UDC 513.881

ON A CLASS OF FUNCTION SPACES

M. D. RAMAZANOV

In the study of properties of solutions of boundary-value problems for partial differential equations, various classes of function spaces have found wide application, which has led to an intensive development of the theory of these spaces. At present the theories of the scales \(\overline W_p^r\), \(\overline H_p^r\), \(\overline B_{p,\theta}^r\), \(\overline L_p^r\) \([1—7]\) have been developed most fully. It has turned out that, for the purposes of linear differential equations, the following properties of function spaces have proved to be the most important:

a) the existence of exact (in the form of necessary and sufficient conditions) characteristics of the trace of a function space on hyperplanes,

a′) the possibility of formulating these conditions (item a)) in terms of the original classes, referred to functions of a smaller number of variables—the closedness of the scale of spaces with respect to embedding theorems,

b) among the existing methods for classifying the smoothness of functions—various generalizations of the modulus of continuity, the rate of approximation by smooth functions, the use of Riesz or Bessel potentials—the last, closely connected with the theory of the Fourier transform, has proved the most convenient for application to equations. All the indicated properties are possessed, for example, by the scales \(W_2^r\) \([2]\) and \(H^\mu\) \([8]\), but for a more complete study of solutions of boundary-value problems these spaces alone are insufficient.

Property a) holds for each of the classes we have named \(\overline W_2^r\), \(\overline H_p^r\), \(\overline B_{p,\theta}^r\), \(\overline L_p^r\); however the scales \(\overline W_p^r\), \(\overline L_p^r\) for \(p\ne 2\) are not closed with respect to embedding theorems \([4]\). \(\overline B_{p,\theta}^r\) and \(\overline H_p^r\) satisfy condition a′), but so far there are no results establishing for them condition b), which makes it difficult to apply these classes in differential equations\(^*\).

In connection with the above, the problem was posed—to construct a scale of spaces, closed with respect to embedding theorems, close in a certain (precisely established in the article) sense to the scales \(\overline W_p^r\), \(\overline H_p^r\), \(\overline B_{p,\theta}^r\), \(\overline L_p^r\) and constructed starting from some basic space by means of Bessel potentials.

The work is devoted to the description of such a scale of spaces, denoted below by \(l^q L_p^\mu\). In § 1 the relations of \(l^q L_p^\mu\) with the classes \(\overline W_p^r\), \(\overline H_p^r\), \(\overline B_{p,\theta}^r\), \(\overline L_p^r\), most fully studied at the—

\(^*\) During the preparation of this work for publication, the author learned that in a joint work of S. M. Nikol’skii, J.-L. Lions, and P. I. Lizorkin \([16]\) a connection is established between the classes \(\overline B_{p,\theta}^r\) and \(\overline H_p^r\) with the Fourier transform.

present time. § 2 contains the theory of multipliers of Fourier transforms in the spaces \(l^q L_p^{\bar\mu}\), and in § 3 sharp embedding and extension theorems are proved.

Results concerning \(l^1 L_p\) spaces were published earlier in the note [9] (in the notation of [9], the spaces \(V_p\)).

§ 1. Definition of the spaces \(l^q L_p\), their relations with other spaces

Let \(f(\bar x)=f(x_1,\ldots,x_n)\) be a finite continuously differentiable function in the \(n\)-dimensional Euclidean space \(E^n\),
\[ (Ff)(\alpha_1,\ldots,\alpha_n)=\int f(\bar x)\exp(i\bar\alpha\bar x)\,d\bar x \]
its Fourier transform. We divide the Euclidean space formed by the set \((\alpha_1,\ldots,\alpha_n)\) of real arguments of the function \((Ff)(\bar\alpha)\) into the domains
\[ \Pi(\bar m)=\Pi(m_1,\ldots,m_n)=\{\bar\alpha:\ 2^{m_j}<|\alpha_j|<2^{m_j+1},\ j=1,\ldots,n\}, \]
where \((m_1,\ldots,m_n)\) is an integer vector running through all possible values, \(m_j=0,\pm1,\pm2,\ldots\) \((j=1,\ldots,n)\). To each domain \(\Pi(\bar m)\) we associate a function \(f_{\bar m}(\bar x)\) whose Fourier transform \((Ff_{\bar m})(\bar\alpha)\) on \(\Pi(\bar m)\) coincides with the function \((Ff)(\bar\alpha)\) and is equal to zero for the remaining \(\bar\alpha\). It can be shown\(^*\) that for a finite continuously differentiable function \(f(\bar x)\) the quantity
\[ |f,l^qL_p|=\left(\sum_{\bar m}|f_{\bar m},L_p|^q\right)^{1/q} \tag{1} \]
is finite for any values \(1<p<\infty\) and \(1\le q<\infty\).

The closure of the finite continuously differentiable functions in the norm (1) forms a Banach space, which we denote by \(l^q L_p\) (or \(l^q L_{p;\bar x}\), if it is necessary to emphasize the arguments of the functions).

Definition. Let \(\mu(\bar\alpha)\) be a measurable function and let \(B\) denote one of the spaces \(L_p, l^qL_p\). We shall say that a function \(f(\bar x)\) belongs to the space \(B^\mu\) if it admits a representation of the form
\[ (Ff)(\bar\alpha)=\mu^{-1}(\bar\alpha)(Fg)(\bar\alpha), \]
and the function \(g(\bar x)\in B\). As the norm of the function \(f(\bar x)\) in the space \(B^\mu\) we take the quantity \(|f,B^\mu|=|g,B|\).

In those cases when
\[ \mu(\bar\alpha)=\sum_{j=1}^{n}|\alpha_j|^{r_j}+1,\qquad r_j>0, \]
instead of \(B^\mu\) we shall agree to write \(B^{\bar r}\), approaching the notation adopted by other authors [2, 3].

We now turn to clarifying the relations of the introduced function spaces with the main classes known at the present time:
\[ W_p^{\bar r},\ H_p^{\bar r},\ B_{p,\theta}^{\bar r},\ L_p^{\bar r}\quad [1—7]. \]
First we establish several auxiliary assertions.

Lemma 1. For any \(1<p<\infty\) and any convex set with piecewise smooth boundary \(\Omega\subset E^n\) we have
\[ |F^{-1}\chi_\Omega Ff,L_p|\le k(p,\Omega)|f,L_p|, \]

\(^*\) This will be established below — Corollary 2 of Theorem 2.

where \(\chi_{\Omega}\) is the operator of multiplication by the characteristic function of the set \(\Omega\), and the constant \(k(p,\Omega)\) does not depend on the function \(\hat f(\bar x)\).

The proof follows from the theorem on multipliers in \(L_p\)-spaces [10].

Lemma 2. For any \(1 \le p < \infty\), \(1 \le q \le \infty\), and \(p' = p/(p-1)\), \(q' = q/(q-1)\), the space conjugate to the \(l^q L_p^\mu\) space is

\[ (l^q L_p^\mu)^* = l^{q'} L_{p'}^{\mu^{-1}} . \]

In this lemma, \(l^{q'}L_p\) for \(q'=1,\infty\) denotes the space of all functions \(f\) with finite \(|f,l^{q'}L_p|\) (in our usual definition (p. 65), the space \(l^{q'}L_p\) was called the closure of finite functions in the norm \(|l^{q'}L_p|\)).

We shall give the detailed proof for functions of one variable, since the remaining cases can be considered analogously. First let us show that \((l^q L_p)^*=l^{q'}L_{p'}\). Let \(\chi_j(\alpha)\) be the characteristic function of the set \(2^j<|\alpha|<2^{j+1}\).

A linear functional \(Af\) on \(l^qL_p\) is represented in a unique way as the sum of functionals \(A_j f\) coinciding with \(Af\) on the subspace of functions of the form \(f_j=F^{-1}\chi_j Ff\) and equal to zero on the subspace of functions of the form \(F^{-1}(1-\chi_j)Ff\), \(Af=\sum_j A_j f\). Each functional \(A_j f\) is simultaneously a functional on \(L_p\), and therefore is represented in the form \(A_j f=(g_j,f)\), where \(|g_j,L_{p'}|<\infty\), \(F^{-1}(1-\chi_j)Fg_j=0\),

\[ Af=\sum_j (g_j,f)=\sum_j (g_j,f_j);\quad \|A\|=\sup_f \sum_j (g_j,f_j)\,|f,l^qL_p|^{-1} \le \]

\[ \le \sup_f \sum_j |g_j,L_{p'}|\,|f_j,L_p|\,|f,l^qL_p|^{-1} \le \sup_f \left(\sum_j |g_j,L_{p'}|^{q'}\right)^{1/q'} \left(\sum_j |f_j,L_p|^q\right)^{1/q}|f,l^qL_p|^{-1} = \]

\[ = |g,l^{q'}L_{p'}|. \]

