UDC 517.513

MATHEMATICS

Yu. E. KHAIKIN

ON THE BOUNDEDNESS OF PSEUDODIFFERENTIAL OPERATORS IN WEIGHTED SPACES

(Presented by Academician V. I. Smirnov on 28 V 1969)

1°. Let \(R^n\) be \(n\)-dimensional Euclidean space of points \(x=(x_1,\ldots\)

\[ \ldots,x_n);\quad \rho^2=|x|^2=\sum_{i=1}^{n}x_i^2;\quad S=\{x:\ |x|=1\};\quad \theta=x\rho^{-1}\in S. \]

If \(\omega=(\omega_1,\ldots,\omega_n)\) is a set of nonnegative integers (a multi-index), then

\[ D_x^\omega=D_{x_1}^{\omega_1}\ldots D_{x_n}^{\omega_n};\qquad D_{x_i}=\partial/\partial x_i;\qquad x^\omega=x_1^{\omega_1}\ldots x_n^{\omega_n}; \]

\[ |\omega|=\sum_{i=1}^{n}\omega_i;\qquad \omega!=\omega_1!\ldots\omega_n! \]

Denote by \(D\) the space of infinitely differentiable functions with compact supports. We define the space \(L_{p,\alpha}\) as the set of functions defined on \(R^n\) with finite norm

\[ \|u\|_{L_{p,\alpha}}=\|\rho^\alpha u\|_{L_p}<\infty,\qquad 1<p<\infty. \]

We shall specify a singular integral operator with symbol \(\Phi(x,\xi)\) in the form

\[ Au=F_{\xi\to x}^{-1}\Phi(x,\xi)F_{y\to \xi}u, \tag{1} \]

where \(F_{y\to \xi}u\) is the Fourier transform of the function \(u\in D\). We shall also use the representation of a singular operator by means of a kernel, namely:

\[ Au=\int_{R^n}K(x,x-y)u(y)\,dy, \tag{1'} \]

where \(K(x,x-y)=f(x,\theta_{xy})|x-y|^{-n}\), \(\theta_{xy}=(x-y)|x-y|^{-1}\).

The boundedness of the one-dimensional singular integral in the space \(L_{p,\alpha}\) \((-1/p<\alpha<1/p';\ 1/p+1/p'=1)\) was proved by K. I. Babenko \((^2)\). E. M. Stein \((^3)\) obtained a theorem on the boundedness of the operator (1′) in the space \(L_{p,\alpha}\) for arbitrary \(n\) and \(\alpha\in(-n/p,n/p')\) under the condition that
\(\operatorname{vrai\,sup}|f(x,\theta)|<\infty\). B. A. Plamenevskii \((^4)\) and the author \((^5)\) independently proved that, for \(\alpha\in(-n/2,n/2)\), \(|\alpha|\ne n/2+k\) \((k=0,1,\ldots)\), the operator (1) with symbol \(\Phi(\xi)\), defined on certain dense subsets in \(L_{2,\alpha}\), is bounded in the space \(L_{2,\alpha}\).

The present paper is devoted to generalizing these results in various directions.

Denote by \(Q_s\) the set of functions \(u\in D\) satisfying the conditions

\[ \int_{R^n}x^\omega u(x)\,dx=0, \]

where \(\omega\) is any multi-index of order \(0\le |\omega|\le s\). By \(Q_{-s}\) we denote the set of functions \(u\in D\) such that \(0\notin \operatorname{supp}u\), and satisfying

conditions

\[ \int_0^\infty \rho^{-q}u(\rho,\theta)\,d\rho=0,\qquad q=1,2,\ldots, \]

The sets \(Q_s\) and \(Q_{-s}\) are dense in the space \(L_{p,\alpha}\).

Theorem 1.1. I. Let \(n/p<\alpha<n/p'\) and

\[ \int_S |f(x,\theta)|^{p'}\,d\theta \leqslant \mathrm{const}<\infty . \tag{2} \]

Then the operator \((1')\) is bounded in the space \(L_{p,\alpha}\).

