Proc. Nat. Acad. Sci. USA Vo. 71, No. 10, pp. 3911-3912, October 1974

LP-multipliers for Noncompact Symmetric Spaces (semi-simple Lie group/maximal function/singular integral operator) J. L. CLERC AND E. M. STEIN Department of Mathematics, Princeton University, Princeton, New Jersey 08540 Contributed by Eliss M. Stein, July 1, 1974

ABSTRACT Let G be a real noncompact semi-simple However, a strong necessary condition is needed in the case of Lie group with finite center and K a maximal compact a sub-group. The symmetric space M = GIK carries a noncompact symmetric space, due to the "holomorphic measure invariant under the action of G. The operators extension" of Fourier transforms of L"-functions (1 < p < 2). which map LP(M) continuously into itself and commute Denote by e the convex hull of the images of p under the with the action of G, can be easily characterized when Weyl group WI by e (0 < + < 1) the t-dilation of e = Cl, p = 2 or p = 1. This note gives some results on "singular and by 5gthe tube over the polygonC:: = + icky integrals" which map LP into itself (1 < p < + co). ga*R THEOREM 1. Let m be a bounded measurable function on a*R Let g = k G) p be a Cartan decomposition of the Lie algebra and suppose that the associated G-invariant operator T on L2(M) of G, and choose a maximal Abelian subspace of p, say a. If extends as a continuous operator on L"(M), for some p, 1 < p < G KAN is an associated Iwasawa decomposition, let H(x) 2. Then in extends as a holomorphic function in the interior of be the unique element of a such that x = k exp H(x) n (k E K, the tube 5,, where t - /p2 1, and is uniformly bounded there. n E N). For any (complex) linear form X on a, let To prove the theorem, notice that T must preserve the class of K-invariant functions on M which are in L", but also Ox (X) =I e(-p + iX)H(xk) dc the related L'-class, because of a simple relation between T and its adjoint on these classes. But now elementary spherical (where p is the half-sum of the restricted roots), be the ele- functions belong to some Lq, q > 2. In fact, using complex mentary spherical functions of the symmetric space M = interpolation and the inequalities of Harish-Chandta' for Vo GIK [endowed with the G-invariant Riemannian metric one can prove the following lemma. d(m,n) induced by the Killing form]. The algebra of G-invariant differential operators on M is LEMMA 1. If X E , then ||,X| <5 c" < + coforany q > q, commutative and the decomposition of the left regular repre- 2 sentation of G on L2(M) yields a simultaneous diagonalitation Taking the lemma for granted, then extend mr by Tt, of this algebra. More precisely, L2(M) = f E, c(v) -dv, = m(A)kx, and the fact that m is holomorphic is clear. G*R A first example of singular integrals-although not a linear where the action of G on E, is equivalent with the class one operator-is given by the Hardy Littlewood maximal function. principal series representation that is induced by the char- Define acter m a n H aiP, where v is a real-valued linear form on a. Now if T is a continuous operator on L2(M) which commutes f*(M)= sup JB(m,r) Kif Jf(n) dnI with the action of G, then there exists a bounded measurable T>O B(mr) function m on a*R invariant by the Weyl group such that where B(m,r) is the (geodesic) ball, with center it, m and radius r. Because of the exponential growth of the measure of a ball as T= fm(v) I c(v)1-2 dv. r tends to infinity, the classical covering lemma used to prove weak-type Ll arid Lv inequalities fails. However the LP result Alternately, this operator can be viewed as a convolution on is still true (1