But now let \(h^{(j)}=|g_j|^{p'-1}\operatorname{sgn} g_j\), and take the function \(f(x)\) in the form

\[ f=\sum_j f_j=\sum_j c_j F^{-1}\chi_j Fh^{(j)} =\sum_j c_j h_j^{(j)},\quad \text{where } c_j=|g_j,L_{p'}|^{q'-1}|h_j^{(j)},L_p|^{-1}. \]

Then

\[ Af=\sum_j (g_j,f_j)=\sum_j c_j(g_j,h_j^{(j)})=\sum_j c_j(g_j,h^{(j)})= \]

\[ =\sum_j c_j \int g_j|g_j|^{p'-1}\overline{\operatorname{sgn} g_j}\,dx =\sum_j c_j |g_j,L_{p'}|^{p'}= \]

\[ =\sum_j c_j |g_j,L_{p'}|\cdot |h^{(j)},L_p|. \]

By Lemma 1, \(|h^{(j)},L_p| \ge k^{-1}|h_j^{(j)},L_p|\). Therefore

\[ Af\ge k^{-1}\sum_j c_j |g_j,L_{p'}|\,|h_j^{(j)},L_p| = k^{-1}\sum_j |g_j,L_{p'}|^{q'} = k^{-1}|g,l^{q'}L_{p'}|^{q'}= \]

\[ = k^{-1}|g,l^{q'}L_{p'}|\,|f,l^qL_p|, \]

i.e.

\[ |g,l^{q'}L_{p'}|\le k\,Af\,|f,l^qL_p|^{-1}\le k\|A\|. \]

Thus, the functional \(Af\) is bounded if and only if \(g\in l^{q'}L_{p'}\), which was to be proved.

Let us now consider the linear functional \(Af\) on the space \(l^q L_p^\mu\). Since \(l^q L_p^\mu\) is isomorphic and isometric to the space \(l^q L_p\), the functional \(Af\) corresponds to a unique functional \(Bf'\), defined on the whole space \(l^q L_p\), where \(f' = F^{-1}\mu^{-1}Ff\),
\[ Af=Bf'=(g,f')=(Fg,Ff')=((Fg)\mu^{-1},Ff)=(F^{-1}\mu^{-1}Fg,f). \]
Thus a one-to-one correspondence preserving the norm is established between the spaces \((l^q L_p^\mu)^*\) and \(l^{q'}L_{p'}^{\mu^{-1}}\).

For functions of one variable the Paley–Littlewood inequality is known: for any \(1<p<\infty\),
\[ c_1|f,L_p|\leq \left|\left(\sum_{j=-\infty}^{\infty}|f_j(x)|^2\right)^{1/2},L_p\right|\leq c_2|f,L_p| \]
with constants \(c_1,c_2>0\), independent of the function \(f(x)\).

The proof of this fact, given in [11], admits a generalization to functions of \(n\) real variables.

Lemma 3. For every \(1<p<\infty\) there exist positive constants \(c_1(p)\) and \(c_2(p)\), depending only on \(p\), such that for every function \(f(\bar x)\), \(\bar x\in E^n\), the inequalities
\[ c_1(p)|f,L_p|\leq \left|\left(\sum_{\bar m}|f_{\bar m}(\bar x)|^2\right)^{1/2},L_p\right| \leq c_2(p)|f,L_p| \tag{2} \]
hold.

We omit the proof, since it follows entirely the ideas of [11]\(^*\).

Theorem 1. \(l^2L_2=L_2\) and \(l^qL_p\ne L_p\) for \(p\ne 2\) or \(p=2\ne q\). If \(q\leq \min(p,2)\), then \(L_p\supset l^qL_p\); if \(q>\max(p,2)\), then \(L_p\subset l^qL_p\); whereas if \(1<p<q<2\) or \(2<q<p<\infty\), then there can be no inclusion relation between the spaces \(L_p\) and \(l^qL_p\).

Proof. Using formula (2), for \(1<p\leq 2\) we have
\[ |f,L_p|\leq c_1^{-1}\left[\int\left(\sum_{\bar m}|f_{\bar m}(\bar x)|^2\right)^{p/2}d\bar x\right]^{1/p}\leq \]
\[ \leq c_1^{-1}\left[\int\sum_{\bar m}|f_{\bar m}(\bar x)|^p\,d\bar x\right]^{1/p} = c_1^{-1}|f,l^pL_p|, \]
and for \(2\leq p<\infty\),
\[ |f,L_p|\leq c_1^{-1}\left[\int\left(\sum_{\bar m}|f_{\bar m}(\bar x)|^2\right)^{p/2}d\bar x\right]^{1/p}\leq \]
\[ \leq c_1^{-1}\left|\sum_{\bar m}|f_{\bar m}(\bar x)|^2,L_{p/2}\right|^{1/2} \leq c_1^{-1}\left(\sum_{\bar m}|f_{\bar m}^2(\bar x),L_{p/2}|\right)^{1/2} = c_1^{-1}|f,l^2L_p|. \]
It is also not difficult to see that if \(q_1<q_2\), then
\[ l^{q_1}L_p\subset l^{q_2}L_p. \]

Hence, for \(1<p\leq 2\),
\[ L_p\supset l^pL_p\supset l^qL_p\quad (q<p), \]

\[ \phantom{.} \]

\(^*\) Inequalities (2) are also given in [12].

and for \(2 \le p < \infty\)
\[ L_p \supset l^2 L_p \supset l^q L_p \quad (q<2). \]

Thus, \(L_p \supset l^q L_p\) for \(q \le \min(p,2)\).

Let us note that for two Banach spaces \(B_1\) and \(B_2\), from \(B_1 \supset B_2\) it follows that \(B_1^* \subset B_2^*\). Using this remark and Lemma 2, we obtain that
\[ L_p \subset l^q L_p \quad \text{for } q \ge \max(p,2). \]

To complete the proof it remains to establish that for \(1<p<q<2\) the sets \(\omega_1=L_p \setminus l^qL_p\) and \(\omega_2=l^qL_p \setminus L_p\) are nonempty.

If
\[ f(\bar{x})=\prod_{j=1}^n \varphi_j(x_j), \]
then
\[ |f,L_p|=\prod_{j=1}^n |\varphi_j(x_j),L_p| \]
and
\[ |f,l^qL_p|=\prod_{j=1}^n |\varphi_j(x_j),l^qL_p|. \]
Therefore it suffices to restrict ourselves to the one-dimensional case \((n=1)\). Let the function \(f(x)\) be such that
\[ f_m(x)=m^{-1/q}\frac{\sin x}{x}\exp(i2^m x) \]
for \(m\ge 1\), and \(f_m(x)=0\) for \(m\le 0\),
\[ |f,L_p|\le c_1\left|\left(\sum_m |f_m|^2\right)^{1/2},L_p\right| = c_1\left(\sum_{m=1}^{\infty} m^{-2/q}\right)^{1/2} \left|\frac{\sin x}{x},L_p\right|<\infty, \]
\[ |f,l^qL_p|=\left(\sum_{m=1}^{\infty} |f_m,L_p|^q\right)^{1/q} =\left(\sum_{m=1}^{\infty} m^{-1}\right)^{1/q} \left|\frac{\sin x}{x},L_p\right|=\infty, \]
i.e. \(f(x)\in\omega_1\).

To construct a function from the set \(\omega_2\), we use a theorem [13] asserting that if \(1<p<2\) and \((Ff)(a)\) is an even, positive, monotonically decreasing function, then \(f(x)\in L_p\) if and only if \((Ff)(a)|a|^{(p-2)/p}\in L_p\).

We define the function \(f(x)\) by its Fourier transform:
\[ (Ff)(\alpha)= \begin{cases} 2, & \text{for } \alpha\in[-2,2],\\ m^{-1/p}|\alpha|^{(1-p)/p}, & \text{for } |\alpha|\in[2^m,2^{m+1}],\quad m=1,2,\ldots, \end{cases} \]
\[ |(Ff)(\alpha)|\alpha|^{(p-2)/p},L_p| \ge \left(\sum_{m=1}^{\infty} m^{-1}\int_{2^m}^{2^{m+1}}\frac{d\alpha}{\alpha}\right)^{1/p} =\infty. \]

For \(m=1,2,\ldots\)
\[ |f_m(x),L_p|= \left|\left(\int_{-2^{m+1}}^{-2^m}+\int_{2^m}^{2^{m+1}}\right) m^{-1/p}|\alpha|^{(1-p)/p}\exp(-i\alpha x)\,d\alpha,L_p\right| \]
\[ = m^{-1/p}\left|\left(\int_{-2}^{-1}+\int_1^2\right) |\beta|^{(1-p)/p}\exp(-i\beta y)\,d\beta,L_p;y\right| =cm^{-1/p}. \]

For \(m=0,-1,-2,\ldots\)
\[ |f_m(x),L_p|= 2\left|\left(\int_{-2^{m+1}}^{-2^m}+\int_{2^m}^{2^{m+1}}\right) \exp(-i\alpha x)\,d\alpha,L_p\right| \le c\,2^{m(1-p)/p}. \]

Therefore

\[ |f(x),l^q L_p|\leq c\left(\sum_{m=-\infty}^{0}2^{mq(p-1)/p}+\sum_{m=1}^{\infty}m^{-q/p}\right)^{1/q}<\infty \]

and \(f(x)\in \omega_2\).

