On bilinear Littlewood-Paley square functions

Publicacions Matem`atiques, Vol 40 (1996), 387–396. ON BILINEAR LITTLEWOOD-PALEY SQUARE FUNCTIONS M. T. Lacey Abstract On the real line, let the Fourier transform of knbe ˆ kn(ξ)=ˆ k(ξ−n) where ˆ k(ξ) is a smooth compactly supported function. Consider the bilinear operators Sn(f,g)(x)=f(x+y)g(x−y)kn(y)dy. If 2 ≤p, q ≤∞, with 1/p +1/q =1/2, I prove that ∞  n=−∞ Sn(f,g)2 2≤C2f2 pg2 q. The constant Cdepends only upon k. 1. The inequalities The principal inequality of this paper has two motivations. In 1964, Alberto Calder´on raised conjectures concerning the following operator acting on two functions fand gdefined on the real line: H(f,g)(x)= 1 πf(x+y)g(x−y)dy/y. This operator, which has come to be known as the bilinear Hilbert transform, serves as the quintessential example of a bilinear operator whose Fourier multiplier has a singularities. This operator will not be addressed in this paper. Research supported in part by an NSF grant. 388 M. T. Lacey The second motivation is the inequality of Littlewood-Paley type below. Denote by SIf(x) the restriction of the Fourier transform of fto an interval I⊂R. There is the inequality ∞  n=−∞ |S[n,n+1)f|21/2p ≤Cpfp,2≤p. This is due to L. Carleson [C]. The restriction to 2 ≤pis sharp, as can be seen by taking the Fourier transform of fto be the indicator of the interval [0,N], for Nlarge. The reader should consult Rubio de Francia’s interesting generalization [RdF]. We extend the inequality above to a bilinear setting, in the case of p= 2. To set notation, consider a smooth bump function with ˆ k(ξ) supported on the unit cube of Rd. For integers n∈Zd, let knbe the function with Fourier transform kn(ξ)=ˆ k(ξ−n) and define Sn(f,g)(x)=Rd f(x+y)g(x−y)kn(y)dy. Theorem 1.1. For all 2≤p, q ≤∞, with 1/p +1/q =1/2, (1.2)  n∈Zd Sn(f,g)2 2≤C2f2 pg2 q. The constant Cis independent of pand q. The proof offered can be extended to other bilinear forms, where for instance the “+y” in the integral is replaced by “−By” where Bis an invertible linear transformation on Rdwhich is not the identity. Notice that we only prove the theorem in its most obvious possible formulation and even then the proof is curiously intricate. The question arises of the boundedness of the square function (n|Sn(f,g)|2)1/2as a map into Lpfor p>2, but I do not know how to address this question. It is natural to ask for possible extensions to a sharp cut off in frequency, namely by taking the knto have Fourier tranform equal to the indicator of the cube [n, n+1). Such a result if true, would be deep as it would already entail the boundedness of the bilinear Hilbert transform as a map into L2. Recent progress has been made on this conjecture in [LT]. Bilinear littlewood-Paley inequalities 389 2. Decomposition of the functions For the proof fand gare decomposed in the space-frequency plane. Define the Fourier transform to be Ff(ξ)= ˆ f(ξ)= 1 (2π)d/2Rd e−2πix·ξf(x)dx, ξ ∈Rd. In the sequel, f,gwill mean f¯gdx, and we shall adopt the notation eξ(x)=e 2πiξ·x. Let ϕbe a smooth, rapidly decreasing function, and set ϕm,n(x)=e m(x)ϕ(x−n) for integers m, n ∈Zd. The particular decomposition of fand gthat is needed is written as (2.1) Φf(x)=  m,n∈Zd f,ϕm,nϕm,n(x). A curious feature of the problem is that a space-frequency decomposition seems to be required, although the definition of Sn(f,g) would not seem to force it upon us. The subsequent section will be devoted to a proof of Lemma 2.2. Adopt the notation of Theorem 1.1, and let ∆(f,g)denote the square function of (1.2). Let ϕbe a function of L2norm 1, with ˆϕsupported in a translate of the cube [−1/4,1/4]d, and (2.3) sup x (1 + |x|)10d|ϕ(x)|=B<∞. Let Φfbe as in (2.1). And let ϕbe a