P. I. LIZORKIN

\((L_p, L_q)\)-MULTIPLIERS OF FOURIER INTEGRALS

(Presented by Academician I. M. Vinogradov on 23 IV 1963)

1. Let the function \(f(x)\) be defined and summable to the power \(p\) in \(E_n\) \(\bigl(f(x)\in L_p,\ 1<p<\infty\bigr)\), and let \(\widetilde f(\lambda)\) be its Fourier transform

\[ \widetilde f(\lambda)=\frac{1}{(2\pi)^{n/2}}\int_{E_n} f(x)e^{-i\lambda x}\,dx,\qquad f\in S^* . \tag{1} \]

Hörmander \((^1)\) showed that every bounded operator \(A\) from \(L_p\) into \(L_q\) \(\bigl(A\in L_p^q\bigr)\), commuting with translations, is represented, for \(p\le q<\infty\), by a convolution \(A\varphi=T*\varphi;\ \varphi\in S,\ T\in S'\), which in Fourier images takes the form \(\widetilde{A\varphi}=\widetilde T\widetilde\varphi\).

Under these conditions the distribution \(\widetilde T\) is called a multiplier of type \((p,q)\), and the set \(\{\widetilde T;\ T\in L_p^q\}\) is denoted by \(M_p^q\). The distribution \(\widetilde T\in M_p^q\) is a locally summable function \((^1)\).

In the present paper we are interested in conditions on a function \(\Phi(\lambda)\) which ensure that it belongs to \(M_p^q\). Conditions of this kind, given in the case \(p=q\) by S. G. Mikhlin \((^2)\), have proved useful in many questions. We follow the method set forth in \((^3)\) and originating with Marcinkiewicz \((^4)\)**. Our results generalize the results of the works mentioned (even for \(p=q\)).

2. Main theorem. Let the function \(\Phi(\lambda)\) be continuous together with the derivative \(\partial^n\Phi/\partial\lambda_1\cdots\partial\lambda_n\) and all preceding derivatives outside the coordinate planes (i.e. for \(|\lambda_j|>0,\ j=1,\ldots,n\)). Then \(\Phi(\lambda)\in M_p^q\), if for the derivatives just mentioned

\[ \left| \lambda_1^{k_1+\beta}\cdots \lambda_n^{k_n+\beta} \frac{\partial^k\Phi}{\partial\lambda_1^{k_1}\cdots\partial\lambda_n^{k_n}} \right|\le M, \tag{2} \]

where \(\beta=1/p-1/q,\ k_j\) takes the value \(0\) or \(1\), \(k=\sum_{j=1}^n k_j=0,1,\ldots,n\), and \(M\) is a constant.

This theorem remains valid if one considers vector-valued functions \(f(x)\) from \(L_p(H)\) (i.e. with values in an arbitrary Hilbert space \(H\) and with finite norm \(\|f\|_{L_p(H)}=\int_{E_n}\|f(x)\|_H\,dx\)). In this case \(\Phi(\lambda)\) is an operator-valued function (i.e. for each \(\lambda\in E_n\), \(\Phi(\lambda)\) is a bounded operator in \(H\)), whose derivatives are understood in the strong sense; condition (2) is written in terms of the operator norm, and it is asserted that

\[ \left\|\widehat{\Phi(\lambda)\widetilde f(\lambda)}\right\|_{L_q(H)} \le c\|f\|_{L_p(H)}*** . \]

First of all we shall reproduce the main steps of the proof of the theorem in the scalar case for \(n=1,\ p<q\).

3. Let \(f(x)\in L_p(l_2)\), i.e. \(f(x)=\{f_k(x)\}_{k=1}^{\infty}\) and

\[ \|f\|_{L_p(l_2)} = \left\{ \int_{-\infty}^{\infty} \left[\sum_1^\infty |f_k(x)|^2\right]^{p/2}dx \right\}^{1/p} <\infty . \]

* \(S\,(S')\) is the space of Schwartz test (generalized) functions.
* The work \((^4)\) concerns series; S. G. Mikhlin derived the mentioned criterion \((^2)\) by passing from the conditions obtained in \((^4)\).
** \(\widehat g\) is the inverse Fourier transform of the function \(g\).

Lemma 1. For every \(p,\ 1<p<\infty\), the convolution transform

\[ g(x)=\mathcal H_{0}f\equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{f(y)\,dy}{|x-y|^{\alpha}} \tag{3} \]

defines a bounded transformation from \(L_p(l_2)\) into \(L_q(l_2)\), where \(1/p-1/q=1-\alpha,\ p<q<\infty\).

