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Paraproducts for bilinear multipliers associated with convex sets

Olli Saari  and  Christoph Thiele Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany [email protected], [email protected]
Abstract.

We prove bounds in the local L2L^{2} range for exotic paraproducts motivated by bilinear multipliers associated with convex sets. One result assumes an exponential boundary curve. Another one assumes a higher order lacunarity condition.

1. Introduction

Given a bounded measurable function mm in the plane, we define the associated bilinear Fourier multiplier operator acting on a pair of one dimensional Schwartz functions (f,g)(f,g) by

Bm(f,g)(x):=2m(ξ,η)f^(ξ)g^(η)e2πi(ξ+η)x𝑑ξ𝑑η.B_{m}(f,g)(x):=\iint_{{\mathbb{R}}^{2}}m(\xi,\eta)\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi+\eta)x}\,d\xi d\eta. (1.1)

The constant multiplier m=1m=1 reproduces the pointwise product that maps (f,g)(f,g) to fgfg, and it is a fundamental question to determine whether and for which mm Hölder’s inequality, true for the pointwise product, extends to BmB_{m}. We focus on bounds

Bm(f,g)p3Cp1,p2,mfp1gp2\|B_{m}(f,g)\|_{p_{3}^{\prime}}\leq C_{p_{1},p_{2},m}\|f\|_{p_{1}}\|g\|_{p_{2}} (1.2)

in the open local L2L^{2} region

2<p1,p2,p3<,1p1+1p2+1p3=1,2<p_{1},p_{2},p_{3}<\infty,\quad\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=1, (1.3)

where p3=p3/(p31)p_{3}^{\prime}=p_{3}/(p_{3}-1) denotes the dual exponent.

A well-understood example is the classical linear Mikhlin multiplier mm in two dimensions. It leads to the theory of paraproducts, for which we refer to the textbooks [21, 23] and the original references therein. Beyond Mikhlin multipliers, the bilinear Hilbert transforms are the most prominent examples. Their essence is captured by choosing mm to be the characteristic function of a half-plane. The bounds (1.2) and (1.3) for this case were established in [13, 14]. They also hold with a constant independent of the slope and location of the associated half-plane [10, 29], and the range of exponents can be extended beyond the local L2L^{2} range as in [8, 15, 16].

Our focus here is on multipliers mm which are characteristic functions of convex sets rather than just half-planes as in the case of the bilinear Hilbert transforms. Certain curved regions to be discussed shortly, most notably the epigraph of a parabola, were considered in [22], and the bilinear disc multiplier has been studied in [11]. The bounds (1.2) and (1.3) are known for the disc. The results in [11] also imply the previously known uniform local L2L^{2} bounds for the half-plane multipliers as can be seen by using the invariance of the bounds under dilation and translation of the convex set and approximating half-planes by large discs. In [5], a lacunary polygon is discussed.

A further list of examples of convex sets illuminating the increasingly complicated structure can be given as follows. We say a line has degenerate direction if it is orthogonal to one of the three vectors (0,1)(0,1), (1,0)(1,0) and (1,1)(1,1). A line through a boundary point of a convex set which does not intersect the interior of the convex set is a tangent. We list:

  1. (1)

    A convex polygon.

  2. (2)

    A convex set of the form {(ξ,η):0ξ1,γ(ξ)η}\{(\xi,\eta):0\leq\xi\leq 1,\ \gamma(\xi)\leq\eta\} for some convex function γ:[0,1][0,1]\gamma:[0,1]\to[0,1] such that all tangent lines at points (ξ,γ(ξ))(\xi,\gamma(\xi)) with 0<ξ<10<\xi<1 have slope between aa and 2a2a for some 0<a<10<a<1.

  3. (3)

    A bounded convex set with C2C^{2} boundary curve such that the curvature of the boundary is nonzero at every boundary point with a degenerate tangent line.

  4. (4)

    A bounded convex set CC such that every boundary point with a degenerate tangent line has tangent lines for every direction in a neighborhood of this degenerate direction.

  5. (5)

    The convex set {(ξ,η):ξ0, 2ξη}\{(\xi,\eta):\xi\leq 0,\ 2^{\xi}\leq\eta\}.