II. Let \(\alpha>n/p'\), \(\alpha\ne n/p'+k\) \((k=1,2,\ldots)\), condition (2) be satisfied, and

\[ \int_{R\leq |x|\leq 2R} |D_z^\omega K(x,z)|_{z=x}|^p\,dx \leqslant \mathrm{const}\cdot R^{-n(p-1)-(\omega)p}, \]

where \(\omega\) is any multi-index of order \(0\leq |\omega|\leq s=[\alpha-n/p']\)* and \(R\in(0,\infty)\). Suppose further that for some \(\varkappa\in(\alpha-n/p',s+1]\) the inequality

\[ \int_{R\leq |x|\leq 2R} |K^+(x,y)|^p\,dx \leqslant \mathrm{const}\cdot R^{-n(p-1)} \left(\frac{|y|}{R}\right)^{\varkappa p}, \]

is satisfied, where

\[ K^+(x,y)=K(x,x-y)- \sum_{|\omega|=0}^{s} D_z^\omega K(x,z)|_{z=x}\frac{y^\omega}{\omega!}. \]

Then the operator \((1')\), defined on the set \(Q(s)\), is bounded in the space \(L_{p,\alpha}\).

III. Let \(\alpha<-n/p\), \(|\alpha|\ne n/p+k\) \((k=1,2,\ldots)\), condition (2) be satisfied, and

\[ \int_S |D_x^\omega K(x,x-y)|_{x=0}|^{p'}\,d\theta \leqslant \mathrm{const}\cdot |y|^{-(n+|\omega|)p'}, \]

where \(\omega\) is a multi-index of order \(0\leq |\omega|\leq s=[|\alpha|-n/p]\). Suppose further that for \(|y|>2|x|\) and \(\varkappa\in(|\alpha|-n/p,s+1]\) the inequality

\[ \int_S |K^-(x,y)|^{p'}\,d\theta \leqslant \mathrm{const}\cdot |y|^{-np'} \left(\frac{|x|}{|y|}\right)^{\varkappa p'}, \]

is satisfied, where

\[ K^-(x,y)=K(x,x-y)- \sum_{|\omega|=0}^{s} D_x^\omega K(x,x-y)|_{x=0}\frac{x^\omega}{\omega!}. \]

Then the operator \((1')\), defined on the set \(Q_{-s}\), is bounded in the space \(L_{p,\alpha}\).

For \(\alpha>n/p'\) the operator \((1')\), defined on the set \(Q_s\), \(s=[\alpha-n/p']\), coincides with the operator

\[ A^+u=\int_{R^n} K^+(x,y)u(y)\,dy. \]

For \(\alpha<-n/p\) the operator \((1')\), defined on the set \(Q_{-s}\), \(s=[|\alpha|-n/p]\), coincides with the operator

\[ A^-u=\int_{R^n} K^-(x,y)u(y)\,dy. \]

\(2^\circ\). Let us consider separately the case of the space \(L_{2,\alpha}\). By definition, \(\Phi(x,\xi)\in L_\infty W_2^l(S)\) if \(\Phi(x,\xi)\) belongs to the space \(W_2^l(S)\)

* Here and below \([r]\) denotes the integer part of the number \(r\).

(the space of S. L. Sobolev—L. N. Slobodetskii on the sphere \(S\)) with respect to the variable \(\xi\) and

\[ \operatorname*{vrai\,sup}_{x\in R^n}\|\Phi(x,\xi)\|_{W_2^l(s)}<\infty . \]

Theorem 2. Let \(\Phi(x,\xi)\in L_\infty W_2^l(S)\), \(l>(n-1)/2+|\alpha|\), \(\alpha\ne n/2+k\) \((k=0,1,\ldots)\). Then the operator (1), defined on the whole space \(L_{2,\alpha}\) for \(|\alpha|<n/2\) and on the set \(Q_s(Q_{-s})\) for \(|\alpha|>n/2\) \((s=[|\alpha|-n/2])\), is bounded in the space \(L_{2,\alpha}\).