second function satisfying these same attributes, with Φgbeing the corresponding decomposition of g. Then, there is a constant CBso that for all 2≤p, q ≤∞, with 1/p + 1/q =1/2, (2.4) ∆(Φf,Φg)2≤CBfpgq. To conclude the theorem, we need to replace Φfby f, and likewise for g. This can be done by way of a general principle, which we formulate in this way. Lemma 2.5. Let Tbe a sublinear map of bounded smooth functions on Rdinto a Banach space X.IfT◦Φis a bounded operator from Lp, 1≤p<∞, into X, with norm N(ϕ), then Tmaps Lpinto Xwith norm at most [0,1)d[0,1)d N(e2πis·ϕ(·−t)) ds dt. 390 M. T. Lacey Proof: The basis of the proof resides in the following resolution of the identity f(x)=ϕ−2 2f,ϕm,nϕm,n(x)dm dn. This equality is initially understood in the sense of inner products, as we show. Let If denote the right hand side above. Then If,g=f,g, indeed If,g=ϕ−2 2f,ϕm,ng,ϕm,ndm dn. Now, the Fourier transform is unitary, so that f,ϕm,n=ˆ f(ξ),e2πin·ξˆϕ(ξ−m), which is the Fourier transform of the function Fm(ξ)= ˆ f(ξ) ˆϕ(ξ−m) evaluated at ξ=−n. Thus, f,ϕm,n= Fm(−n). Likewise, g,ϕm,nis the Fourier transform of the function Gm(ξ)=ˆg(ξ) ˆϕ(ξ−m) evaluated at −n. From these considerations, it follows that If,g=ϕ−2 2 Fm, Gmdm =ϕ−2 2Fm,G mdm =ϕ−2 2ˆ f(ξ)ˆg(ξ)|ˆϕ(ξ−m)|2dξ dm =ˆ f,ˆg =f,g, as claimed. Recalling the definition of Φ, we then see that it is a discrete form of I. To be more explicit, let ϕs,t be the expansion of (2.1) associated to the function es(x)ϕ(x−t), and assume ϕ2= 1. Then If(x)=[0,1]d[0,1]d Φs,tf(x)ds dt. Moreover, for fassumed smooth and compactly supported, the expansions If(x) and Φs,tfare absolutely convergent. Hence we can interpret Bilinear littlewood-Paley inequalities 391 If(x) in a pointwise sense. For such an f, it follows from our assumptions that Tf(x)X=T(If)X ≤[0,1]d[0,1]dTΦs,tfXds dt ≤fp[0,1]d[0,1]d N(es(·)ϕ(·−t)) ds dt. This is the bound claimed in the lemma. Smooth and compactly supported function are dense in Lp, for 1 ≤p<∞, proving the lemma. The previous two lemmas prove Theorem 1.1 for 2 <p<∞as the smooth functions are dense in Lp. That leaves the inequality for the square function ∆ as a map on L2×L∞. Here, one can take f∈L2 to be bounded smooth and compactly supported. For an arbitrary g∈ L∞, one can take bounded smooth and compactly supported gnso that gn∞≤g∞and ∆(f,gn)L2 −→ ∆(f,g) on all compact subsets of Rd. Moreover, as the proof shows, each gncan be written as an average of expansions Φ. Hence the limiting case of L2×L∞follows. 3. The Proof of Lemma 2.2 One needs a clear understanding of Sµas a Fourier multiplier, obtained by expanding Sµ(f,g) in frequency in a formal way. This requires that fand gbe expanded in different frequency variables, say αand β respectively. Sµ(f,g)(x)=f(x+y)eβ(x−y)Fβg(β)dβkµ(y)dy =F−1 βeβ(−y)f(x+y)kµ(y)dy Fβg(β)(x) =F−1 βF−1 αeβ−α(−y)kµ(y)dy Fαf(α) (x)Fβg(β) (x) (3.1) =F−1 βF−1 α kµ(β−α)Fαf(α)(x)Fβg(β)(x) where Fαdenotes the Fourier transform with frequency variable α. The interchange of integrals in this formal calculation can be rigorously verified for smooth and compactly supported functions fand g. 392 M. T. Lacey To recap, we are to establish the inequality (2.4) above. A central part of this is to diagonalize the sum (3.2) Sµ(Φf,Φg)(x) = m,n∈Zd m,n∈Zd f,ϕm,ng,ϕm,nSµ(ϕm,n,ϕ  m,n)(x). To simplify the notation of the proof, we specialize to the case in which ϕ=ϕ, and the Fourier transform of ϕis supported in [−1/4,1/4]d. The function ϕsatisfies in addition the remaining hypotheses of the lemma. The reader will easily supply the necessary changes in the proof for the general case, as they are only evident in the next paragraph. It follows from (3.1) that for µ, m, m∈Zd, (3.3) Sµ(ϕm,n,ϕ m,n)(x)≡0ifm−m/∈{µ+e|e∈{0,1}d}. This