  6. (6)

    The convex set {(ξ,η):0ξ1,γ(ξ)η}\{(\xi,\eta):0\leq\xi\leq 1,\ \gamma(\xi)\leq\eta\}, where γ\gamma is a monotone increasing convex function mapping [0,1][0,1] to [0,28][0,2^{-8}] such that the set {γ(2j),j}\{\gamma(2^{-j}),j\in{\mathbb{N}}\} is multi-lacunary. See Definition 1.3 below for the definition of multi-lacunarity.

  7. (7)

    A general bounded convex set CC.

The first four examples are known to satisfy bounds (1.2) and (1.3).

The first example is easily reduced to the bilinear Hilbert transforms. The reduction works by iteratively cutting the multiplier by a line parallel to a degenerate direction, which results in two new multipliers, whose operators can be expressed through the original bilinear multiplier operator and pre- or post-composition with a linear multiplier such as the Riesz projection.

The second example is a perturbation of the bilinear Hilbert transforms. Thanks to the comparable upper and lower bounds on the slope, the methods for the bilinear Hilbert transform can be adapted to this situation [22, 11]. The third example includes the case of the parabola and the disc [22, 11]. It requires an additional technique to put together infinitely many pieces as in the second example as well as an additional bound for a central piece near each degenerate direction. These additional pieces are what we call exotic paraproducts. The techniques of [22, 11] still apply at this level of generality. Likewise, the fourth example can be handled with similar methods. Such multipliers are more general away from the critical directions but have strongly regulated behavior at the degenerate directions.

In this paper, we study relaxation of the additional conditions near the degenerate directions. Beginning with the proof of the first bounds for the bilinear Hilbert transform, the common approach to understand bilinear multipliers associated with characteristic functions has been to decompose them into smoother multipliers, paraproducts, which are singular only at a single boundary point instead of a one-dimensional set. As this boundary point comes closer to a degenerate tangent direction, the relevant paraproducts undergo a deformation. At a boundary point with degenerate tangent direction, one encounters an entirely exotic paraproduct, whose structure is closely tied to the behaviour of the boundary in the vicinity of that point. In the present paper, in particular due to the local L2L^{2} range, it seems prudent to consider rougher exotic paraproducts which correspond to characteristic functions of certain staircase sets.

One of the results in the present paper provides bounds for the exotic paraproduct associated with case five of the previous list.

Theorem 1.1.

Let p1,p2,p3p_{1},p_{2},p_{3} be as in (1.3). Define

m(ξ,η):=j1[(j+1),j)(ξ)1[2j,1)(η).m(\xi,\eta):=\sum_{j\in{\mathbb{N}}}1_{[-(j+1),-j)}(\xi)1_{[2^{-j},1)}(\eta).

Then the operator (1.1) satisfies the a priori estimate (1.2).

Here we are also able to complete the passage from a paraproduct estimate to a multiplier bound as in [22, 11] and reduce the bounds for case five to bounds for case two.

Corollary 1.2.

Let p1,p2,p3p_{1},p_{2},p_{3} be as in (1.3). Let CC be the convex set

{(ξ,η):ξ0, 2ξη<1}.\{(\xi,\eta):\xi\leq 0,\ 2^{\xi}\leq\eta<1\}.

Then BmB_{m} as in (1.1) with m=1Cm=1_{C} satisfies the a priori bounds (1.2).

The particular cut-offs at 0 and 11 are not important in this theorem. One also obtains bounds for similar convex sets with constraints of the type aξηa^{\xi}\leq\eta with a2a\neq 2 by applying translation and isotropic dilation to the multiplier. Hence the number 22 has no fundamental importance in the corollary.

Our second result proves bounds for exotic paraproducts related to the sixth case of our list.

Definition 1.3 (Multi-lacunarity).

Let bb be a non-negative integer. We call a finite set XX of real numbers (0,b)(0,b)-lacunary, if it consists of a single element.

Let dd be a non-negative integer and assume we have already defined (d,b)(d,b)-lacunarity. We call a finite set XX of real numbers (d+1,b)(d+1,b)-lacunary, if it can be partitioned into two sets LL and OO such that LL is (d,b)(d,b)-lacunary and for any pair of different points ξ,ξ\xi,\xi^{\prime} in OO we have

dist(ξ,ξ)2bdist(ξ,L).\operatorname{dist}(\xi,\xi^{\prime})\geq 2^{-b}\operatorname{dist}(\xi,L).
Theorem 1.4.