o and m' = g'No. We may assume p is finite, and write f*(m) < f*1i(m) + (x * If (m), where f* has the same definition as f*, but the Then formally Tf(m) = f K(m,m )f(m')dm' = (k * f) (m). sup is taken only over balls of radius less than one, and X is the biinvariant function given by x(n) = inf(c, B(oAn) If k is a (finite) measure, then Tmaps LI(M) into itself and all where o is the origin in M(o = e.K), and c is some adequate the operators on L1(M) which commute with T are obtained that way. A characterization of the invariant LP-operators I Harish-Chandra (1958) "Spherical functions on a semisimple- (p 1,2,co) is not available, even in the Euclidean case. Lie group, I," Amer. J. Math. 80, 241-310. 3911 Downloaded by guest on October 4, 2021 3912 Mathematics: Clerc and Stein Proc. Nat. Acad. Sci. USA 71 (1974) strictly positive constant. Now the L' inequality for f*i is For ki, that is the local part, we use a method of Coifman and handled along classical lines, whereas liX * If IL .c< IfIlI Weiss.4 (1 < p < + co) follows from the observation that X belongs to < LEMMA 3. Suppose H sh2(a,H) k (exp H) is a convolutor any L5(1 < p + co) (the distribution function of x is 1/a, aEA + for a small), together with the following version of a well- (in the Abelian sense) ofLv(a), then k is a convolutor of LP(M). known phenomenon for convolutions for semi-simple groups. In order to prove that II sh2(a,H) ki (exp H) is a convolu- aEA+ LEMMA 2. Suppose k is a K-biinvariant function on G, which tor of Lv(a), it suffices to prove that IJ sh(a,H) II (aH) belongs to LYo for all po > 1, then aEA + aid + k(exp H) has an (Abelian) Fourier transform on a* which Ilk *flp < Cp Hlfllp (1 < p < + co). satisfies the Mikhlin conditions. But easy computation shows that it is given by m(X) = (I Da)( (ai.m) (X), The same idea can be used to obtain a multiplier theorem; where Da is the derivative in we restrict ourselves to the complex case, that is, we assume g the direction dual to a. Using is a complex Lie algebra; if is Leibnitz's rule, and the assumption on m, one shows that m'- is k a compact real form of g, then bounded, and satisfies the g = k + ik is a Cartan decomposition. If a is a maximal Abelian Mikhlin conditions (notice that dim a = n -2 card A+, so that N> degree (H Da) + subspace of i.k, the k = a + ia is a (complex) Cartan sub- ai+ algebra of g. If A + is the set of positive roots of (gfk) with 1/2 dim a). respect to some ordering on a, then we have the Iwasawa The behavior at infinity is handled via Lemma 1. We need decomposition g = k e a ( E + go, and p = E + a, first an "integration-by-parts" formula. aidsA+ aEA+ because dimR g, = 2. For K-biinvariant functions, the Haar LEMMA 4. The Fourier transform of d(o,m)2 k(m) is measure is C. II sh2a(H)- dH. The elementary spherical aEA + X 1 ( rI1 (ax) -l a( 1I (ax)mwX) functions are indexed by complex-valued forms on a, and if v aEA + aEA + is a nonsingular one, where A is the Laplacian associated with the scalar product on a*R. Z ei(w( )H)e(w) II (aP,) From the lemma it follows easily that wewEW aEA + Ze(w(P)H V)TI(,) II H AR 2.(E[(f)X)M(X))ei(Hfx)dX wEW aEA + |HINk(expH)= aEA + The Plancherel measure is supported by a*R and up to some We may suppose that H belongs to the dominant Weyl positive constant is given by c(v) -2 II (vla) 2. Chamber a+. Noticing that ANI2 is a aE + ll(aX)m(X) holomorphic Let N be the smallest even integer greater than n/2, where function in S",0 which is dominated in any tube 3t by some n M. constant times XI -N/2+k, where k = card A+, we may change Hdim the contour of integration to a* + it,, where t is close to 1. So THEOREM 3. Suppose m is a holomorphic function in the open k(exp H) = HI-N (aII sh(aH))-1e-t(H1P)K(H) tube 31°, invariant by the Weyl group, and such that for any t < 1, aEA + and any differential operator P(D) with constant coefficients and where fIKI(H)I2 dH < C, by Plancherel's formula for degree d < N. there exists a constant C such that L2(a). (*) P(D) m (X) < + X EG 3 But now Holder inequality allows us to estimate f k(exp I C(1 IXI)), H1>1 HEa+ Then the associated G-invariant operator T extends to a bounded H) P (HI sh(a,H))2 dH, where p > 1 (of course one has to operator on L" (1 < p < + co). choose t close to is close to For the proof, we may assume that m has an exponential 1, as p 1). as tends to within fixed tube < We remark that the proof of the theorem can be extended decay t infinity any 53(O + in the < 1), as long as the estimates do not involve specific bounds to give a sufficient condition for multipliers range q < p < q' (with 1 < q < = this result can be for the decay. So the kernel k is well-defined on a as the inverse 2,1/q + 1/q' 1); Fourier transform of m. Let be a K biinvariant C'-function viewed as a partial converse to Theorem 1. so We m in the interior of the tube on G such that sp = 1 in a neighborhood of the identity, and 'p assume that is holomorphic 3,, with t = - and for differential = 0 outside a larger neighborhood, and let k = kyp + k(1 -'p). 2/q 1, any operator P(D) = k, + k2. with constant coefficient of degree d < N- the inequality (*) is satisfied. Then the associated G invariant operator extends 2 This particular lemma follows easily from the theorem on p. 206 to a bounded operator on LY, (q < p < q'). of Stein, E. M. (1970), "Analytic continuation of group repre- sentations," Advan. Math. 4, 172-207, together with an addi- tional convexity argument. 3 See Warner, G. (1972) Harmonic Analysis on Semi-Simple Lie 4 See their paper (1973) "Operators associated with representa- Groups, II (Springer-Verlag, New York), p. 329. tions ...," Studia Math. 47, 285-303. Downloaded by guest on October 4, 2021