The proof of Lemma 1 reduces to applying Theorem 2 from [3]. In what follows we shall use the theorem on the Fourier transform of a convolution:

\[ \widetilde{h*f}=\tilde h\cdot \tilde f, \]

regarding \(h\) as a distribution from \(S'\) and \(h*f\) as \(\{h*f_k\}_{1}^{\infty}\). The Fourier transform of the function \(f(x)\in L_p(l_2)\) may here be understood in the sense

\[ \tilde f(\lambda)=\{\tilde f_k(\lambda)\}_{1}^{\infty}. \]

Lemma 2. Let \(f\in L_p(l_2)\), \(1<p<\infty\), and let \(\Phi(\lambda)\) be defined by the formula

\[ \Phi(\lambda)=\int_{-\infty}^{\infty}\frac{d\rho(t)}{|t-\lambda|^{\beta}},\qquad \beta=\frac1p-\frac1q,\qquad p<q<\infty, \]

where \(\rho(t)\) is a function of bounded variation. Then the transformation \(\mathcal K\), defined in Fourier images by the formula

\[ \widetilde{\mathcal K f}=\Phi(\lambda)\tilde f(\lambda), \]

is a bounded mapping from \(L_p(l_2)\) into \(L_q(l_2)\).

Proof. Let \(f\in S\) (i.e., the functions \(f_k(x)\), \(k=1,2,\ldots\), are infinitely differentiable and decrease at infinity faster than any power of \(|x|\)). We have

\[ \begin{aligned} \mathcal K f &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(\lambda)\tilde f(\lambda)e^{i\lambda x}\,d\lambda \\ &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde f(\lambda)e^{i\lambda x} \left(\int_{-\infty}^{\infty}\frac{d\rho(t)}{|\lambda-t|^{\beta}}\right)d\lambda \\ &=\int_{-\infty}^{\infty} \left(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \frac{1}{|\lambda-t|^{\beta}}\tilde f(\lambda)e^{i\lambda x}\,d\lambda\right)d\rho(t) \\ &=\int_{-\infty}^{\infty}\mathcal H_t f\,d\rho(t). \end{aligned} \tag{4} \]

Here by \(\mathcal H_t\) we have denoted the transform defined by the formula

\[ \widetilde{\mathcal H_t f}=\frac{1}{|\lambda-t|^{\beta}}\tilde f(\lambda). \]

It maps \(L_p(l_2)\) continuously into \(L_q(l_2)\). Indeed, from formula (3) we have

\[ \widetilde{\mathcal H_0 f}=\widehat{|x|^{-\alpha}*f} =D(\alpha)|\lambda|^{-1+\alpha}\tilde f,\qquad 1-\alpha=\frac1p-\frac1q\ *. \]

The transform \(\mathcal H_t\) differs from \(\mathcal H_0\) (up to the multiplicative constant \(D(\alpha)\)) only by a shift of the first factor by \(t\). Since \(\mathcal H_0\) is bounded, \(\mathcal H_t\) is also bounded (independently of \(t\)). From relation (4) we easily obtain

\[ \|\mathcal K f\|_{L_q(l_2)} \leq \int_{-\infty}^{\infty}\|\mathcal H_t f\|_{L_q(l_2)}\,|d\rho(t)| \leq c\|f\|_{L_p(l_2)}, \]

and Lemma (2) follows from the density of \(S\) in \(L_r(l_2)\), \(1<r<\infty\).

Lemma 3. Let \(f\in L_p(l_2)\), \(1<p<\infty\), and let \(\Phi_m(\lambda)\) be a sequence of functions representable in the form

\[ \Phi_m(\lambda)=\int_{-\infty}^{\infty}\frac{d\rho_m(t)}{|\lambda-t|^{\beta}},\qquad \beta=\frac1p-\frac1q,\qquad p<q<\infty, \tag{5} \]

where \(\rho_m(t)\) are functions of bounded variation for which

\[ \operatorname*{var}_{-\infty<t<\infty}\rho_m(t)\leq M. \]

Then the mapping

\[ \widetilde{(\mathcal K f)}_m=\Phi_m(\lambda)\tilde f_m(\lambda) \]

is a bounded mapping from \(L_p(l_2)\) into \(L_q(l_2)\).

\[ {}^{*}\ \text{The factor }D(\alpha)=-\sqrt{2\pi}/2\cos\frac{\pi\alpha}{2}\,\Gamma(\alpha)\text{ does not vanish for the }\alpha\text{ under consideration.} \]