Let b,d2b,d\geq 2 be integers and let p1,p2,p3p_{1},p_{2},p_{3} be as in (1.3). Assume that we have sequences

(ηj)j,(ζj)j,(ξj)j(\eta_{j})_{j\in{\mathbb{N}}},\ (\zeta_{j})_{j\in{\mathbb{N}}},\ (\xi_{j})_{j\in{\mathbb{N}}}

such that for all jj

2jηjζj<22j,\displaystyle 2^{-j}\leq\eta_{j}\leq\zeta_{j}<2^{2-j}, (1.4)
0ξj+26jξj1.\displaystyle 0\leq\xi_{j}+2^{6-j}\leq\xi_{j-1}. (1.5)

Assume the image XX of the sequence (ξj)(\xi_{j}) be (d,b)(d,b)-lacunary. Then the multiplier operator BmB_{m} as in (1.1) with

m(ξ,η)=j1(0,ξj)(ξ)1(ηj,ζj)(η)m(\xi,\eta)=\sum_{j}1_{(0,\xi_{j})}(\xi)1_{(\eta_{j},\zeta_{j})}(\eta) (1.6)

satisfies the estimate (1.2) with constant

Cp1,p2,m=Cp1,p2,b,dC_{p_{1},p_{2},m}=C_{p_{1},p_{2},b,d}

that depends on mm only through bb and dd.

In this case, however, we are not able to complete the program as the convex set lacks the regularity relevant for techniques in [22, 11] to work. Moreover, the bounds (1.2) and (1.3) for the general case seven appear entirely beyond the techniques known to us. We are without any bias towards validity or invalidity of these bounds.

Standard paraproducts come with exponentially growing sequences ξj\xi_{j}, ηj\eta_{j} and ζj\zeta_{j}, all other configurations with an increasing sequence ξj\xi_{j} and an interlaced pair of increasing sequences (ηj,ζj)(\eta_{j},\zeta_{j}) may be considered exotic. We emphasize that the point here rests in the growth conditions on these sequences, the characteristic functions in place of smoother versions found elsewhere in the literature being a minor modification in the considered range of exponents.

Previous research on various assumptions on the sequences can be found in [25, 24, 18, 19, 20], partially in connection with bilinear Hilbert transforms on curves, but none of these references appears to go beyond the case d=1d=1 of multi-lacunarity. The multi-lacunarity assumption also appears in similar context elsewhere in harmonic analysis. For example, it provides a sharp condition on sets of directions for which the directional maximal operator is unbounded [28, 1]. These are deep facts building on a long history of related results. Directional Hilbert transforms in the plane with multi-lacunarity assumptions are considered in [6]. Multi-lacunary sets of directions in higher dimensions appear in [3, 26, 7].


Acknowledgement. The authors were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813 as well as SFB 1060. Part of the research was carried out while the authors were visiting the Oberwolfach Research Institute for Mathematics, the workshop Real and Harmonic Analysis.

2. Proof of Theorem 1.1

Throughout this section, we fix p1,p2,p3p_{1},p_{2},p_{3} as in (1.3). We work through a sequence of lemmata, at the end of which we are ready to prove Theorem 1.1 and Corollary 1.2. Define the VrV^{r} norm for a function hh on {\mathbb{Z}} or {\mathbb{R}} by

hVr:=supx|h(x)|+supN,x0<x1<<xN(n=1N|h(xn)h(xn1)|r)1/r.\|h\|_{V^{r}}:=\sup_{x}|h(x)|+\sup_{N,\ x_{0}<x_{1}<\dots<x_{N}}\left(\sum_{n=1}^{N}|h(x_{n})-h(x_{n-1})|^{r}\right)^{1/r}.

For a measurable function nn, we define the linear multiplier operator

Mnf(x):=f^(ξ)n(ξ)e2πixξ𝑑x,M_{n}f(x):=\int_{\mathbb{R}}\widehat{f}(\xi)n(\xi)e^{2\pi ix\xi}\,dx, (2.1)

and we define the multiplier norms

nMp=supfp=1Mnfp.\|n\|_{M^{p}}=\sup_{\|f\|_{p}=1}\|M_{n}f\|_{p}\ .

