UDC 517.432

MATHEMATICS

K. TELFER

MULTIPLIER TRANSFORMATIONS FOR PSEUDODIFFERENTIAL OPERATORS IN (L_p)

(Presented by Academician V. M. Smirnov on 10 X 1969)

1. By a multiplier transformation we shall mean an operation that assigns to every pseudodifferential (ps.d.) operator (K) with symbol (K(\xi,x)), i.e.

[
(Ku)(x)=(2\pi)^{-m/2}\int_{R^m} K(\xi,x)e^{i\langle \xi,x\rangle}\hat u(\xi)\,d\xi,
\tag{1}
]

a new ps.d. operator (\Phi K) with symbol (\varphi(\xi,x)K(\xi,x)), i.e.

[
(\Phi Ku)(x)=(2\pi)^{-m/2}\int_{R^m}\varphi(\xi,x)K(\xi,x)e^{i\langle \xi,x\rangle}\hat u(\xi)\,d\xi.
\tag{2}
]

Here (R^m) is (m)-dimensional Euclidean space; (x) and (\xi) are points of (R^m); (\langle \xi,x\rangle=\xi_1x_1+\cdots+\xi_mx_m); (\hat u) is the Fourier transform of the function (u\in S), where (S) is the space of rapidly decreasing functions. For symbols of ps.d. operators homogeneous of degree 0 (with respect to (\xi)), an operator of the form (1) is a singular integral (s.i.) operator.

The question is posed: under what conditions does the transformation (K\to\Phi K) preserve the continuity of ps.d. operators in various function spaces? For the spaces (L_2(R^m)) and (W_2^s(R^m)) of Sobolev–Slobodetskii this problem was investigated in papers ((^{1-5})). The purpose of the present note is to report two results on multiplier transformations in the spaces (L_p(R^m)).

2. We shall give a description of the function spaces in terms of which the conditions of Theorems 1 and 2 will be expressed.

a) Introduce the rings (\mathfrak A_l={\xi\in R^m: 2^{l-2}\le |\xi|\le 2^{l+2}}), (l=0,\pm1,\ldots). By (L_p^{\alpha,\beta}) ((\alpha,\beta\ge 0)) we denote the space of all generalized functions (\varphi(\xi)) belonging to (L_p^\alpha(\mathfrak A_l)) (see ((^{1}))) for every (l), and for which the seminorm (|\varphi|_{L_p^{\alpha,\beta}}) is finite, where

[
|\varphi|{L_p^{\alpha,\beta}}^p
=
\sum^p.}^{\infty}2^{l\beta p}|\varphi|_{L_p^\alpha(\mathfrak A_l)
]

If (\alpha=N) is an integer, then (|\varphi|_{L_p^{N,\beta}}) is equivalent to the integral seminorm

[
\left(
\sum_{|\nu|=N}\int_{R^m}|\xi|^{|\beta p|}\,|D^\nu\varphi(\xi)|^p\,d\xi
\right)^{1/p}.
]

b) By (L_{\infty;x}(L_p^{\alpha,\beta};\xi)) we shall denote the space of all functions (\varphi(\xi,x)) belonging for almost all (x\in R^m) to the space (L_p^{\alpha,\beta}) (with respect to—

-tively variable (\xi)) and for which the seminorm is finite

[
|\varphi|{L}(L_{p;\xi}^{\alpha,\beta})
\equiv \operatorname*{vrai\,sup}{x\in R^m}
|\varphi(\cdot,x)|.}
]

c) Let (S^{m-1}) be the unit sphere of the space (R^m); (\theta) its points. By (W_p^\alpha(S^{m-1})) we denote the Sobolev–Slobodetskii space of functions (\varphi(\theta)) defined on the sphere (S^{m-1}). The space (L_{\infty;x}(W_{p;\theta}^{\alpha}(S^{m-1}))) is defined by analogy with item b).