The proof is based on the following estimate

\[ \sum_{m=0}^{m_k}\bigl\|F_{\xi\to x}^{-1}Y_{mk}(\xi)F_{y\to \xi}u\bigr\|_{L_{2,\alpha}}^2 \leq \operatorname{const}\cdot k^{\,n-2+2|\alpha|}\|u\|_{L_{2,\alpha}}^2, \]

where \(Y_{mk}(\xi)\) is a spherical function of order \(k\).

For \(\alpha=0\), theorem 2 becomes the theorem of S. G. Mikhlin \(\left({}^{1}\right)\)—M. S. Agranovich \(\left({}^{7}\right)\). There is an assertion analogous to theorem 2 in the spaces \(L_2\) with weight function \(\lambda(x)=(1+|x|)^\alpha\) and with weight function \(\lambda(x)\) equal to \(|x|^\alpha\) for \(|x|\leq 1\) and to unity for \(|x|>1\).

\(3^\circ.\) Let \(H^\mu\) be the function space defined in \(\left({}^{6}\right)\), endowed with the norm

\[ \|u\|_{H^\mu}=\left(\int_{R^n}\mu^2(\xi)\left|F_{x\to \xi}u\right|^2\,d\xi\right)^{1/2}<\infty . \]

We shall say that \(a(x,\xi)\in H^\mu L_\infty\) if \(a(x,\xi)\) belongs to the space \(H^\mu\) with respect to the variable \(x\) and

\[ \operatorname*{vrai\,sup}_{\xi\in R^n}\|a(\cdot,\xi)\|_{H^\mu}<\infty . \]

The class of functions \(L_\infty H^\mu\) is defined analogously.

We define the space \(L_2(\mu)\) as the set of functions given on \(R^n\) for which the norm

\[ \|u\|_{L_2(\mu)}=\left(\int_{R^n}|\mu(x)u(x)|^2\,dx\right)^{1/2}<\infty \]

is finite.

Theorem 3. The pseudodifferential operator

\[ Bu=F_{\xi\to x}^{-1}b(x,\xi)F_{y\to \xi}u \tag{3} \]

is bounded in the space \(L_2\) if \(b(x,\xi)\in H^\mu L_\infty\) and

\[ \int_{R^n}\mu^{-2}(\tau)\,d\tau<\infty . \tag{4} \]

Condition (4) is, in a certain sense, sharp. Namely, for any function \(\mu\) not satisfying condition (4), one can find a symbol \(b(x)\in H^\mu\) such that the operator (3) is not bounded in \(L_2\). A close result for singular operators, refining the theorem of S. G. Mikhlin—M. S. Agranovich mentioned above, was obtained in \(\left({}^{11}\right)\).

Theorem 4. Let a pseudodifferential operator with symbol \(b(x,\xi)\) be bounded in the space \(L_2(\nu)\), and let the function \(\varphi(x,\xi)\) belong to \(L_\infty H^\mu\).

Then the pseudodifferential operator with symbol \(\varphi(x,\xi)b(x,\xi)\) is bounded in the space \(L_2(\nu)\), if

\[ \int_{R^n}\nu^2(x+y)\mu^{-2}(y)\,dy\leq \operatorname{const}\cdot \nu^2(x). \]

The fact that the continuity conditions for pseudodifferential operators in \(L_2\) can at the same time be multiplier conditions was first noted in the works of M. Sh. Birman and M. Z. Solomyak \(\left({}^{9,10}\right)\).

Theorem 3 refines, and Theorem 4 generalizes, analogous assertions from works \((^9,{}^{10})\), obtained by methods of the theory of double operator integrals.