If nn is the characteristic function of an interval II, we write MIM_{I} instead of MnM_{n}. We also consider the Hardy–Littlewood maximal operator {\mathcal{M}}.

Our first Lemma 2.1 concerns a bilinear operator, which has almost the form of a bilinear multiplier. The occurrence of an external parameter rr in the exponential makes the difference. The main case is α=0\alpha=0, whereas the side product case α=1\alpha=1 will be used to estimate certain error terms.

Lemma 2.1.

Let ρ,ϕ\rho,\phi be Schwartz functions supported in [2,2][-2,2]. Let rr\in{\mathbb{R}} and α{0,1}\alpha\in\{0,1\}. Let ϵ>0\epsilon>0 be small enough so that

|121p1|<12+ϵ.\left\lvert\frac{1}{2}-\frac{1}{p_{1}}\right\rvert<\frac{1}{2+\epsilon}.

Define the operator

B(f,g)(x):=2f^(ξ)g^(η)e2πi(ξx+ηr)j2αjρ(ξ+j)ϕ(2jη)dξdηB(f,g)(x):=\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi x+\eta r)}\sum_{j\in{\mathbb{N}}}2^{-\alpha j}\rho(\xi+j){\phi}(2^{j}\eta)d\xi d\eta

and the averages

Ag(r)(j):=g^(η)ϕ(2jη)e2πiηr𝑑η.Ag(r)(j):=\int\widehat{g}(\eta)\phi(2^{j}\eta)e^{2\pi i\eta r}d\eta.

Then there is a constant Cρ,ϕ,p1,ϵC_{\rho,\phi,p_{1},\epsilon} such that for all Schwartz functions ff and gg, we have

B(f,g)p1Cρ,ϕ,p1,ϵfp1Ag(r)V2+ϵ.\|B(f,g)\|_{p_{1}}\leq C_{\rho,\phi,p_{1},\epsilon}\|f\|_{p_{1}}\|Ag(r)\|_{V^{2+\epsilon}}. (2.2)
Proof.

We decompose ρ\rho into a Fourier series on an interval of length four

ρ(ξ)=1[2,2](ξ)k/4ρ^ke2πikξ.\rho(\xi)=1_{[-2,2]}(\xi)\sum_{k\in{\mathbb{Z}}/4}\widehat{\rho}_{k}e^{2\pi ik\xi}.

Splitting BB correspondingly and using the rapid decay of ρ^k\widehat{\rho}_{k}, we see it suffices to prove bounds analogous to (2.2) on

2f^(ξ)g^(η)e2πi(ξx+ηr)j2αj1[2,2](ξ+j)e2πik(ξ+j)ϕ(2jη)dξdη,\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi x+\eta r)}\sum_{j\in{\mathbb{N}}}2^{-\alpha j}1_{[-2,2]}(\xi+j)e^{2\pi ik(\xi+j)}\phi(2^{j}\eta)d\xi d\eta,

uniformly in k/4k\in{\mathbb{Z}}/4.

We split the sum over jj into four sums depending on the congruence class of jj modulo four, and we notice that the factor e2πikje^{2\pi ikj} is constant in jj in each of the four sums. Using further that the translation xx+kx\to x+k leaves the Lp1L^{p_{1}} norm invariant, we see it suffices to prove bounds analogous to (2.2) on

2f^(ξ)g^(η)e2πi(ξx+ηr)j,jj0mod42αj1[2,2](ξ+j)ϕ(2jη)dξdη\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi x+\eta r)}\sum_{j\in{\mathbb{N}},\ j\equiv j_{0}\operatorname{mod}4}2^{-\alpha j}1_{[-2,2]}(\xi+j)\phi(2^{j}\eta)d\xi d\eta

for a fixed parameter j0j_{0}.

We identify this expression as a linear multiplier of the form (2.1) applied to ff. The multiplier symbol is

n(ξ)=j,jj0mod42αj1[2,2](ξ+j)Ag(r)(j),n(\xi)=\sum_{j\in{\mathbb{N}},\ j\equiv j_{0}\operatorname{mod}4}2^{-\alpha j}1_{[-2,2]}(\xi+j)Ag(r)(j),

and it thus suffices to show

nMp1Cρ,ϕ,p1,ϵAg(r)V2+ϵ.\|n\|_{M^{p_{1}}}\leq C_{\rho,\phi,p_{1},\epsilon}\|Ag(r)\|_{V^{2+\epsilon}}. (2.3)

We apply the following well-known control of the multiplier norm by variation norms proven by Coifman, Rubio de Francia and Semmes [4].