Corollary. The assertion of Theorem 1 remains valid if in its formulation the spaces \(L_p,\ l^q L_p\) are replaced by the spaces \(L_p^\mu,\ l^q L_p^\mu\), respectively.

Between the spaces \(L_p\) and \(l^q L_p\) one can also establish relations of another kind, taking into account the possibility of different smoothness of one and the same function in different spaces.

Theorem 2. Let \(\nu(x)\) be an even positive function, monotonically increasing as \(|x|\to\infty\), \(\nu(0)=1\), and

\[ \int_{0}^{\infty}\nu^{-1}(x)\,dx=1. \]

We have

\[ l^q L_p \supset L_p^{\mu_1(\bar\alpha)}, \tag{3} \]

where

\[ \mu_1(\bar\alpha)=\prod_{j=1}^{n}\mu_1(\alpha_j),\quad \mu_1(\alpha_j)=1,\quad \text{if } q\geq \gamma=\max(p,2), \]

\[ \mu_1(\alpha_j)=\bigl[\nu(\ln|\alpha_j|)\bigr]^{(\gamma-q)/\gamma q},\quad \text{if } q<\gamma, \]

\[ L_p \supset l^q L_p^{\mu_2(\alpha)}, \tag{4} \]

where

\[ \mu_2(\bar\alpha)=\prod_{j=1}^{n}\mu_2(\alpha_j),\quad \mu_2(\alpha_j)=1,\quad \text{if } q\leq \delta=\min(p,2), \]

and

\[ \mu_2(\alpha_j)=\bigl[\nu(\ln|\alpha_j|)\bigr]^{(q-\delta)/\delta q},\quad \text{if } q>\delta. \]

Proof. If formula (3) is proved, then (4) will follow from Lemma 2. Indeed, passing in (3) to the conjugate spaces, we obtain

\[ l^{q'}L_{p'} \subset L_{p'}^{\mu_1^{-1}(\alpha)}. \]

But the condition \(q\geq \gamma\) is transformed into the condition \(q'\leq \delta=\min(p',2)\). Therefore \(\mu_1(\bar\alpha)\) is replaced by \(\mu_2(\alpha)\), and we in fact obtain formula (4).

Consider (3) for \(q<\gamma\):

\[ \left(\sum_{\bar m}|f_{\bar m},L_p|^q\right)^{1/q} = \left(\sum_{\bar m}\left[|f_{\bar m},L_p|\,\mu_1(\bar m)\right]^q\mu_1^{-q}(\bar m)\right)^{1/q}\leq \]

\[ \leq \left(\sum_{\bar m}|f_{\bar m}\mu_1(\bar m),L_p|^\gamma\right)^{1/\gamma} \left(\sum_{\bar m}[\mu_1(\bar m)]^{-\gamma q/(\gamma-q)}\right)^{(\gamma-q)/\gamma q}. \]

But

\[ |f_{\bar m}\mu_1(\bar m),L_p| \leq C|F^{-1}\chi_{\bar m}(\bar\alpha)\mu_1(\bar\alpha)Ff,L_p|, \]

since the function

\[ \chi_{\bar m}(\bar\alpha)\,\mu_1^{(\bar m)}\,\mu_1^{-1}(\bar\alpha) \]

is a multiplier in \(L_p\).

Therefore, according to Theorem 1, we have

\[ |f,l^q L_p|\leq C|f,l^\gamma L_p^{\mu_1}|\leq C_1|f,L_p^{\mu_1}|, \tag{5} \]

which was to be established.

Corollary 1. With any measurable function \(\lambda(\bar\alpha)\) we have

\[ l^q L_p^{\lambda(\bar\alpha)}\supset L_p^{\mu_1(\bar\alpha)\lambda(\bar\alpha)},\quad L_p^{\lambda(\bar\alpha)}\supset l^q L_p^{\mu_2(\bar\alpha)\lambda(\bar\alpha)}. \tag{6} \]

One can calculate that, for a function to belong to the class \(L_p^{\mu_1(\alpha)}\), it is sufficient that the following conditions be simultaneously fulfilled:

\[ \int |f(\bar{x})|^p (|\bar{x}|+1)^\varepsilon\,d\bar{x}<\infty \]
with some positive \(\varepsilon\);

\[ f(\bar{x})\in L_p^{\mu_3(\bar{\alpha})}, \tag{6'} \]

where

\[ \mu_3(\bar{\alpha})=\prod_{j=1}^n \mu_3(\alpha_j),\qquad \mu_3(\alpha_j)= \begin{cases} 1, & \text{for } |\alpha_j|<1,\\ \mu_1(\alpha_j), & \text{for } |\alpha_j|>1. \end{cases} \]

In particular, we obtain

Corollary 2. Finite functions from the space \(L_p^\varepsilon\) (with arbitrary \(\varepsilon>0\)) belong to the space \(l^qL_p\) for all \(1\le q\le \infty\).

Corollary 3. For any \(\varepsilon>0,\ 1\le q\le\infty,\ 1<p<\infty\), and finite function \(f(\bar{x})\), the following inequalities hold:

\[ |f,l^qL_p|\le C_1(\varepsilon,q,p)|f,L_p^\varepsilon| \le C_2(\varepsilon,p)|f,W_p^{2\varepsilon}|\le \]

\[ \le C_3(\varepsilon,q,p)|f,B_{p,q}^{3\varepsilon}| \le C_4(\varepsilon,q,p)|f,H_p^{4\varepsilon}|\le \]

\[ \le C_5(\varepsilon,p)|f,L_p^{5\varepsilon}| \le C_6(\varepsilon,q,p)|f,l^qL_p^{6\varepsilon}|. \]

The first of these inequalities was established in Corollary 2; the second through fifth inequalities are known or follow easily from known results [14, 5].

Only the last inequality needs verification, and it is sufficient to consider it for functions of one variable:

\[ |f,L_p^{5\varepsilon}| \le C_7\left[\,|F^{-1}\chi_{(-1,1)}Ff,L_p^{\mu_2^{-1}(\alpha)}| +\right. \]

\[ \left. +|F^{-1}(1-\chi_{(-1,1)})Ff,L_p^{5\varepsilon}|\right] \le C_8|f,l^\delta L_p^{(\alpha^{6\varepsilon}+1)\mu_2^{-1}(\alpha)}| \le \]

\[ \le C_9|f,l^qL_p^{6\varepsilon}|. \]
Here \(\delta=\min(p,2)\).

§ 2. Some bounded operators in the spaces \(l^qL_p\)

Let the linear operator \(A\) act on functions from the space \(l^qL_p\) according to the formula

\[ g(\bar{x})=(Af)(\bar{x})=\bigl[F^{-1}\Phi(\bar{\alpha})Ff\bigr](\bar{x}). \]

In the case when \(A\) is a bounded operator mapping \(l^{q_1}L_{p_1}\) into \(l^{q_2}L_{p_2}\), we shall call the function \(\Phi(\bar{\alpha})\) a multiplier from \(l^{q_1}L_{p_1}\) into \(l^{q_2}L_{p_2}\). Analogously, a multiplier from the space \(L_{p_1}\) into the space \(L_{p_2}\) is defined.

Lemma 4. Every multiplier from \(L_{p_1}\) to \(L_{p_2}\) is simultaneously a multiplier from \(l^qL_{p_1}\) to \(l^qL_{p_2}\).

Proof. Let the operator \(Af=F^{-1}\Phi Ff\) be bounded from \(L_{p_1}\) to \(L_{p_2}\). Then

\[ |Af,l^qL_{p_2}| =\left(\sum_{\bar{m}} |(Af)_{\bar{m}},L_{p_2}|^q\right)^{1/q} = \left(\sum_{\bar{m}} |A(f_{\bar{m}}),L_{p_2}|^q\right)^{1/q} \le \|A\|\,|f,l^qL_{p_1}|. \]

Corollary. Suppose

\[ \Phi(\bar{\alpha})=\Phi_1(\bar{\alpha})\prod_{j=1}^n |\alpha_j|^\gamma \Gamma^\gamma(\ln|\alpha_j|)^\beta, \]

where \(\Phi_1(\bar\alpha)\) is a multiplier from \(l^{q_1}L_{p_1}\) to \(l^{q_1}L_{p_1}\), \(0 \le \gamma < \dfrac{1}{p}\), \(\beta \ge 0\). Then \(\Phi(\bar\alpha)\) is a multiplier from \(l^{q_1}L_{p_1}\) to \(l^{q_2}L_{p_2}\), where

\[ 0 \le \frac{1}{p_1}-\frac{1}{p_2}\le \gamma,\qquad \frac{1}{q_2}-\frac{1}{q_1}\le \beta . \]

The proof that \(\Phi_1(\bar\alpha)|\alpha_1,\ldots,\alpha_n|^\gamma\) is a multiplier from \(l^{q_1}L_{p_1}\) to \(l^{q_1}L_{p_2}\) follows from Lemma 4 and the theorem on multipliers from \(L_{p_1}\) to \(L_{p_2}\) [12]. And the fact that

\[ F^{-1}\prod_{j=1}^{n}|\nu(\ln|\alpha_j|)|^\beta Ff \]

is a bounded operator from \(l^{q_1}L_{p_2}\) to \(l^{q_2}L_{p_2}\) was in fact proved in Theorem 2—the first part of formula (5).

