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On Fourier Multiplier Transformations of Banach-Valued Functions by Terry R

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On Fourier Multiplier Transformations of Banach-Valued Functions by Terry R TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 285, Number 2, October 1984 ON FOURIER MULTIPLIER TRANSFORMATIONS OF BANACH-VALUED FUNCTIONS BY TERRY R. MCCONNELL1 ABSTRACT. We obtain analogues of the Mihlin multiplier theorem and Little- wood-Paley inequalities for functions with values in a suitable Banach space B. The requirement on B is that it have the unconditionality property for martingale difference sequences. 1. Introduction. A number of authors have considered the extension of the Calderón-Zygmund theory of singular integrals to vector-valued functions (see, e.g., [1, 2, and 6]). Recently Burkholder [5] has obtained definitive results on transforms of Banach space-valued martingales, which are the analogues of singular integrals in probability theory, and these results have been applied to the study of singu- lar integrals by Burkholder and McConnell [6] and by Bourgain [2 and 3]. In the present paper we use martingale methods to obtain extensions of the Mihlin- Hörmander multiplier theorem and the Littlewood-Paley inequalities to the case of Banach-valued functions. For suitable real-valued test functions / defined on R" let / denote the Fourier transform of /, (1.1) /(A) = (2tt)-" / e~iXxf(x)dx, and / the inverse Fourier transform. To each bounded Borel function m on Rn we may associate an operator Tm defined on the test functions / by (1.2) Tmf(x) = (m/)v(x). The Calderón-Zygmund singular integral transformations provide an interesting class of examples (see, e.g., [22, p. 39]) and the fact that such operators are bounded on the Lp spaces for 1 < p < co suggests the following question: under what condi- tions on m does Tm defined in (1.2) extend to a bounded linear transformation of the Lp spaces? A general though by no means complete answer is given by Theorem A, an improvement due to Hörmander [11] of the Mihlin multiplier theorem [17]. THEOREM A. Let k be an integer greater than n/2 and m aCk function defined on R". Assume that there is a constant c such that for all multi-indices a = (ax,... ,an) satisfying \a\ = \ax\ + • • • + |a„| < k we have (1.3) |A|lQl|dam(A)/dAa| < c. Received by the editors August 29, 1983. The results of this paper were announced at the 808th meeting of the AMS, Evanston, November 11-12, 1983. 1980 Mathematics Subject Clasdfication. Primary 42A45, 46E40; Secondary 42A61. Key words and phrases. Fourier multiplier, martingale transform, Lp inequalities, vector-valued function, unconditionality. Supported in part by NSF Grant MCS 82-02286. ©1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 739 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 740 T. R. McCONNELL Then for I < p < co the operator Tm admits an extension to Lp(Rn) satisfying (1-4) ||Tm/||p<Cp||/||p, where the cp are constants depending only on n,p and c. There seem to be two distinct standard methods of proving Theorem A and related results. In the first approach, inequality (1.4) is obtained for the wider class of convolution operators whose kernel K satisfies Hörmander's condition (1.5) / \K(x-y)-K(x)\dx<c. J\x-y\>2\y\ Condition (1.5) implies the boundedness of the Fourier transform of K, and (1.4) for p = 2 then follows by Plancherel's identity. The extension to other values of p is accomplished with the aid of the Calderón-Zygmund decomposition. The second approach relies on inequalities for the Littlewood-Paley g-functions and, again, Plancherel's identity in the case p = 2 plays a pivotal role. For an exposition of both of these methods of proof see [8]. Let B be a real or complex Banach space and let Lp(Rn; B) denote the space of ß-valued Bochner integrable functions / defined on R" and satisfying 11/11?