Theorem 2.2.

Let nn be a measurable function on {\mathbb{R}}, then

nMpCp,rnVr,\|n\|_{M^{p}}\leq C_{p,r}\|n\|_{V^{r}},

provided 1<p<1<p<\infty and |1/21/p|1/r|1/2-1/p|\leq 1/r.

For α=0\alpha=0, inequality (2.3) follows if one identifies nn as a step function constant on intervals of length 44, taking precisely the value Ag(r)(j)Ag(r)(j) in the jj-th interval, counted in natural order, and taking the value 0 outside the union of these intervals.

For α=1\alpha=1, we observe that for any sequence a(k)a(k) tending to zero, the variation norm of the sequence b(k)=2ka(k)b(k)=2^{-k}a(k) is bounded by a constant times the supremum norm of aa, which in turn is controlled by the variation norm of aa. This fact following from a plain application of the triangle inequality completes the proof of Lemma 2.1. ∎

The next lemma passes to a localized version of actual bilinear multipliers. The parameter rr in the exponent disappears, but we introduce a new localization parameter ss. In what follows, we use the translation operator defined by

Tsh(x)=h(xs).T_{s}h(x)=h(x-s).
Lemma 2.3.

Let χ,ρ,ϕ\chi,\rho,\phi be Schwartz functions supported in [2,2][-2,2] and

B(f,g)(x):=Tsχ(x)2f^(ξ)g^(η)e2πi(ξ+η)xjρ(ξ+j)ϕ(2jη)dξdη.B(f,g)(x):=T_{s}\chi(x)\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi+\eta)x}\sum_{j\in{\mathbb{N}}}\rho(\xi+j)\phi(2^{j}\eta)d\xi d\eta. (2.4)

Given the averages Ag(r)(j)Ag(r)(j) and a parameter ϵ\epsilon as in Lemma 2.1, there is a constant Cχ,ρ,ϕ,p1,ϵC_{\chi,\rho,\phi,p_{1},\epsilon} so that for all Schwartz functions ff and gg we have

B(f,g)p1Cχ,ρ,ϕ,p1,ϵfp12(AgV2+ϵ)(s).\|B(f,g)\|_{p_{1}}\leq C_{\chi,\rho,\phi,p_{1},\epsilon}\|f\|_{p_{1}}{\mathcal{M}}_{2}(\|Ag\|_{V^{2+\epsilon}})(s).
Proof.

We apply the support assumption on χ\chi and the fundamental theorem of calculus as well as the product rule to write (2.4) as

s2xTsχ(r)2f^(ξ)g^(η)e2πi(ξx+ηr)jρ(ξ+j)ϕ(2jη)dξdηdr\displaystyle\int_{s-2}^{x}T_{s}\chi^{\prime}(r)\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi x+\eta r)}\sum_{j\in{\mathbb{N}}}\rho(\xi+j)\phi(2^{j}\eta)d\xi d\eta dr (2.5)
+s2xTsχ(r)2f^(ξ)g^(η)2πiηe2πi(ξx+ηr)jρ(ξ+j)ϕ(2jη)dξdη.\displaystyle\ +\int_{s-2}^{x}T_{s}\chi(r)\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)2\pi i\eta e^{2\pi i(\xi x+\eta r)}\sum_{j\in{\mathbb{N}}}\rho(\xi+j)\phi(2^{j}\eta)d\xi d\eta. (2.6)

We first discuss the term (2.5). We estimate the integral in rr with the L1L^{1} norm to obtain the bound