will diagonalize the sums in (3.2) in the frequency variables, which is to say mand m. Below, for notational convienence, we specialize to the case where m=m+µ. For the diagonalization in space, let n, n∈Zd, and observe that Sµ(ϕm,n,ϕ m+µ,n)(x) =e 2m+µ(x)eµ(−y)kµ(y)ϕ(x+y−n)ϕ(x−y−n)dy. In this integral, kµ(y) is the convolution kernel associated to the operation Sµ. Hence, ˜ k(y)=e µ(−y)kµ(y) is independent of µ. The integral above is then ψn,n−n(x)=˜ k(y)ϕ(x+y−n)ϕ(x−y−n)dy. Below, we will denote the difference n−nby η. The Fourier transform of ˜ k(y) is smooth, hence |˜ k(y)|≤Cm(1 ∧|y|−d)m,m>1. With the assumption stated on the decay of ϕ, (3.4) |ϕ(x)|≤Bϕ(1 + |x|)−10d, it follows that for n, η ∈Zd,ψn,η satisfies |ψn,η(x)|≤Bϕ(1 + |η|)−5d(1 + |x−n|+|x−n−η|)−5d. Bilinear littlewood-Paley inequalities 393 Provided a function ψhas a sufficently fast decay, the map of L2(Rd) into &2(Zd) given by f→{f,em(·)ψ(·−n)|m, n ∈Zd}is bounded and invertible. See for instance Section 3.4 of [D] for a discussion of this. In particular, with the estimate on ψn,η, we have the inequality below. (3.5)  m,n∈Zd am,nem(x)ψn,η(x)2 ≤Bϕ(1 + |η|)−4d  m,n∈Zd |am,n|2  1/2 . The rapid decay in ηpermits us to diagonalize the sum in (3.2) in the space variables nand n. Let us consider, with µ∈Zdfixed,  η∈Zd m,n∈Zd f,ϕm,ng,ϕm+µ,n+ηSµ(ϕm,n,ϕ m+µ,n+η)2 = η∈Zd m,n∈Zd f,ϕm,ng,ϕm+µ,n+ηe2m+µ(x)ψn,η(x)2 (3.6) ≤ η∈Zd m,n∈Zd f,ϕm,ng,ϕm+µ,n+ηe2m+µ(x)ψn,η(x)2 ≤CB ϕ η∈Zd (1 + |η|)−4d  m,n∈Zd |f,ϕm,ng,ϕm+µ,n+η|2   1/2 ≤CB ϕ  η∈Zd (1 + |η|)−6d m,n∈Zd |f,ϕm,ng,ϕm+µ,n+η|2   1/2 . We have invoked (3.5), and used a Cauchy-Schwartz inequality to control the sum over η. With rapid decay in |η|, that variable no longer plays an interesting role. Do not forget that the square of the term above must be summed over µas well. Thus we fix η, sum over µand observe that the summing 394 M. T. Lacey variables mand µseparate.  n∈Zd m,µ∈Zd |f,ϕm,ng,ϕm+µ,n+η|2 (3.7) =  n∈Zd m,µ∈Zd |f,ϕm,ng,ϕµ,n+η|2 = n∈ZdUU m em(x)f,ϕm,n µ eµ(y)g,ϕµ,n+η 2 dx dy. In the last line U=[0,1)d. From the definition of ϕm,n observe that  m∈Zd em(x)f,ϕm,n= m∈Zd ϕ(x−m−n)f(x−m), where on the right, we have periodized the function ϕ(x−n)f(x). It is clear that the right hand side has bounded Lp([0,1)d) norm, if f∈ Lp(Rd). To be unambigious, let pbe the conjugate index to p,thus 1<p <2. Then, to simplify notation, we set n= 0 and estimate as follows.  m∈Zd em(x)f,ϕm,0Lp([0,1)d) ≤ m∈Zd f(x−m)ϕ(x−m)Lp([0,1)d) ≤A m∈Zd (1 + |m|)−10df(x−m)Lp([0,1)d) ≤A  m∈Zd (1 + |m|)−pd  1/p ×  m∈Zd (1 + |m|)−9df(x−m)p Lp([0,1)d)  1/p ≤A(1 + |x|)−9df(x)Lp(Rd). Bilinear littlewood-Paley inequalities 395 Clearly the same observations apply for any other choice of n= 0, and to gas well. As 1/p +1/q =1/2, it follows that we can bound the expression in (3.7), yielding the inequalities  n∈Zd m,µ∈Zd |f,ϕm,ng,ϕm+µ,n+η|2 ≤C n∈Zd f(x)(1 + |x−n|)−9d2 p×g(x)(1 + |x−n−η|)−9d2 q (3.8) ≤C  n∈Zd f(x)(1 + |x−n|)−9dp p  2/p ×  n∈Zd g(x)(1 + |x−n−η|)−9dq q  2/q ≤Cf2 pg2 q. Recall that we were to bound the sum over µof Sµ(Φf,Φg)2 2,as expanded in (3.2). As pointed out in (3.3), the sum in (3.2) diagonalized in mand m. The discussion leading to (3.5) shows that the sum effectively diagonalizes in the variables nand nas well. Finally, (3.6) and (3.8) conclude the proof. The constants that interceed depend only on the decay condition on ϕ, (3.4), which is as Lemma 2.2 requires. References [C] L. Carleson, On the Littlewood-Paley Theorem, Inst. MittagLeffler, Report (1967). [D] I. Daubechies,“Ten Lectures on Wavelets,” CBMS Regional Conference Series in Applied Mathematics, SIAM, Philapelphia, 1992. [LT] M. Lacey and C. Thiele, Bounds for the bilinear Hilbert transform on Lp,p>2, Preprint (1996).