Theorem 3. Let the function \(\Phi(\bar\alpha)=\Phi(\alpha_1,\ldots,\alpha_n)\) be defined for real \(\alpha_j\) \((j=1,\ldots,n)\) and, for each set \((k_1,\ldots,k_n)\), where \(k_j\) takes the values \(0\) or \(1\), be analytically continuable from the set

\[ \prod_{j=1}^{n}\bigl[0<(-1)^{k_j}\alpha_j<\infty\bigr] \]

to complex values \(\bar\alpha=(\alpha_1,\ldots,\alpha_n)\) forming the set

\[ \omega_{\bar k}^{\varepsilon}=\prod_{j=1}^{n}\left(|\arg[(-1)^{k_j}\alpha_j]|<\varepsilon\right) \]

with some \(\varepsilon>0\).

If, for all possible sets \(\bar k=(k_1,\ldots,k_n)\), the function \(\Phi(\bar\alpha)\) remains bounded by some constant \(C\) on the sets \(\omega_{\bar k}^{\varepsilon}\), then it is a multiplier from \(l^qL_p\) to \(l^qL_p\), and the norm of the operator \(Af=F^{-1}\Phi Ff\) does not exceed \(Ck(p,\varepsilon)\), where \(k(p,\varepsilon)\) is some absolute constant.

Proof. Taking into account the theorem on multipliers from \(L_p\) to \(L_p\) [12], it is enough to establish boundedness, for real \((\alpha_1,\ldots,\alpha_n)\), of the expressions

\[ \left|d_{j_1}\cdots d_{j_k}D_{\alpha_{j_1}\cdots \alpha_{j_k}}^{k}\Phi(\bar\alpha)\right|, \]

where \(j_l\ne j_m\) for \(l\ne m\), \(k\) takes the values \(1,2,\ldots,n\), and all possible sets \((j_1,\ldots,j_k)\) are considered. We shall show, for example, the boundedness of \(|\alpha_1D_{\alpha_1}\Phi(\bar\alpha)|\). The remaining expressions can be considered analogously:

\[ D_{\alpha_1}\Phi(\bar\alpha) = \frac{1}{2\pi i}\int_{0}^{2\pi} \frac{\Phi(\alpha_1+\varepsilon\alpha_1\exp(i\varphi),\alpha_2,\ldots,\alpha_n)\,\varepsilon\alpha_1\,d\exp(i\varphi)} {[\varepsilon\alpha_1\exp(i\varphi)]^2} = \]

\[ =(2\pi\varepsilon\alpha_1)^{-1}\int_{0}^{2\pi} \exp(-i\varphi)\, \Phi(\alpha_1+\varepsilon\alpha_1\exp(i\varphi),\alpha_2,\ldots,\alpha_n)\,d\varphi, \]

\[ \left|\alpha_1D_{\alpha_1}\Phi(\bar\alpha)\right| \le (2\pi\varepsilon)^{-1}\int_{0}^{2\pi} \left|\Phi(\alpha_1+\varepsilon\alpha_1\exp(i\varphi),\alpha_2,\ldots,\alpha_n)\right|\,d\varphi \le C/\varepsilon . \]

This also yields the estimate of the operator norm.

Let \(\bar x'=(x_1,\ldots,x_l)\), \(\bar x''=(x_{l+1},\ldots,x_n)\), \((\bar x',\bar x'')=\bar x\). We shall denote the Fourier transform operator with respect to the variables \(\bar x'\) by \(F'\).

The function \(\Phi(\bar\alpha',\bar x'')=\Phi(\alpha_1,\ldots,\alpha_l,x_{l+1},\ldots,x_n)\) will be called a multiplier from \(l^qL_{p,\bar x'}\) to \(l^qL_{p,\bar x'}\) if the operator

\[ Af(\bar x')=\bigl[F'^{-1}\Phi F'f\bigr](\bar x') \]

is a bounded operator from \(l^qL_{p,\bar x'}\) to \(l^qL_{p,\bar x'}\).

Theorem 4. Let the function \(\Phi(\bar\alpha',\bar x'')\) be defined for real \((\bar\alpha',\bar x'')\) and, for any \(\bar x''\) and each set \((k_1,\ldots,k_l)\), where \(k_j\) takes the values \(0\) or \(1\), admit analytic continuation from the set

\[ \prod_{j=1}^{l}\left[0<(-1)^{k_j}\alpha_j<\infty\right] \]

to complex values \(\bar\alpha'=(\alpha_1,\ldots,\alpha_l)\), forming the set

\[ \omega_{\bar k'}^{\varepsilon} = \prod_{j=1}^{l} \left(\left|\arg[(-1)^{k_j}\alpha_j]\right|<\varepsilon\right) \]

with some \(\varepsilon>0\). If, for all possible sets \(\bar k'=(k_1,\ldots,k_l)\) and uniformly in \(\bar\alpha'\in\omega_{\bar k'}^{\varepsilon}\), the function \(\Phi(\bar\alpha',\bar x'')\) has finite \(l^q L_{p,\bar x''}\)-norm, then the function \(\Phi(\bar\alpha',\bar x'')\) is a multiplier from \(l^q L_{p,\bar x''}\) into \(l^q L_{p,\bar x}\).

Proof. For simplicity we shall assume that \(\bar\alpha'=\alpha_1\) is a one-dimensional variable and that \(\Phi(\alpha_1,\bar x'')=0\) for \(\alpha_1<0\), although the arguments given below also apply in the general case:

\[ g(\bar x)=Af(\bar x')=[F'^{-1}\Phi F'f](\bar x), \]

\[ g_{\bar m}(\bar x) = \int_{\Pi(\bar m)} d\bar\alpha\, \exp(-i\bar\alpha\bar x)(F'f)(\alpha_1) \int d\bar y''\,\Phi(\alpha_1,\bar y'')\exp(i\bar\alpha''\bar y'') = \]

\[ = \int_{\Pi(m_1)} d\alpha_1\, \exp(-i\alpha_1x_1)(F'f)(\alpha_1)\Phi_{\bar m''}(\alpha_1,\bar x''), \]

where

\[ \Phi_{\bar m''}(\alpha_1,\bar x'') = \int_{\Pi(\bar m'')} d\bar\alpha''\, \exp(-i\bar\alpha''\bar x'')(F''\Phi)(\bar\alpha), \]

\[ \left|g_{\bar m},L_{p,\bar x}\right| = \left| \int_{\Pi(m_1)} d\alpha_1\, \exp(-i\alpha_1x_1)(F'f)(\alpha_1)\Phi_{\bar m''}(\alpha_1,\bar x''),L_{p,\bar x} \right| = \]

\[ = \left| \Phi_{\bar m''}(2^{m_1+1},\bar x'') \int_{2^{m_1}}^{2^{m_1+1}} (F'f)(\alpha_1)\exp(-i\alpha_1x_1)\,d\alpha_1 - \right. \]

\[ \left. - \int_{2^{m_1}}^{2^{m_1+1}} \frac{\partial}{\partial\alpha_1}\Phi_{\bar m''}(\alpha_1,\bar x'') \int_{2^{m_1}}^{\alpha_1} (F'f)(\beta)\exp(-i\beta x)\,d\beta\,d\alpha_1, L_{p,\bar x} \right| \le \]

\[ \le \left|\Phi_{\bar m''}(2^{m_1+1},\bar x''),L_{p,\bar x''}\right| \cdot \left|f_{m_1}(x_1),L_{p,x_1}\right| + \]

\[ + \int_{2^{m_1}}^{2^{m_1+1}} d\alpha_1 \left| \frac{\partial}{\partial\alpha_1}\Phi(\alpha_1,\bar x''),L_{p,\bar x''} \right| \cdot \left| \int_{2^{m_1}}^{\alpha} (F'f)(\beta)\exp(-i\beta,x)\,d\beta,L_{p,x} \right|. \]

Using Lemma 1, we obtain

\[ \left|g_{\bar m}(\bar x),L_{p,\bar x}\right| \le \left|f_{m_1}(x_1),L_{p,x_1}\right| \cdot \left[ \left|\Phi_{\bar m''}(2^{m_1+1},\bar x''),L_{p,\bar x''}\right| + \right. \]

\[ \left. + k(p) \int_{2^{m_1}}^{2^{m_1+1}} d\alpha_1 \left| \frac{\partial}{\partial\alpha_1}\Phi(\alpha_1,\bar x''),L_{p,\bar x''} \right| \right]. \]

The function \(\Phi_{\bar m''}(\alpha_1,\bar x'')\) can be continued in \(\alpha_1\) as a single-valued analytic function from the half-plane \(x_1>0\).