4°. By \(L_{p,\alpha,\beta,\gamma}\) we denote the completion of the space of functions \(u \in D\) \((0 \notin \operatorname{supp} u)\) with respect to the norm

\[ \|u\|_{L_{p;\alpha,\beta,\gamma}} = \|\rho^\alpha(-\Delta)^{\beta/2}(-\delta)^{\gamma/2}u\|_{L_p}, \]

where \(\Delta\) is the Laplace operator, \(\delta\) is the spherical part of \(\Delta\), and \(\alpha,\beta,\gamma\) are real numbers.

Let us formulate some properties of the spaces \(L_{p,\alpha,\beta,\gamma}\).

1) Let \(-n/p < \alpha-\beta < \alpha < n/p'\). Then, for functions \(u \in D\), the inequalities

\[ c_1\|\rho^\alpha(-\Delta)^{\beta/2}u\|_{L_p} \leq \|(-\Delta)^{\beta/2}\rho^\alpha u\|_{L_p} \leq c_2\|\rho^\alpha(-\Delta)^{\beta/2}u\|_{L_p}, \]

hold, where \(c_1,c_2\) are constants independent of the function \(u\).

2) Let \(0<\beta\ne n/p+k,\ k=0,1,\ldots\). Then for all functions \(u\in D\) \((0\notin \operatorname{supp}u)\) the inequalities

\[ c_1\|(-\Delta)^{\beta/2}u\|_{L_p} \leq \|(-\Delta_\rho)^{\beta/2}u\|_{L_p} + \|\rho^{-\beta}(-\delta)^{\beta/2}u\|_{L_p} \leq c_2\|(-\Delta)^{\beta/2}u\|_{L_p}, \tag{5} \]

are valid, where \(\Delta_\rho\) denotes the radial part of the Laplace operator.

In the proof of inequality (5) the estimate

\[ \|\rho^{-\beta}u\|_{L_p} \leq c\|(-\Delta)^{\beta/2}u\|_{L_p}, \]

is used, where \(0<\beta\ne n/p+k,\ k=0,1,\ldots\), previously proved by V. P. Il’in \((^8)\) for \(0<\beta<n/p\).

With the aid of the theorem from 1° and the formulated properties of the spaces \(L_{p,\alpha,\beta,\gamma}\), one proves theorems on the boundedness of operators whose symbols, for large \(|\xi|\), have the form

\[ \sigma(x,\xi)=\sum_{k=1}^{N}|\xi|^{\lambda_k}\sigma_k(x,\xi), \]

where \(\lambda_k\) are real numbers, and \(\sigma_k(x,\xi)\) are functions positively homogeneous of degree zero, sufficiently smooth and, for large \(|x|\), independent of \(x\).

The author expresses sincere gratitude to B. G. Mazy’a for proposing the problem and for assistance in carrying out the work.

Leningrad Electrotechnical Institute
named after V. I. Ulyanov (Lenin)

Received
22 V 1969

REFERENCES

1. S. G. Mikhlin, Multidimensional singular integrals and integral equations, Moscow, 1962.
2. K. I. Babenko, DAN, 62, No. 1 (1948).
3. E. M. Stein, Proc. Am. Math. Soc., 8, No. 2 (1957).
4. B. A. Plamenevskii, DAN, 179, No. 5 (1968).
5. Yu. E. Khaikin, Vestn. LGU, No. 13 (1969).
6. L. R. Volevich, B. P. Paneakh, UMN, 20, 1 (1965).
7. M. S. Agranovich, UMN, 20, 5 (1965).
8. V. P. Il’in, Matem. sborn., 54, No. 3 (1961).
9. M. Sh. Birman, M. Z. Solomyak, Problems of Mathematical Physics, issue 2, 1967.
10. M. Sh. Birman, M. Z. Solomyak, Vestn. LGU, No. 1 (1969).
11. V. G. Mazy’a, Yu. E. Khaikin, Vestn. LGU, No. 19 (1969).