= / |/(x)|pB(ix<co, JRn where ||B denotes the norm of B. For functions / in L1(Rn;ß)nL°°(Rn;ß) with finite-dimensional range we may define the Fourier transform as in (1.1) and the usual Fourier inversion formula holds. Given a bounded real-valued function m, define Tmf as in (1.2). One of the goals of this paper is to show that for B-valued functions / the conclusion (1.4) of Theorem A holds under a strengthened version of (1.3) for a large class of Banach spaces B. The class of Banach spaces, to be described fully below, includes all the Lp spaces for 1 < p < co. However, the classical methods of proof described above break down unless B has an equivalent norm under which it is a Hilbert space. The difficulty is that the Fourier transform is not, in general, a bounded operator on L2(R"; B). The allowable Banach spaces in our version of Theorem A are the f-convex spaces introduced by Burkholder [5]. A real or complex Banach space B is f-convex if there exists on B x B a real-valued function c; having the following properties: (1.6) ç(x, •) is convex for each x G B, (1.7) ç(x,y) = ç(y,x), (1-8) ç{x,y)<\x + y\B if Mb < 1 < Mb, and (1.9) s(0,0)>0. For example, the spaces LP(R;R) and lp are ç-convex for 1 < p < oo, while L'(R;R) and I1 are not (see [5] for these facts and much further information concerning these spaces). In the light of recent results, including those of the present paper, it appears likely that ç-convexity is the correct condition to impose on a Banach space B in order License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FOURIER MULTIPLIER TRANSFORMATIONS 741 for the results of classical harmonic analysis to carry over to the B-valued case. For example, the results of Burkholder and McConnell [6] and of Bourgain [3] taken together show that the Hubert transform is a bounded operator on LP(R; B) if and only if B is ç-convex. (We obtain here an independent proof that f-convexity implies boundedness of the Hubert transform on LP(R; B). See Remark 3.1 below.) More recently, Bourgain has obtained results which imply the full analogue of Theorem A for f-convex Banach lattices [2]. Let us turn now to the precise statements of our results. THEOREM 1.1. Let B be a ç-convex Banach space and m a Cn+1 function defined on Rn. Assume there is a constant cm such that for all multi-indices a satisfying \a\ < n + I we have (1.10) |A|lal|dam(A)/dAQ|<cm. Then for 1 < p < oo the operator Tm has an extension to LP(R"; B) which satisfies (1.11) ||Tm/||p<cp||/||p, Kp<co. The constants cp depend only on cm,n and p. Note that (1.10) is more restrictive than (1.3) in Theorem A. We have been able to weaken (1.10) only at the expense of imposing the additional assumption that m have support in some dyadic rectangle (a dyadic rectangle is a rectangle of the form [ki2~ll,(ki + l)2_il] x • • • x [kn2~ln, (kn + l)2~in] for integers kx,...,kn and *1, • • • , In)- THEOREM 1.2. Let B be a ç-convex Banach space and k an integer greater than n/2. Let m be a Ck function on Rn\{0} such that (1.10) holds for all multi- indices a with \a\ < k. Assume, in addition, that m is supported in some dyadic rectangle. Then, for 1 < p < co the operator Tm has an extension to Lp(Rn;B) which satisfies (1.11). We suspect that the conclusion of Theorem 1.2 remains true even without the assumption on the support of m, but we have been unable to prove this. (How- ever, see Remark 3.2 below where we obtain a new proof of Theorem A.) A more general open question is whether the convolution operators with kernel satisfying Hörmander's condition (1.5) are bounded operators on Lp(Rn;B) for 1 < p < oo and B f-convex. Our final result is an extension of the Littlewood-Paley inequalities to the .re- valued case. To decribe the classical results fix an ordering of the dyadic rectangles {h}k>i, and, for / in L^R™;^) H L°°(R";5) having finite-dimensional range, define the Littlewood-Paley function S(f) by (oo li ' X>(*)li fc=i where fk(x) — jj f(X)élXx dX. In the real case (B = R) Littlewood and Paley [14, 15 and 16] proved that there are constants cp depending only on n and p such that (1-12) c^wfw^wsim^cjfw, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 742 T. R. McCONNELL for 1 < p < co. Let (Sk)k>i be any sequence of numbers from {±1} and define an operator T/£<¡by oo T(e)f(x) = y£tekfk(x). k=l An immediate consequence of (1.12) in the real case is that Tte) satisfies (1.13) c-i/||p<||T(£)/||p<cp||/||p for 1 < p < co. In fact, Khintchine's inequalities imply that (1.12) and (1.13) are equivalent. The situation in the ß-valued case is quite different: Inequalities (1.12) hold if and only if B is isomorphic to a subspace of a Hubert space, while inequalities (1.13) hold if and only if B is f-convex.
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