Proof. First the boundedness of the “truncated” operator \(\mathcal K_N\) is proved:
\[ \widetilde{(\mathcal K_N \mathbf f)}_m= \begin{cases} \Phi_m(\lambda)\,\widetilde f_m(\lambda),& m\leqslant N,\\ 0,& m>N. \end{cases} \]
By the example of the preceding lemma, \(\mathcal K_N\) is represented in the form of a superposition
\[ \mathcal K_N\mathbf f= \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} \mathcal H(t_1,\ldots,t_N)\mathbf f\,dp_1(t_1)\cdots dp_N(t_N) \tag{6} \]
of bounded operators \(\mathcal H(t_1,\ldots,t_N)\), defined by the equality
\[ \mathcal H((t_1,\ldots,t_N)\mathbf f= \begin{cases} \dfrac{1}{|\lambda-t_m|^\beta}\,\widetilde f_m,& m\leqslant N,\\ 0,& m>N. \end{cases} \]
Since
\[ \mathcal H(t_1,\ldots,t_N) =\mathcal M(t_1,\ldots,t_N)\,\mathcal H(0,\ldots,0)\,\mathcal M(-t_1,\ldots,-t_N), \]
where \(\mathcal M(t_1,\ldots,t_N)\) is an isometry of \(L_r(l_2)\), \(1<r<\infty\), given by the formula
\[ f_m(x)\to e^{it_mx}f_m(x),\qquad t_m=0\ \text{for }m>N, \]
and \(\mathcal H(0,\ldots,0)\) coincides with the restriction of \(\mathcal H_0\) to the subspace of dimension \(N\), the norm of \(\mathcal H(t_1,\ldots,t_N)\) is bounded (uniformly with respect to \(N\) and independently of \(t_j\)).

If now
\[ \operatorname*{var}_{-\infty<t<\infty}\rho_m(t)=1,\qquad m=1,2,\ldots, \]
then from (6) the boundedness of \(\mathcal K_N\), independently of \(N\), follows, and as \(N\to\infty\) we obtain the assertion of the lemma. In the general case one must normalize the functions \(\rho_m\) (to unit total variation) by multiplying by certain constants \(a_m\); since
\[ c_m\geqslant\delta>0\quad(\text{for }\operatorname*{var}\rho_m(t)\leqslant M), \]
the proof is not affected.

1. The proof of our main theorem is based on Lemma 3 and on the following proposition (see \((^5,^3)\)).

Theorem 1 (on decompositions). Let \(f(x)\in L_p\), \(1<p<\infty\), and let the Fourier transform of the function \(f_m(x)\) be concentrated in the interval
\[ 2^m<|\lambda|\leqslant 2^{m+1} \]
and coincide there with \(\widetilde f(x)\). Then there exist constants \(c_1\) and \(c_2\), independent of \(f\), such that
\[ c_1\|f\|_{L_p}\leqslant \left\{\int_{-\infty}^{\infty} \left(\sum_{-\infty}^{\infty}|f_m(x)|^2\right)^{p/2}\right\}^{1/p} \leqslant c_2\|f\|_{L_p}. \]

Finally, suppose that we are given a function \(\Phi(\lambda)\), differentiable away from the origin and such that
\[ |\Phi(\lambda)|\,|\lambda|^\beta\leqslant M,\qquad |\Phi'(\lambda)|\,|\lambda|^{1+\beta}\leqslant M. \tag{7} \]
We shall prove that \(\Phi(\lambda)\in M_p^q\). The mapping \(\mathcal L\) defined by \(\Phi\) will be represented in the form
\[ \mathcal L=A_3A_2A_1, \]
where the linear operators \(A_1,A_2,A_3\) act according to the scheme
\[ L_p\xrightarrow{A_1}L_p(l_2)\xrightarrow{A_2}L_q(l_2)\xrightarrow{A_3}L_q^* \]
and are defined as follows:
\[ A_1\mathbf f=\mathbf f=\{f_m\}_{-\infty}^{\infty} \quad(f_m\ \text{is connected with }f\text{ as in Theorem 1}); \]
\[ \mathbf g=A_2\mathbf f=\{\Phi_m\widetilde f_m\}_{-\infty}^{\infty},\qquad \Phi_m(\lambda)= \begin{cases} \Phi(\lambda),& \text{for }2^m<|\lambda|\leqslant 2^{m+1},\\ 0,& \text{for }\lambda\notin(2^m,2^{m+1}]; \end{cases} \]
\[ \widetilde{A_3\mathbf g}=\Phi(\lambda)\widetilde f(\lambda) \quad (=\Phi_m(\lambda)\widetilde f_m(\lambda),\ |\lambda|\in(2^m,2^{m+1}];\ m=0,\pm1,\ldots). \]
The boundedness of the operators \(A_1\) and \(A_3\) follows from Theorem 1, and it remains for us to prove the boundedness of the operator \(A_2\) under conditions (7). According to Lemma 3, for this it is enough to verify that the functions \(\Phi_m(\lambda)\) are representable in the form (5). Thus the question is reduced to the solvability of the integral equation
\[ \int_{-\infty}^{\infty}\frac{dp_m(t)}{|\lambda-t|^\beta} =\Phi_m(\lambda)= \begin{cases} \Phi(\lambda),& \text{for }2^m<|\lambda|\leqslant 2^{m+1},\\ 0,& \text{for }|\lambda|\notin(2^m,2^{m+1}] \end{cases} \tag{8} \]