1. We formulate the results of the note.

Theorem 1. Suppose that the following two conditions are satisfied:

[
\text{(I)}\quad \varphi(\xi,x)\in L_{\infty;x}\bigl(W_{2;\xi}^{\alpha}(R^m)\bigr)
\quad \text{for some } \alpha>m/2.
]

[
\text{(II)}\quad \varphi(\xi,x)\in L_{\infty;x}\bigl(L_{2;\xi}^{m/2+\beta,\beta}\bigr)
\quad \text{for some } \beta>0.
]

If (|1/p-1/2|<\beta/m) ((1<p<\infty)), then the following conclusion is valid:

For every operator (K) of the form (1) with locally summable symbol (K(\xi,x)), continuous in (L_p(R^m)), the operator (\Phi K) of the form (2) is also continuous in (L_p(R^m)). Moreover

[
|\Phi K|{L_p\to L_p}
\leq
C\left(|\varphi|
+
|\varphi|{L\infty(L_2^{m/2+\beta,\beta})}\right)
|K|_{L_p\to L_p}.
]

Theorem 2. Let the function (\varphi(\xi,x)) be homogeneous of degree (0) in (\xi). If (^*)

[
\varphi(\theta,x)\in L_{\infty;x}\bigl(W_{2;\theta}^{\alpha}(S^{m-1})\bigr),
\quad \text{then, for } \alpha>(m-1)/2+m|1/p-1/2|
]

[
(1<p<\infty)
]

the conclusion of Theorem 1 is valid. Moreover

[
|\Phi K|{L_p\to L_p}
\leq
C|\varphi|.}|K|_{L_p\to L_p
]

1. In the space (L_2(R^m)) the multiplier problem formulated above is included in the theory of double operator integrals (({}^1,{}^4)), the use of which leads to various results. In studying the same problem in the spaces (L_p(R^m)), the general methods of Hilbert space applied in (({}^1\text{--}{}^4)) are unsuitable. However, some elements of the method of integral sums developed in (({}^1)) can be used in proving Theorems 1 and 2. The idea consists in approximating the multipliers (\varphi(\xi,x)) by functions of a special form—piecewise-polynomial approximations corresponding to suitable decompositions of the space (R^m). This idea is combined with the application of new results of Petre (({}^6)) and Littman, McCarthy, and Rivière (({}^7)) on multipliers of Fourier integrals and on estimates of Littlewood–Paley type.

2. Since the symbol of the identity operator is equal to (K(\xi,x)\equiv 1), our results contain the following assertion.

Corollary. Under the conditions of Theorems 1 and 2, a ps.d. (s.i.) operator with symbol (\varphi(\xi,x)) is bounded in (L_p(R^m)).

We note that the boundedness criterion for a s.i. operator in (L_p(R^m)) that follows as a special result from Theorem 2, for (p<2), somewhat improves the result of S. G. Mikhlin ((({}^8),) Theorem V.1.26). For (p>2) the conditions of Theorem 2 are more restrictive than the conditions of S. G. Mikhlin; this is connected with the fact that the latter are not of multiplier character.

Other boundedness conditions for ps.d. operators in (L_p(R^m)) were indicated, in the case (p=2), by L. Hörmander (({}^9)); from this V. M. Kagan (({}^{10})) derived an analogous criterion for arbitrary (p). Both results likewise do not have multiplier character. Kagan’s conditions and Theorem 1 of the present work do not cover one another.

(^*) The restriction of the function (\varphi(\xi,x)) to the sphere (S^{m-1}) (with respect to the first variable) will be denoted by (\varphi(\theta,x)).

The author expresses sincere gratitude to M. Z. Solomyak for posing the problem and for his attention to the work.

Leningrad State University
named after A. A. Zhdanov

Received
1 X 1969

REFERENCES

1. M. Sh. Birman, M. Z. Solomyak, in: Problems of Mathematical Physics, vol. 2, L., 1967, p. 26.
2. M. Sh. Birman, M. Z. Solomyak, Vestn. LGU, No. 13, 21 (1967).
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6. J. Peetre, Ricerche di Mat., 15, fasc. 1°, 3 (1966).
7. W. Littman, Ch. McCarthy, N. Rivière, Studia Math., 30, No. 2, 193 (1968).
8. S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Moscow, 1963.
9. L. Hörmander, in: Pseudo-differential Operators, Moscow, 1967, p. 297.
10. V. M. Kagan, Izv. Vyssh. Uchebn. Zaved., Matematika, No. 6 (73), 35 (1968).