Tsχ(r)2f^(ξ)g^(η)e2πi(ξx+ηr)jρ(ξ+j)ϕ(2jη)dξdηL1(r)Lp1(x)Tsχ(r)2f^(ξ)g^(η)e2πi(ξx+ηr)jρ(ξ+j)ϕ(2jη)dξdηLp1(x)L1(r)\left\|\left\|T_{s}\chi^{\prime}(r)\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi x+\eta r)}\sum_{j\in{\mathbb{N}}}\rho(\xi+j)\phi(2^{j}\eta)d\xi d\eta\right\|_{L^{1}(r)}\right\|_{L^{p_{1}}(x)}\\ \leq\left\|T_{s}\chi^{\prime}(r)\left\|\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi x+\eta r)}\sum_{j\in{\mathbb{N}}}\rho(\xi+j)\phi(2^{j}\eta)d\xi d\eta\right\|_{L^{p_{1}}(x)}\right\|_{L^{1}(r)}

for the Lp1(x)L^{p_{1}}(x) norm of (2.5). We used Minkowski’s inequality and the condition 1<p11<p_{1} here. Next we estimate the inner norm with Lemma 2.1. Setting α=0\alpha=0 and ϵ>0\epsilon>0 small, we obtain an upper bound by

Cρ,ϕ,p1,ϵTsχ(r)fp1Ag(r))V2+ϵL2(r)Cρ,ϕ,χ,p1,ϵfp1(AgV2+ϵ)(s).C_{\rho,\phi,p_{1},\epsilon}\left\|T_{s}\chi^{\prime}(r)\left\|f\right\|_{p_{1}}\left\|Ag(r))\right\|_{V^{2+\epsilon}}\right\|_{L^{2}(r)}\\ \leq C_{\rho,\phi,\chi,p_{1},\epsilon}\|f\|_{p_{1}}{\mathcal{M}}(\|Ag\|_{V^{2+\epsilon}})(s).

The last inequality follows by recognizing a smooth average over the support of TsχT_{s}\chi^{\prime}, which is near ss, and dominating it by the Hardy–Littlewood maximal function. This completes the bound for first term (2.5).

We rewrite the second term (2.6) as

2πis2xTsχ(r)2f^(ξ)g^(η)e2πi(ξx+ηr)j<02jρ(ξj)ϕ~(2jη)dξdη,2\pi i\int_{s-2}^{x}T_{s}\chi(r)\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi x+\eta r)}\sum_{j<0}2^{j}\rho(\xi-j)\tilde{\phi}(2^{-j}\eta)d\xi d\eta,

where the new Schwartz function ϕ~(η)=ηϕ(η)\tilde{\phi}(\eta)=\eta\phi(\eta) depends on ϕ\phi only. We can proceed exactly as with the first term (2.5), but now applying Lemma 2.1 with α=1\alpha=1 and the Schwartz function ϕ~\tilde{\phi} instead of α=0\alpha=0 and the Schwartz function ϕ\phi. This results in the desired bound and hence completes the proof of Lemma 2.3. ∎

Finally, we can pass to a standard bilinear multiplier estimate. This entails getting rid of the localization present in the previous estimate. The multiplier below is a smooth model of the one in Theorem 1.1.

Lemma 2.4.

Let ρ\rho and ϕ\phi be Schwartz functions supported in [2,2][-2,2] and

B(f,g)(x):=2f^(ξ)g^(η)e2πi(ξ+η)xjρ(ξ+j)ϕ(2jη)dξdη.B(f,g)(x):=\int_{{\mathbb{R}}^{2}}\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i(\xi+\eta)x}\sum_{j\in{\mathbb{N}}}\rho(\xi+j)\phi(2^{j}\eta)d\xi d\eta.

There is a constant Cϕ,ρ,p1,p2C_{\phi,\rho,p_{1},p_{2}} such that for any Schwartz functions ff and gg

B(f,g)p3Cϕ,ρ,p1,p2fp1gp2.\|B(f,g)\|_{p_{3}^{\prime}}\leq C_{\phi,\rho,p_{1},p_{2}}\|f\|_{p_{1}}\|g\|_{p_{2}}.
Proof.