Therefore

\[ \frac{\partial}{\partial a_1}\Phi(a_1,\bar{x}'')=(2\pi i\varepsilon a_1)^{-1} \int_0^{2\pi}\exp(-i\varphi)\Phi_{\bar{m}''}(a_1+ \]

\[ +a_1\varepsilon\exp(i\varphi),\bar{x}'')\,d\varphi \]

with some \(\varepsilon>0\). Hence,

\[ \begin{aligned} |g(\bar{x}),l^qL_{p,\bar{x}}| &\leq \Biggl\{\sum_{\bar{m}}2^q|f_{m_1}(x_1),L_{p,x_1}|^q \Biggl[ |\Phi_{\bar{m}''}(2^{m_1+1},\bar{x}''),L_{p,\bar{x}''}|^q \\ &\quad +k^q(p)(2\pi\varepsilon)^{-q} \left( \int_{2^{m_1}}^{2^{m_1+1}} \left| \frac{\partial}{\partial a_1} \int_0^{2\pi}d\varphi\, |\Phi_{\bar{m}''}(a_1+a_1\varepsilon\exp(i\varphi),\bar{x}''),L_{p,\bar{x}''}| \right|^q \right) \Biggr]\Biggr\}^{1/q} \\ &\leq \Biggl\{ \sum_{m_1}|f_{m_1}(x_1),L_{p,x_1}|^q2^q \Biggl[ \sum_{\bar{m}''}|\Phi_{\bar{m}''}(2^{m_1+1},\bar{x}''),L_{p,\bar{x}''}|^q \\ &\quad +k^q(p)(2\pi\varepsilon)^{-q}(2\pi)^{q-1} \left( \int_{2^{m_1}}^{2^{m_1+1}} a_1^{q/(q-1)}\,da_1 \right)^{q-1} \int_{2^{m_1}}^{2^{m_1+1}}da_1 \int_0^{2\pi}d\varphi \sum_{\bar{m}''} |\Phi_{\bar{m}''}(a_1+ \\ &\quad +a_1\varepsilon\exp(i\varphi),\bar{x}''),L_{p,\bar{x}''}|^q \Biggr] \Biggr\}^{1/q} \\ &\leq C(p,q,\varepsilon)\max_{(a_1,\varphi)} |\Phi(a_1+a_1\varepsilon\exp(i\varphi),\bar{x}''),l^qL_{p,\bar{x}''}|\cdot |f(x_1),l^qL_{p,x_1}|, \end{aligned} \]

where \(C(p,q,\varepsilon)\leq C_1(p,\varepsilon)\). This is what had to be proved.

Example. A multiplier from \(l^qL_{p,\bar{x}'}\) into \(l^qL_{p,\bar{x}}\) is the function

\[ \Phi(\bar{\alpha}',\bar{x}'')=\lambda^{(n-l)/p}\exp[-\lambda(|x_{l+1}|+\cdots+|x_n|)]. \]

Here

\[ \lambda=\sum_{j=1}^n|\alpha_j|^{m_j}+1, \]

and \(m_j\) \((j=1,\ldots,l)\) are arbitrary nonnegative numbers.

It is not difficult to see that \(\Phi(\bar{\alpha}',\bar{x}'')\) can be extended to a single-valued function analytic on some set \(\omega_{\bar{k}'}\). It remains to verify the boundedness of the \(l^qL_{p,\bar{x}''}\)-norm:

\[ \Phi(\bar{\alpha}',\bar{x}'')=\prod_{j=l+1}^n\varphi(\bar{\alpha}',x_j), \qquad \text{where }\ \varphi(\bar{\alpha}',x_j)=\lambda^{1/p}\exp(-\lambda|x_j|). \]

Therefore

\[ |\Phi(\bar{\alpha}',\bar{x}''),l^qL_{p,\bar{x}''}| = \prod_{j=l+1}^n |\varphi(\bar{\alpha}',x_j),l^qL_{p,x_j}| = |\varphi(\bar{\alpha}',y),l^qL_{p,y}|^{\,n-l}. \]

For simplicity we shall assume \(\varphi(\bar{\alpha}',y)=0\) for \(y<0\) (in exact calculations one should first split the function \(\varphi(\bar{\alpha}',y)\) into two summands concentrated on the half-axes \(y>0\) and \(y<0\)). Denote by \(\lambda_\varepsilon\) the \(\lambda\) corresponding to the arguments \(\alpha_j+\alpha_j\varepsilon\exp(i\psi_j)\), \(j=1,\ldots,l\),

\[ [F_y\varphi(\bar{\alpha}',y)](\beta)=\lambda^{1/p}(\lambda-j\beta)^{-1}. \]

To avoid complicating the notation, in what follows we shall take into account only positive \(\beta\):

\[ J_m=\left|\varphi_m(\bar{\alpha}' + \alpha'\varepsilon \exp(i\psi'),y),L_p,y\right| = \left|\int_{2^m}^{2^{m+1}} 2\lambda_\varepsilon^{1/p} \frac{\exp(-i\beta y)}{\lambda_\varepsilon-i\beta}\,d\beta,L_p,y\right|. \]

Making the changes of variables \(\beta=\gamma 2^{[\log_2|\lambda_\varepsilon|]}\), \(z=y2^{[\log_2|\lambda_\varepsilon|]}\), and denoting
\(\theta_\varepsilon=\lambda_\varepsilon 2^{-[\log_2|\lambda_\varepsilon|]}\) (where \([a]\) denotes the integer part of the number \(a\)), we obtain

\[ J_m= \left|\int_{2^{m-[\log_2|\lambda_\varepsilon|]}}^{2^{m-[\log_2|\lambda_\varepsilon|]+1}} d\gamma\, \theta_\varepsilon^{1/p}(\theta_\varepsilon-i\gamma)^{-1} \exp(-i\gamma z),L_p,z\right|. \]

Let us note that \(1\leq \operatorname{Re}\theta_\varepsilon\leq 2\), \(\arg\theta_\varepsilon=\arg\lambda_\varepsilon\in(-\varepsilon,\varepsilon)\). Therefore the function
\[ \rho(\gamma)=\theta_\varepsilon^{1/p}(\theta_\varepsilon-i\gamma)^{-1}(1+\gamma^2)^{1/2} \]
is a multiplier from \(L_p\) into \(L_p\), and it suffices for us to prove the finiteness of

\[ \left(\sum_{m=-\infty}^{\infty} \left|\psi_{m-[\log_2|\lambda_\varepsilon|]}(z),L_p,z\right|^q \right)^{1/q} = \left(\sum_{m=-\infty}^{\infty} \left|\psi_m(z),L_p,z\right|^q \right)^{1/q}, \]

where \((F\psi)(\gamma)=(1+\gamma^2)^{-1/2}\).

We could establish that \(\psi(z)\) belongs to the space \(l^qL_p\) by applying Corollary 1 of Theorem 2, but in view of the importance of this example we shall carry out direct computations. Obviously, it is enough to establish that \(\psi(z)\in l^1L_p\),

\[ (F\psi)(\gamma)=(1+\gamma^2)^{-\varepsilon/2}(1+\gamma^2)^{(\varepsilon-1)/2} =(1+\gamma^2)^{-\varepsilon/2}(F\zeta)(\gamma) \quad\text{with }\varepsilon=1/2p. \]

The function \(\zeta(z)\in L_p\) by the theorems of Hardy and Littlewood [13], since \((F\zeta)(\gamma)\) is even, monotonically decreasing as \(|\gamma|\to\infty\), and
\[ (F\zeta)(\gamma)\,|\gamma|^{(p-2)/p}\in L_p,\gamma \]
(for any \(1<p<\infty\)). Hence \(\psi(z)\in L_p^\varepsilon\), \(\varepsilon=1/2p\), and for such functions we have already proved the finiteness of
\[ \sum_{m=1}^{\infty}\left|\psi_m,L_p\right| \]
in the proof of Theorem 2. It remains to consider
\[ \sum_{m=-\infty}^{0}\left|\psi_m,L_p\right|: \]

\[ \left|\psi_m,L_p\right| = \left|\left(\int_{-2^{m+1}}^{-2^m}+\int_{2^m}^{2^{m+1}}\right) d\gamma\,\exp(-i\gamma z)(1+\gamma^2)^{-1/2},L_p,z\right| \leq \]

\[ \leq C_1 \left|\left(\int_{-2^{m+1}}^{-2^m}+\int_{2^m}^{2^{m+1}}\right) d\gamma\,\exp(-i\gamma z),L_p,z\right| \leq C_2\left|\frac{\sin 2^m z}{z},L_p,z\right| \leq C_3 2^{m(1-p)/p} \]

and therefore

\[ \sum_{m=-\infty}^{0}\left|\psi_m,L_p\right| \leq C_3\sum_{m=-\infty}^{0}2^{m(1-p)/p}<\infty. \]

§ 3. EMBEDDING THEOREMS

In this section it is assumed that the functions \(\mu(\bar{\alpha})\) have power growth, and the spaces
\[ l^qL_{p,\bar{x}}^{\bar{k}},\qquad (k_j\geq 0) \]
are considered. For brevity we shall denote the spaces
\[ l^qL_{p,x_1,\ldots,x_{j-1},x_j,x_{j+1},\ldots,x_n}^{0,\ldots,0,k_j,0,\ldots,0} \]
by
\[ l^qL_{p,\bar{x}}^{k_j}. \]

Lemma 5. \(l^{q} L_{p,\bar x}^{\bar k}=\displaystyle\bigcap_{j=1}^{n} l^{q}L_{p,\bar x}^{k_j}\).

