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Some Banach Space Embeddings of Classical Function Spaces BULL. AUSTRAL. MATH. SOC. 46E30, 60G50 VOL. 43 (1991) [73-77] SOME BANACH SPACE EMBEDDINGS OP CLASSICAL FUNCTION SPACES N.L. CAROTHERS AND S.J. DILWORTH Banach space embeddings of the Orlicz space Lp + Lq and the Lorentz space LPt q into the Lebesgue—Bochner space LT{1,) are demonstrated for appropriate ranges of the parameters. 1. INTRODUCTION The purpose of this paper is to explain how the existence of isomorphic embed- dings of particular classical Banach function spaces into the Lebesgue-Bochner spaces Lr(l,) may be deduced quite easily from some known stochastic and analytic facts. Our approach has at its foundation a moment inequality of Rosenthal [0] for sums of independent random variables. Indeed, our Theorem 1 below, which describes an embedding of the Orlicz space LT + L. into Lr(l,) in the range 0 < r < a < oo, is in effect a disguised form of a particular version (Theorem A below) of Rosenthal's inequality. Theorem A is a special case of a recent, very general result of Johnson and Schechtman [7], although the idea of using Rosenthal's inequality to define Banach space embeddings dates back at least to [6] (see also [8]). From Theorem 1 just a few short steps are required to deduce the existence of an embedding of the Lorentz space Lp<g into Lq(l,) in the range q < p < a. The latter implies the surprising theorem of Schutt, obtained by rather different finite-dimensional methods, on the existence of an embedding of LViq into Lq for 1 ^ q < p < 2. Schutt's theorem motivated our work to a great extent, and it is to be hoped that our approach helps to explain his theorem by exhibiting it in a different light. 2. NOTATION AND PRELIMINARY RESULTS For 0 < p < oo, Lp denotes the usual Lebesgue space of real-valued functions on [0, oo) with its usual norm, denoted by ||-|| . For 0 < r < a < oo and 0 < t < oo, Received 12 January 1990 The first author was supported in part by BGSU Faculty Research Committee Grant and the second author in part by NSF DMS-8801731. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/91 SA2.00+0.00. 73 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 30 Sep 2021 at 13:32:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700028781 74 N.L. Carothers and S.J. Dilworth [2] LT + t • L, denotes the Orlicz space LT + L, equipped with the norm ||/||x,r.(-«.x, = K(t,f;Lr,L.), where (1) Kit, f; Lr, L.) = ird{\\g\\r +1 \\h\\. : f = g + h} is the standard if-functional of interpolation theory. The Lorentz space LP), is the q 9 p space of real-valued functions on [0, oo) with the norm ||/|L „ = [/0°° f*{t) d(i ^ )] > where /* denotes the decreasing rearrangement of |/|. This form of the XPi, norm is not particularly suitable for our purposes, however, and we shall instead use the following formula which is derived from the Lions-Peetre if-method of interpolation (for example [1]): ne n (2) H/IL,~ E 2- "K(2 ,f;Lr,L.y\ Ln=—oo J provided 1/p = (1 — 6)/r + 6/s. (Throughout, A ~ B means that there is a constant c > 0, depending only on the parameters p, q, et cetera, such that (1/c) A < B ^ cA.) The Lebesgue-Bochner space LT(l,), 0 < r, a < oo, is the space of sequences (/n)^Li °f functions on [0, 1] equipped with the norm -,1/r Given a family of Banach spaces and 0 < p < oo, denotes / oo X1/? x the space of sequences (a'n)^L_OO) n € Xn, with the norm I ^Z ll^nlT I • When all the Xn 's are the same space X, we write lp(X). We turn now to state the moment inequality of Rosenthal [9] in a form which illuminates its linear character. First some notation: given a sequence of functions (/nC=i on l°» !] >let £ ®/n denote the function / on [0, oo) defined by f{n - 1 + t) = n fn(t) for n ^ 1 and 0 < t < 1. Identify [0, 1] with the (measure-equivalent) product space il = [0, 1]N , and denote a typical element of fl by («i, 82, • • •), so that Si, 82, ... are independent coordinates. The version of Rosenthal's inequality which we shall use may be stated succinctly in the following form. THEOREM A. Let 0 < p < 2. Then (3) n=l This version of Rosenthal's inequality was first obtained by Johnson and Schecht- man [7]. (See also [2] or [4] for an approach more in keeping with the present paper.) Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 30 Sep 2021 at 13:32:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700028781 [3] Banach space embeddings 75 3. THE MAIN RESULTS THEOREM 1- (See [4, Theorem 3.1]) Let 0 < r < s < oo. Tien Lr + L, is linearly isomorphic to a subspace of Lr(lB). /2 PROOF: Since 2r/s < 2, we can apply (3) with /„ replaced by |/n|* to obtain 2/. £®/n n=! n=l LT+L. 1/2 2r/, 5 Hence the mapping X) © /« —> (/n( n))^=i defines a linear isomorphism from Lr + £, into LT{1.). " D For our results on LP}q the following simple proposition is the key. PROPOSITION 1. Let 0 < r < 3 < oo and let 0 < t < oo. Then Lr +t • L. is linearly isometric to Lr + L,. PROOF: By dilation of (1) by A > 0, we have 1 1 K(t, /; Lr, L.) = inf^ /' ||*(A*)||r + tX '' \\h(Xx)\\. :f=g + h}. T r Select A = t"K—r) > so that t = X<-'- ^ '. Then l r K(t, f; Lr, L.) = X ' int{\\g(Xx)\\r + \\h(Xx)\\. : f = g + h} l = X l*K{\, f(Xx); Lr, L.) = <•/<-'-'Jif(l, /(*"/<—•'*); Lr, L.). p r) Thus the mapping f(x) -> f*A*- )/(<W('~ x) defines a h'near isometry from ir+<-L, onto ir + L.. D REMARK. The above dilation argument rests on the fact that the function spaces are defined not on [0, 1] but on [0, oo). COROLLARY 1. LetO<r<p<a<oo and let 0 < 9 < oo. TAen LPi, is linearly isomorphic to a subspace of lq(Lr + L,). n PROOF: (2) says that Lp<q is isomorphic to a subspace of X) ®{Lr + 2 • L,)\ . L-oo \q By Proposition 1 each LT + t • L, is linearly isometric to LT + L,, and so the desired result follows. U Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 30 Sep 2021 at 13:32:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700028781 76 N.L. Carothers and S.J. Dilworth [4] THEOREM 2 . Let 0 < q < p < a. Then Lp<q is linearly isomorphic to a subspace ofLq(l.). PROOF: Applying Corollary 1 with r = q shows that Lp>q embeds into lg{Lg + L,). Hence, by Theorem 1, Lp>q embeds into lq(Lq(l,)) which is of course isomorphic to Lq{l,). D COROLLARY 2 . [10] Let 0 < q < p < 2. Then Lp>q is linearly isomorphic to a subspace of Lq. PROOF: Applying Theorem 2 with a = 2 shows that Lp>q embeds into Lq(l2)- It is a well-known consequence of Khintchine's inequality that the mapping {fn(s))™=i —* ^2 fn{a)rn(t), where (rn(t))'^_1 are the Rademacher functions, defines an isomorphic n 2 embedding of Lq{li) into -L9([0, I] ) (which is isomorphic to Lq), and the desired conclusion follows. U REMARK. It is also proved in [10] that ifl <p<g<2 then LPiq is not isomorphic to a subspace of Lx. We indicate here a short proof of this fact. Since Lp>q has "type" p in this range (see for example [8] for the definition of this notion), it follows from the main result of [5] that if LPiq embeds into L\, then lp embeds into LPiq. But by [3, Theorem 2.8], lp does not embed into LPiq. REFERENCES [l] J. Bergh and J. Lofstrom, Interpolation spaces (Springer-Verlag, New York, 1976). [2] N.L. Carothers and S.J. Dilworth, 'Inequalities for sums of independent random vari- ables', Proc. Amer. Math. Soc. 104 (1988), 221-226. [3] N.L. Carothers and S.J. Dilworth, 'Subspaces of Lp,,', Proc. Amer. Math. Soc. 104 (1988), 537-545. [4] S.J. Dilworth, 'A scale of linear spaces related to the Lp scale', Illinois J. Math, (to appear). [5] S. Guerre and M. Levy, 'Espaces lp dans les sous-espaces de L\', Trans. Amer. Math. Soc. 279 (1983), 611-616. [6] W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, 'Symmetric structures in Banach spaces', Mem. Amer. Math. Soc. 217 (1979). [7] W. B. Johnson and G. Schechtman, 'Sums of independent random variables in rearrange- ment invariant function spaces', Ann. Probab. 17 (1989), 789-808. [8] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II. Function Spaces (Springer-Verlag, New York, 1979). [9] H.P. Rosenthal, 'On the subspaces of Lp (p > 2) spanned by sequences of independent random variables', Israel J. Math. 8 (1970), 273-303. [10] C. Schiitt, 'Lorentz spaces that are isomorphic to subspaces of L\', Trans.
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