=\exp(-z)[(2-\cos z+\sin z)\varphi_0(x,y)+ \]

\[ +(2-2\cos z+\sin z)\varphi_1(x,y)+(1-\cos z)\varphi_2(x,y)]. \]

Now from formula (2.4) it is clear that, with respect to the variable \(z\), the function \(f(x,y,z)\) is infinitely differentiable. Therefore, if \(\varphi_k(x,y)\in S_p^{(r_1,r_2)}H(R_2)\) \((k=0,1,\ldots,s-1)\), then \(f\in S_p^{(r_1,r_2,r_3)}H(R_3^0)\) for any \(r_3>0\). Conditions (2.1), generally speaking, are satisfied in the sense of convergence in the \(p\)-mean. Let us show this for the function (2.5). We have

\[ \left(\iint_{R_2}|f(x,y,z)-\varphi_0(x,y)|^p\,dx\,dy\right)^{1/p} = \left(\iint_{R_2}\left|[\exp(-z)(\cos z+\sin z)-1]\times\right.\right. \]

\[ \left.\left.\times\varphi_0(x,y)+\exp(-z)\sin z\cdot \varphi_1(x,y)\right|^p\,dx\,dy\right)^{1/p} \le cz\|\varphi_1\|_{L_p(R_2)}+ \]

\[ +c\left(\iint_{R_2}|\exp(-z)(\cos z-1)+\exp(-z)-1|^p|\varphi_0|^p\,dx\,dy\right)^{1/p} + \]

\[ +cz\|\varphi_0\|_{L_p(R_2)} \le cz\left(\|\varphi_0\|_{L_p(R_2)}+\|\varphi_1\|_{L_p(R_2)}\right)\to 0 \quad \text{as } z\to 0. \]

Further, since

\[ \frac{\partial f}{\partial z} = \exp(-z)[-2\sin z\cdot\varphi_0(x,y)+(\cos z-\sin z)\varphi_1(x,y)], \]

then, as above,

\[ \left(\iint_{R_2}\left|\frac{\partial f}{\partial z}-\varphi_1\right|^p\,dx\,dy\right)^{1/p} \le cz\left(\|\varphi_0\|_{L_p(R_2)}+\|\varphi_1\|_{L_p(R_2)}\right)\to 0 \]

as \(z\to 0\), and the theorem is proved.

Remark 1. The theorem is also true in terms of \(S_p^{(r_1,r_2,r_3)}W\).

Remark 2. In the case of extension to the whole space \(R_3\), it is necessary in formula (2.2), instead of the factor \(\exp(-z)\), to take the factor \(\exp(-z^2)\). For example, for \(s=3\)

\[ f(x,y,z)=\exp(-z^2)[(3-2\cos z)\varphi_0(x,y)+(1-\cos z)\varphi_2(x,y)+ \]

\[ +\sin z\cdot\varphi_1(x,y)]. \]

3. Let us show that for \(n=2,\ p=2\) the corresponding extension is carried out in the form of a solution of a certain differential equation. Let

\[ L^{(r,s)}u\equiv (-1)^{r+1}\frac{\partial^{2r}u}{\partial x^{2r}} + (-1)^{s+1}\frac{\partial^{2s}u}{\partial y^{2s}} + (-1)^{s+r+1}\frac{\partial^{2s+2r}u}{\partial x^{2r}\partial y^{2s}} =0, \tag{3.1} \]

where \(r,s\) are natural numbers (see \((1^6)\)).

We pose the following problem for equation (3,1): to find in the upper half-plane \(R_2^0=\{y>0,\ -\infty<x<\infty\}\) a bounded solution of equation (3,1) under the condition that

\[ \frac{\partial^k u(x,0)}{\partial y^k}=\varphi_k(x)\quad (k=0,1,\ldots,s-1),\qquad \lim_{y\to+\infty}u(x,y)=0. \tag{3,2} \]

We note that conditions (3,2) are understood in the sense of mean-square convergence. Applying the Fourier method of separation of variables, the solution of problem (3,1)—(3,2) (for \(s=1\)) can be written in the form

\[ u(x,y)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \Phi_0(\lambda)\exp(-\varkappa y)e^{i\lambda x}\,d\lambda, \]

where \(\Phi_0\) is the Fourier transform of the function \(\varphi_0(x)\), \(\varkappa=|\lambda|^r(1+\lambda^{2r})^{-1/2}\).

Theorem 2. If the function \(\varphi_0(x)\in S_2^{(r_1)}H\equiv H_2^{(r_1)}(R_1)\), \((r_1>0)\), then the solution of the boundary-value problem (3,1)—(3,2) for \(s=1\) belongs to the class \(S_2^{(r_1,r_2)}H[R_2(0,1)]\) for any \(r_2>0\), and

\[ \int_{-\infty}^{\infty}|u(x,y)-\varphi_0(x)|^2\,dx\to 0 \quad \text{as } y\to +0, \]

where \(R_2(0,1)=\{0<y<1,\ -\infty<x<\infty\}\).

Remark. If \(\varphi_0\in W_2^{(r_1)}(R_1)\), \(r_1\) is an integer, then \(u\in S_2^{(r_1,r_2)}W[R_2(0,1)]\) for any integer \(r_2\).

1. For \(s=2\) the solution of the boundary-value problem has the form

\[ u(x,y)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \exp(-v_0y)\left[\Phi_0(\lambda)(\cos v_0y+\sin v_0y)+ \Phi_1(\lambda)\frac{\sin v_0y}{v_0}\right]e^{i\lambda x}\,d\lambda, \tag{4,1} \]

where \(\Phi_0,\Phi_1\) are the Fourier transforms of the functions \(\varphi_0\) and \(\varphi_1\), respectively,

\[ v_0=|\lambda|^{r/2}(1+\lambda^{2r})^{-1/4}. \]

Theorem 3. If the functions \(\varphi_0,\varphi_1\) belong to the class \(S_2^{(r_1)}H(R_1)\), \(r_1>0\) \(\bigl(S_2^{(r_1)}W(R_1),\ r_1\) an integer\(\bigr)\), then the solution (4,1) belongs to the class \(S_2^{(r_1,r_2)}H[R_2(0,1)]\) \(\bigl(S_2^{(r_1,r_2)}W[R_2(0,1)]\bigr)\) for any \(r_2>0\).

CITED LITERATURE

1. S. M. Nikol’skii, a) Matem. sborn., 33 (75), 2, 261 (1958); b) DAN, 146, No. 3, 543 (1962); c) DAN, 146, No. 3, 767 (1962); d) Proceedings of the Second All-Union Conference on the Constructive Theory of Functions, Baku, 1962 (in press).