Proof. Let \(f(\bar x)\in l^{q}L_{p,\bar x}^{\bar k}\). Then

\[ (Ff)(\bar\alpha)= \left(\sum_{l=1}^{n}|\alpha_l|^{k_l}+1\right)^{-1}(Fg)(\bar\alpha) = (|\alpha_j|^{k_j}+1)^{-1} \frac{|\alpha_j|^{k_j}+1}{\sum_{l=1}^{n}|\alpha_l|^{k_l}+1} (Fg)(\bar\alpha), \]

where \(g(\bar x)\in l^{q}L_{p,\bar x}\). Since the function

\[ \Phi_j(\bar\alpha)= (|\alpha_j|^{k_j}+1)\times \left(\sum_{l=1}^{n}|\alpha_l|^{k_l}+1\right)^{-1} \]

is a multiplier from \(l^{q}L_{p,\bar x}\) into \(l^{q}L_{p,\bar x}\) (Theorem 3), the function \(g_j(\bar x)\), defined by the formula
\[ (Fg_j)(\bar\alpha)=\Phi_j(\bar\alpha)(Fg)(\bar\alpha), \]
belongs to the space \(l^{q}L_{p,\bar x}\). Hence \(f(\bar x)\in l^{q}L_{p,\bar x}^{k_j}\) and

\[ l^{q}L_{p,\bar x}^{\bar k}\subseteq \bigcap_{j=1}^{n} l^{q}L_{p,\bar x}^{k_j}. \]

Now let
\[ f(\bar x)\in \bigcap_{j=1}^{n} l^{q}L_{p,\bar x}^{k_j}. \]
Then in the space \(l^{q}L_{p,\bar x}\) there will be found such functions \(g_j(\bar x)\) that the equalities
\[ (Ff)(\bar\alpha)(|\alpha_j|^{k_j}+1)=(Fg_j)(\bar\alpha),\qquad j=1,\ldots,n \]
hold. Hence
\[ (Ff)(\bar\alpha)\left(\sum_{j=1}^{n}|\alpha_j|^{k_j}+n\right) = \sum_{j=1}^{n}(Fg_j)(\bar\alpha)=(Fg)(\bar\alpha), \]
where the function
\[ g(\bar x)=\sum_{j=1}^{n}g_j(\bar x)\in l^{q}L_{p,\bar x}, \]
i.e.
\[ f(\bar x)\in l^{q}L_{p,\bar x}^{\bar k} \quad\text{and}\quad l^{q}L_{p,\bar x}^{\bar k}\supseteq \bigcap_{j=1}^{n} l^{q}L_{p,\bar x}^{k_j}. \]

Theorem 5. Suppose that on the hyperplane \(x_n=0\) a function
\[ g(\bar x')\in l^{q}L_{p,\bar x'}^{\bar k'} \]
is given, \(k_j\ge 0\) \((j=1,\ldots,n-1)\). Then for any \(l>0\) there exists a function \(f_l(\bar x)\) such that \(f_l(\bar x',0)=g(\bar x')\) in the sense of the norm of the space \(l^{q}L_{p,\bar x'}^{\bar k'}\), and
\[ f_l(\bar x)\in l^{q}L_{p,\bar x}^{\bar m}, \]
where
\[ \bar m'=\bar k'\left(1+\frac{1}{pl}\right),\qquad m_n=l+\frac{1}{p}. \]

Proof. Consider the function \(f_l(\bar x)\), defined by the formula

\[ (F'f_l)(\bar\alpha',x_n)= \begin{cases} (F'g)(\bar\alpha')\exp(-\lambda x_n), & \text{for } x_n>0,\\[6pt] (F'g)(\bar\alpha')\displaystyle\sum_{s=1}^{\left[l+\frac{1}{p}\right]+1} a_s\exp(\lambda x_n/s), & \text{for } x_n<0, \end{cases} \]

where the constants \(a_s\) are chosen so as to ensure the continuity of the function \((F'f_l)(\bar\alpha',x_n)\), together with \(\left[l+\dfrac1p\right]\) derivatives with respect to \(x_n\), at \(x_n=0\)*). Here

\[ \lambda=\left(\sum_{j=1}^{n-1}|\alpha_j|^{k_j}+1\right)^{1/l}. \]

Let us note that

\[ (F'g)(\bar\alpha') = \left(\sum_{j=1}^{n-1}|\alpha_j|^{k_j}+1\right)^{-1} (F'h)(\bar\alpha') = \lambda^{-l}(F'h)(\bar\alpha'), \]

where \(h(\bar x)\in l^qL_{p,\bar x'}\). Therefore, for \(x_n>0\),

\[ (F'f)(\bar\alpha',x_n) = \lambda^{-l}\exp(-\lambda x_n)(F'h)(\bar\alpha') = \lambda^{-\lambda-\frac1p}\bigl[\lambda^{1/p}\exp(-\lambda x_n)(F'h)(\bar\alpha')\bigr]. \]

For \(x_n<0\) the function has an analogous expression. But, according to the example discussed, the function
\[ F'^{-1}\bigl[\lambda^{1/p}\exp(-\lambda x_n)(F'h)(\bar\alpha')\bigr](\bar x) \]
for \(x_n>0\) (and the similar one for \(x_n<0\)) belongs to \(l^qL_{p,\bar x}\). Hence the function \(f_l(\bar x)\in l^qL_{p,\bar x}\) and has the smoothness ensured to it by the multiplier

\[ \lambda^{-l-\frac1p} = \left(\sum_{j=1}^{n-1}|\alpha_j|^{k_j}+1\right)^{-1-\frac1{pl}}, \]

i.e.

\[ f_l(\bar x)\in l^qL_{p,\bar x',x_n}^{\bar k'\left(1+\frac1{pl}\right),0}. \]

If we show that \(f_l(\bar x)\in l^qL_{p,\bar x',x_n}^{0,l+\frac1p}\), then the assertion of the theorem will follow from Lemma 5. In what follows we shall use the following notation:

\[ \left[l+\frac1p\right]=a,\qquad l+\frac1p=a+\gamma,\qquad 0\le\gamma<1. \]

Let

\[ (Ff_l)(\bar\alpha)(|\alpha_n|^{a+\gamma}+1)=(Fz_1)(\bar\alpha) \]

and it is necessary to show that \(z_1(\bar x)\in l^qL_{p,\bar x}\). Consider

\[ Fz_2(\bar\alpha)=(\alpha_n^a+i)Ff_l(\bar\alpha), \]

\[ (Fz_1)(\bar\alpha) = (|\alpha_n|^\gamma+1)\bigl[(|\alpha_n|^\gamma+1)^{-1}(\alpha_n^a+i)^{-1}(|\alpha_n|^{a+\gamma}+1)\bigr](Fz_2)(\bar\alpha). \]

Since

\[ \frac{(\alpha_n)^{a+\gamma}+1}{(|\alpha_n|^\gamma+1)(\alpha_n^a+i)} \]

is a multiplier from \(l^qL_{p,\bar x}\) into \(l^qL_{p,\bar x}\), it is enough to establish that the function \(z_2(\bar x)\in l^qL_{p,\bar x',x_n}^{0,\gamma}\). But

\[ (F'z_2)(\bar\alpha',x_n) = \bigl[(-iD_{x_n})^a+i\bigr](F'f_l)(\bar\alpha',x_n), \]

*) The proof of the possibility of such a choice of \(a_s\) belongs to Whitney and Hestenes and is presented in [15].

\[ (F' z_2)(\bar{\alpha}', x_n)= \begin{cases} (i^a+i\lambda^{-a})\lambda^{\frac1p-\gamma}\exp(-\lambda x_n)(F'h)(\bar{\alpha}'), & x_n>0,\\[4pt] \displaystyle \sum_{s=1}^{a+1} a_s\left[(-i)^a s^{-a}+i\lambda^{-a}\right]\lambda^{\frac1p-\gamma}\exp(\lambda x_n/s)(F'h)(\bar{\alpha}'), & x_n<0. \end{cases} \]