The solution of this equation is given by the formula

\[ \rho'_m(\lambda)=b\left\{\int_{2^m}^{2^{m+1}} \frac{\Phi'(t)}{|t-\lambda|^\alpha}\operatorname{sign}(t-\lambda)\,dt +\Phi(2^m)\frac{\operatorname{sign}(2^m-\lambda)}{|2^m-\lambda|^\alpha} -\Phi(2^{m+1})\frac{\operatorname{sign}(2^{m+1}-\lambda)}{|2^{m+1}-\lambda|^\alpha}\right\}. \]

A calculation shows that the function \(\rho'_m(\lambda)\) is summable and, for a certain constant \(c\) independent of \(\Phi\), the inequality

\[ \operatorname*{var}_{-\infty<\lambda<\infty}\rho_m(\lambda) = \int_{-\infty}^{\infty}|\rho'_m(t)|\,dt \leq cM \]

holds.

The proof of the theorem in the case under consideration is complete.

1. In the \(n\)-dimensional case, as the “elementary” mapping \(\mathcal H_0\) (see Lemma 1) from \(L_p(l_2)\) into \(L_q(l_2)\), one should consider the mapping

\[ \mathcal H_0 f= \frac{1}{(2\pi)^{n/2}}\int_{E_n} \frac{\operatorname{sign}(x_1-y_1)\cdots \operatorname{sign}(x_n-y_n)} {|x_1-y_1|^\alpha\cdots |x_n-y_n|^\alpha} f(y)\,dy, \tag{9} \]

where \(1/p-1/q=1-\alpha,\quad p\leq q<\infty\). The introduction here of the factor

\[ \prod_1^n \operatorname{sign}(x_i-y_i) \]

is caused by the desire to include also the singular case \(\alpha=1\)

\[ \left(\text{in this case the integral in (9) is understood in the sense of } \lim_{\sum\delta_i^2\to 0}\int_{|x_i-y_i|>\delta;\ i=1,\ldots,n}\cdots dy\right). \]

Instead of the sequence (5) one has to consider a sequence with \(n\) “inputs”

\[ \Phi_{m_1,\ldots,m_n}(\lambda) = \sum \underbrace{\int\cdots\int}_{s} \frac{d\rho_{m_{j_1},\ldots,m_{j_s}}(t_{j_1},\ldots,t_{j_s})} {|\lambda_{j_1}-t_{j_1}|^\beta\cdots|\lambda_{j_s}-t_{j_s}|^\beta}, \tag{10} \]

where the summation ranges over all subspaces of \(E_n\) of dimension \(s\), \(1\leq s\leq n\) \(\bigl((t_{j_1},\ldots,t_{j_s})\) are the coordinates of a point of the subspace over which the integration is performed\(\bigr)\); \(\rho_{m_{j_1},\ldots,m_{j_s}}\) are finite measures whose total variations are uniformly bounded. The splitting theorem takes the form

\[ c_1\|f\|_{L_p} \leq \left\{\int_{E_n}\left( \sum_{m_1=-\infty}^{\infty}\cdots \sum_{m_n=-\infty}^{\infty} |f_{m_1,\ldots,m_n}(x)|^2 \right)^{p/2}\right\}^{1/p} \leq c_2\|f\|_{L_p}, \tag{11} \]

where the function \(f_{m_1,\ldots,m_n}\) is defined by the requirement that its Fourier transform be concentrated in the region \(2^{m_j}<|\lambda_j|\leq 2^{m_j+1}\), \(j=1,\ldots,n\), and coincide there with \(\hat f(\lambda)\). The arguments of item 4 remain valid with equation (5) replaced by equation (10); the solution of the latter is likewise written explicitly and satisfies the required conditions.

1. The transfer of the proof to \(H\)-valued functions is carried out following the example of the work \({}^3\). It should be noted that the splitting theorem proved in \({}^3\) for \(n=1\) is also valid in the case of arbitrary \(n\) in the form (11) (and for \(H\)-valued functions).

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
10 IV 1963

REFERENCES

\({}^1\) L. Hörmander, Acta Math., 104, 1–2, 93 (1960).
\({}^2\) S. G. Mikhlin, DAN, 109, No. 4, 701 (1956).
\({}^3\) J. Schwartz, Comm. Pure and Appl. Math., 14, No. 4, 789 (1961).
\({}^4\) J. Marcinkiewicz, Studia Math., 8, 78 (1939).
\({}^5\) D. Guy, Trans. Am. Math. Soc., 7, 1 (1957); 8, 1 (1958).