From the continuity of \((F' z_2)(\bar{\alpha}', x_n)\) we obtain

\[ i^a+i\lambda^{-a}=\sum_{s=1}^{a+1} a_s\left[\left(-\frac{i}{s}\right)^a+i\lambda^{-a}\right] \]

and one may write

\[ (F'z_2)(\bar{\alpha}',x_n)= \sum_{s=1}^{a+1} a_s\left[\left(-\frac{i}{s}\right)^a+i\lambda^{-a}\right](F'u_s)(\bar{\alpha}',x_n), \]

where

\[ (F'u_s)(\bar{\alpha}',x_n)= \lambda^{\frac1p-\gamma}(F'h)(\bar{\alpha}') \begin{cases} \exp(-\lambda x_n), & x_n>0,\\ \exp(\lambda x_n/s), & x_n<0. \end{cases} \]

All the functions \(\left(-\dfrac{i}{s}\right)^a+\lambda^{-a}\) are multipliers from \(l^q L_{p,\bar{x}'}\) into \(l^q L_{p,\bar{x}'}\); therefore, in order to have \(z_2(\bar{x})\in l^q L^0{}_{p,\bar{x}',x_n}^{\gamma}\), it is enough to establish that

\[ u_s(\bar{x})\in l^q L^0{}_{p,\bar{x}',x_n}^{\gamma} \qquad (s=1,\ldots,a+1). \]

We shall show this for one of the functions \(u_s(\bar{x})\), taking for simplicity \(u_1(\bar{x})\):

\[ (F'u_1)(\bar{\alpha}',x_n)= \lambda^{\frac1p-\gamma}\exp(-\lambda |x_n|)(F'h)(\bar{\alpha}'), \]

\[ (|\alpha_n|^\gamma+1)(Fu_1)(\bar{\alpha}) = 2\,\frac{\lambda^{1-\gamma+\frac1p}(|\alpha_n|^\gamma+1)} {\lambda^2+\alpha_n^2}\,(F'h)(\bar{\alpha}') =(Fv)(\bar{\alpha}), \]

and we shall prove that \(v(\bar{x})\in l^q L_{p,\bar{x}}\). We have

\[ |v,l^q L_{p,\bar{x}}| = 2\left( \sum_m \left| \int_{\Pi(m')} (F'h)(\bar{\alpha}')\exp(-i\bar{\alpha}'\bar{x}') \psi_{m_n}(\bar{\alpha}',x_n)\,d\bar{\alpha}', \,L_{p,\bar{x}}^q \right| \right)^{1/q}, \]

where

\[ \psi_{m_n}(\bar{\alpha}',x_n) = \int_{\Pi(m_n)} \exp(-i\alpha_n x_n) \frac{\lambda^{1-\gamma+\frac1p}(|\alpha_n|^\gamma+1)} {\lambda^2+\alpha_n^2}\,d\alpha_n. \]

Just as in the proof of Theorem 4, we establish that

\[ |v,l^q L_{p,\bar{x}}| \le C(p,\varepsilon)|h,l^q L_{p,\bar{x}'}| \max_{(\bar{\alpha}',\bar{\varphi}')} \left|\psi(\bar{\alpha}'+\varepsilon\exp(i\bar{\varphi}')\bar{\alpha}',x_n), l^q L_{p,x_n}\right|, \]

where

\[ \psi(\bar{\alpha}',x_n) = \int \lambda^{1-\gamma+\frac1p}(|\alpha_n|^\gamma+1) (\lambda^2+\alpha_n^2)^{-1} \exp(-i\alpha_n x_n)\,d\alpha_n. \]

As in the example considered earlier, denote by \(\lambda_\varepsilon\) the \(\lambda\) corresponding to the argument \(\bar{\alpha}'+\bar{\alpha}'\varepsilon\exp(i\bar{\varphi}')\), and put \(\theta_\varepsilon=\lambda_\varepsilon 2^{-[\log_2|\lambda_\varepsilon|]}\).

Then

\[ J_m=\left|\psi_{mn}\left(\bar a' + \bar a'\varepsilon \exp(i\bar\varphi'), x_n\right), L_p,x_n\right| =\left|\int_{\Pi(m_n)} d\alpha_n\, \frac{\lambda^{1-\gamma+\frac1p}\left(|\alpha_n|^\gamma+1\right)} {\lambda^2+\alpha_n^2} \times \right. \]

\[ \left. {}\times \exp(-i\alpha_n x_n), L_p,x_n\right|. \]

Making the change of variables

\[ \alpha_n=\beta\,2^{[\log_2|\lambda_\varepsilon|]},\qquad y=x_n\,2^{[\log_2|\lambda_\varepsilon|]},\qquad l=m-[\log_2|\lambda_\varepsilon|], \]

we obtain

\[ J_m= \left| \left(\int_{-2^{l+1}}^{-2^l}+\int_{2^l}^{2^{l+1}}\right) \exp(-i\beta y)\, \frac{|\beta|^\gamma+2^{-\gamma[\log_2|\lambda_\varepsilon|]}} {\theta_\varepsilon^2+\beta^2} \,d\beta,\ L_p,y \right| |\theta_\varepsilon|^{1-\gamma+\frac1p} \le \]

\[ \le C_1(p,\varepsilon)\,|\zeta_l(y),L_p,y|, \]

where

\[ (F\zeta)(\beta)=\frac{1+|\beta|^\gamma}{1+\beta^2} =\Phi^{-1}(\beta,\bar\alpha',\varepsilon)\, \frac{|\beta|^\gamma+2^{-\gamma[\log_2|\lambda_\varepsilon|]}} {\theta_\varepsilon^2+\beta^2}, \]

and the function

\[ \Phi(\beta,\bar\alpha',\varepsilon)= \frac{1+\beta^2}{\theta_\varepsilon^2+\beta^2}\, \frac{|\beta|^\gamma+2^{-\gamma[\log_2|\lambda_\varepsilon|]}} {1+|\beta|^\gamma} \]

is a multiplier from \(l^q L_p,x_n\) into \(l^q L_p,x_n\), uniformly in \(\bar\alpha'\), \(\bar\varphi'\):

\[ \left|\psi\left(\bar\alpha'+\bar\alpha'\varepsilon\exp(i\bar\varphi'),x_n\right), l^qL_p,x_n\right| \le C_1^{1/q}(p,\varepsilon) \left(\sum_{l=-\infty}^{\infty}|\zeta_l(y),L_p,y|^q\right)^{1/q}. \]

But

\[ (1+|\beta|^\gamma)(1+\beta^2)^{-1} = \left[(1+|\beta|^\gamma)(1+\beta^2)^{-1/2}\right](1+\beta^2)^{-1/2}, \]

and since \(0\le\gamma<1\), the first factor is a multiplier from \(l^qL_p,y\) into \(l^qL_p,y\), while the finiteness of the \(l^qL_p,y\)-norm of the function
\(F^{-1}[(1+\beta^2)^{-1/2}]\) was already computed in the example. Theorem 5 is proved.

Theorem 6. If \(f(\bar x)\in l^qL_{p,\bar x}^{\bar m}\) and \(m_n>\dfrac1p\), then \(f(\bar x',0)\in l^qL_{p,\bar x'}^{\bar k'}\), where

\[ \bar k'=\bar m'\left(1+\frac1{pm_n}\right). \]

Proof. There exists a function \(g(\bar x)\in l^qL_{p,\bar x}^{\bar m}\) such that

\[ (Ff)(\bar\alpha)=\left(\lambda+|\alpha_n|^{m_n}\right)^{-1}(Fg)(\bar\alpha), \]

where, for brevity, the notation

\[ \lambda=\sum_{j=1}^{n-1}|\alpha_j|^{m_j}+1 \]

has been introduced. Then

\[ (F'f)(\bar\alpha',0)= \int_{-\infty}^{\infty}\frac{(Fg)(\bar\alpha)\,d\alpha_n} {\lambda+|\alpha_n|^{m_n}} -\sum_{m_n}\int_{\Pi(m_n)} (Fg)(\bar\alpha)\left(\lambda+|\alpha_n|^{m_n}\right)^{-1}d\alpha_n, \]

\[ \left|f(\bar x',0),l^qL_{p,\bar x'}^{\bar k'}\right| = \left\{ \sum_{\bar m'}\int_{\Pi(\bar m')} \left| \sum_{m_n}\int_{\Pi(m_n)} (Fg)(\bar\alpha)\left(\lambda+\right. \right. \]

\[ \left. \left. {}+|\alpha_n|^{m_n}\right)^{-1} \left(\sum_{j=1}^{n-1}|\alpha_j|^{k_j}+1\right) d\alpha_n\,d\bar\alpha',L_p,\bar x' \right|^q \right\}^{1/q} = \]

\[ =\left\{\sum_{\bar m'}\left|\int_{\Pi(\bar m')} d\bar\alpha'\sum_{m_n}\int_{\Pi(m_n)} (Fh)(\bar\alpha)\,(\lambda+|\alpha_n|^{m_n})^{-1} \lambda^{1-\frac{1}{pm_n}}\,d\alpha_n,L_{p,\bar x'}\right|^q\right\}^{1/q}. \]

Here
\[ (Fh)(\bar\alpha)=\Phi_1(\bar\alpha)(Fg)(\bar\alpha),\quad \Phi_1(\bar\alpha)= \left(\sum_{j=1}^{n-1}|\alpha_j|^{m_j\left(1-\frac{1}{pm_n}\right)}+1\right)\times \]
\[ \times\left(\sum_{j=1}^{n-1}|\alpha_j|^{m_j}+1\right)^{\frac{1}{pm_n}-1} \]
is a multiplier from \(l^q L_{p,\bar x}\) into \(l^q L_{p,\bar x}\); therefore

\[ h(\bar x)\in l^q L_{p,\bar x}; \]

\[ |f(\bar x',0),l^q L_{p,\bar x'}^{\bar k'}|= \left\{\sum_{\bar m'}\left|\sum_{m_n}\int_{\Pi(\bar m')}d\bar\alpha'\, \exp(-i\bar\alpha'\bar x')\int dx_n\,(F'h_{m_n})(\bar\alpha',x_n)\times\right.\right. \]
\[ \left.\left.\times \Phi_{m_n}(\bar\alpha',x_n),L_{p,\bar x'}\right|^q\right\}^{1/q} \le \left\{\sum_{\bar m'}\left[\sum_{m_n}\int dx_n\left| \int_{\Pi(\bar m')}d\bar\alpha'\,\exp(-i\bar\alpha'\bar x')\times\right.\right.\right. \]
\[ \left.\left.\left.\times (F'h_{m_n})(\bar\alpha',x_n)\Phi(\alpha',x_n),L_{p,\bar x'}\right|^q\right]\right\}^{1/q}, \tag{7} \]
where
\[ \Phi(\bar\alpha',x_n)= \int \lambda^{1-\frac{1}{pm_n}}(\lambda+|\alpha_n|^{m_n})^{-1} \exp(-i\alpha_n x_n)\,d\alpha_n. \]

Further, repeating the arguments set out in the proof of Theorem 4, we obtain

\[ \left|\int_{\Pi(\bar m')}d\bar\alpha'\,\exp(-i\bar\alpha'\bar x') (F'h_{m_n})(\bar\alpha',x_n)\Phi_{m_n}(\bar\alpha',x_n),L_{p,\bar x'}\right| \le \]
\[ \le |h_{\bar m}(\bar x),L_{p,\bar x}|\,C_1(p,\varepsilon) \left[|\Phi_{m_n}(2^{\bar m'}x_n)|+\right. \]
\[ \left.+\int_{\Pi(\bar m')} \frac{d\bar\alpha'}{\bar\alpha'}\int_0^{2\pi}d\bar\varphi'\, |\Phi_{m_n}(\bar\alpha'+\bar\alpha'\varepsilon\exp(i\bar\varphi'),x_n)|\right]. \]

In formula (7) we estimate \(\int dx_n\) and \(\sum_{m_n}\) by means of Hölder’s inequality, setting \(p'=p/(p-1)\), \(q'=q/(q-1)\):

\[ |f(\bar x',0),l^q L_{p,\bar x'}^{\bar k'}| \le C_2(p,\varepsilon)\left\{ \sum_{\bar m'}\sum_{m_n}|h_{\bar m}(\bar x),L_{p,\bar x}|^q \left[|\Phi(2^{\bar m'},x_n),l^{q'}L_{p',x_n}|+\right.\right. \]
\[ \left.\left. +\int_{\Pi(\bar m')}\frac{d\bar\alpha'}{\bar\alpha'}\int_0^{2\pi}d\bar\varphi'\, |\Phi(\bar\alpha'+\bar\alpha'\varepsilon\exp(i\bar\varphi'),x_n),l^{q'}L_{p',x_n}|^q \right]\right\}^{1/q} \le \]
\[ \le C_3(p,\varepsilon)|h(\bar x),l^q L_p|\, \max_{(\bar\alpha',\bar\varphi')} |\Phi(\bar\alpha'+\bar\alpha'\varepsilon\exp(i\bar\varphi'),x_n),l^{q'}L_{p',x_n}|. \]

Thus, it remains to estimate the \(l^{q'}L_{p',x_n}\) norm of the function
\(\Phi(\overline{\alpha}' + \overline{\alpha}'\varepsilon \exp i(\overline{\varphi}'), x_n)\).
Make the changes of variables
\(\alpha_n=\beta\,2^{[\log_2|\lambda_\varepsilon|/m_n]}\),
\(y=x_n2^{[\log_2|\lambda_\varepsilon|/m_n]}\),
\(\lambda_\varepsilon/|\lambda_\varepsilon|=\theta_\varepsilon\),
\(\log_2|\lambda_\varepsilon|/m_n-[\log_2|\lambda_\varepsilon|/m_n]=\chi\),
\(0\leq \chi<1\). Then

\[ \left|\Phi(\alpha' + \alpha'\varepsilon \exp(i\varphi'),x_n),\,l^{q}L_{p',x_n}\right|= \]

\[ =\left(\sum_{l=-\infty}^{\infty} \left| \left(\int_{-2^l+1}^{-2^l}+\int_{2^l}^{2^{l+1}}\right) d\beta\, \frac{\theta_\varepsilon^{\,1-\frac1{pm_n}}\,2^{\left(m_n-\frac1p\right)\chi}} {\theta_\varepsilon 2^{m_n\chi}+|\beta|^{m_n}} \exp(-i\beta y),\,L_{p',y} \right|^{q'} \right)^{1/q'} \leq \]

\[ \leq C(p,\varepsilon)\left( \sum_{l=-\infty}^{\infty} \left| \left(\int_{-2^l+1}^{-2^l}+\int_{2^l}^{2^{l+1}}\right) d\beta\, \frac{\exp(-i\beta y)}{1+|\beta|^{m_n}},\,L_{p'} \right|^{q'} \right)^{1/q'}, \]

because the function
\[ (1+|\beta|^{m_n})\theta_\varepsilon^{\,1-\frac1{pm_n}} 2^{\left(m_n-\frac1p\right)\chi} \left(\theta_\varepsilon 2^{m_n\chi}+|\beta|^{m_n}\right)^{-1} \]
is a multiplier from \(l^{q'}L_{p',x_n}\) to \(l^{q'}L_{p',x_n}\), uniformly in
\((\overline{\alpha}',\overline{\varphi}')\). It suffices to prove convergence of the series

\[ \sum_{l=-\infty}^{\infty} \left| \int_{2^l}^{2^{l+1}} \frac{\exp(-i\beta y)\,d\beta}{1+\beta^{m_n}},\,L_{p'} \right|. \]

For \(l\leq 0\),

\[ \left| \int_{2^l}^{2^{l+1}} \frac{\exp(-i\beta y)\,d\beta}{1+\beta^{m_n}},\,L_{p'} \right| \leq C_1 \left| \int_{2^l}^{2^{l+1}} \exp(-i\beta y)\,d\beta,\,L_{p'} \right| \leq C_2(p)2^{l/p}, \]

and convergence of the series

\[ \sum_{l=-\infty}^{0} \left| \int_{2^l}^{2^{l+1}} \frac{\exp(-i\beta y)\,d\beta}{1+\beta^{m_n}},\,L_{p'} \right| \]
is established.

It is known that \(m_n>\dfrac1p\). Let \(m_n=\dfrac1p+2\delta\). Then

\[ (1+|\beta|^{m_n})^{-1} =(1+|\beta|^\delta)^{-1} \left[ (1+|\beta|^\delta)(1+|\beta|)^{\frac1p+\delta} (1+|\beta|^{m_n})^{-1} \right]\times \]

\[ \times(1+|\beta|)^{-\frac1p-\delta}. \tag{8} \]

The function
\[ F^{-1}\left[(1+|\beta|)^{-\frac1p-\delta}\right](x)\in L_{p'}, \]
since \((1+|\beta|)^{-\frac1p-\delta}\) is an even function, monotonically decreasing as \(|\beta|\to\infty\), and

\[ \left| |\beta|^{(p'-2)/p'}(1+|\beta|)^{-\frac1p-\delta},\,L_{p',\beta} \right| = \left[ 2\int_{0}^{\infty}\beta^{p'-2}(1+\beta)^{-p'(1+\delta)}\,d\beta \right]^{1/p'} <\mathrm{const} \]

(by the Hardy–Littlewood theorem [13]). The middle function in formula (8) is a multiplier from \(L_{p'}\) to \(L_{p'}\). Hence
\([F^{-1}(1+|\beta|^{m_n})](x)\in L_{p'}\), and for such functions we have already shown the finiteness of

\[ \sum_{l=1}^{\infty} \left| \int_{2^l}^{2^{l+1}} \frac{\exp(-i\beta y)}{1+\beta^{m_n}}\,d\beta,\,L_{p'} \right|. \]

The theorem is proved.

6. Differential equations

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Received by the editors
September 7, 1965.

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