It suffices to prove the dual estimate

h(x)B(f,g)(x)𝑑xCϕ,ρ,p1,p2fp1gp2hp3\int_{\mathbb{R}}h(x)B(f,g)(x)\,dx\leq C_{\phi,\rho,p_{1},p_{2}}\|f\|_{p_{1}}\|g\|_{p_{2}}\|h\|_{p_{3}}

for all Schwartz functions hh. Fixing a suitable normalized non-negative Schwartz function χ\chi supported on [2,2][-2,2], we write

h(x)B(f,g)(x)𝑑x=3h(x)Tsχ3(x)B(fTs+tχ,g)(x)𝑑s𝑑t𝑑x,\int_{\mathbb{R}}h(x)B(f,g)(x)\,dx=\int_{{\mathbb{R}}^{3}}h(x)T_{s}\chi^{3}(x)B(fT_{s+t}\chi,g)(x)\,dsdtdx, (2.7)

which can be done because Tsχ(x)𝑑s\int_{\mathbb{R}}T_{s}\chi(x)\,ds is a constant function as is the integral over ss of Tsχ3(x)T_{s}\chi^{3}(x). Next we seek an estimate for

TsχB(fTs+tχ,g)p1\|T_{s}\chi B(fT_{s+t}\chi,g)\|_{p_{1}} (2.8)

for arbitrary real tt.

For |t|4|t|\leq 4, we use Lemma 2.3 and an epsilon depending only on p1p_{1} to obtain the upper bound

TsχB(fTs+tχ,g)p1Cχ,ρ,ϕ,p1fTs+tχp1(AgV2+ϵ)(s).\|T_{s}\chi B(fT_{s+t}\chi,g)\|_{p_{1}}\leq C_{\chi,\rho,\phi,p_{1}}\|fT_{s+t}\chi\|_{p_{1}}{\mathcal{M}}(\|Ag\|_{V^{2+\epsilon}})(s).

For |t|>4|t|>4, we write the Fourier expansion

B(fTs+tχ,g)(x)=3f^(ζ)χ^(ξζ)g^(η)e2πi[(ξ+η)x(s+t)(ξζ)]jρ(ξ+j)ϕ(2jη)dξdηdζ.B(fT_{s+t}\chi,g)(x)\\ =\int_{{\mathbb{R}}^{3}}\widehat{f}(\zeta)\widehat{\chi}(\xi-\zeta)\widehat{g}(\eta)e^{2\pi i[(\xi+\eta)x-(s+t)(\xi-\zeta)]}\sum_{j\in{\mathbb{N}}}\rho(\xi+j)\phi(2^{j}\eta)d\xi d\eta d\zeta. (2.9)

Integrating by parts twice, we can write the ξ\xi integral in (2.9) as

β=131(xst)2χ^β(ξζ)e2πi[(ξ+η)x(t+s)(ξζ)]j<0ρβ(ξj)dξ,\sum_{\beta=1}^{3}\frac{1}{(x-s-t)^{2}}\int_{{\mathbb{R}}}\widehat{\chi}_{\beta}(\xi-\zeta)e^{2\pi i[(\xi+\eta)x-(t+s)(\xi-\zeta)]}\sum_{j<0}\rho_{\beta}(\xi-j)d\xi,

where the three pairs of Schwartz functions (χβ,ρβ)(\chi_{\beta},\rho_{\beta}), β=1,2,3\beta=1,2,3, are determined by the product rule. Set

χ~(x)=β=13|χβ(x)|+|χ(x)|.\tilde{\chi}(x)=\sum_{\beta=1}^{3}|\chi_{\beta}(x)|+|\chi(x)|.

Note that |xst||x-s-t| is comparable to |t||t| when x[s2,s+2]x\in[s-2,s+2], which is the essential domain of integration in (2.7). Inserting this into (2.9) and using Lemma 2.3, we obtain for (2.8) the upper bound

Cχ,ρ,ϕ,p1t2fTs+tχ~p1(AgV2+ϵ)(s).C_{\chi,\rho,\phi,p_{1}}t^{-2}\|fT_{s+t}\tilde{\chi}\|_{p_{1}}{\mathcal{M}}(\|Ag\|_{V^{2+\epsilon}})(s).

Combining the two cases of small and large tt, we obtain for (2.8) the bound

Cχ,ρ,ϕ,p1(1+t2)1fTs+tχ~p1(AgV2+ϵ)(s).C_{\chi,\rho,\phi,p_{1}}(1+t^{2})^{-1}\|fT_{s+t}\tilde{\chi}\|_{p_{1}}{\mathcal{M}}(\|Ag\|_{V^{2+\epsilon}})(s).

Turning back to (2.7), we apply Hölder’s inequality on the integral in xx and a trivial bound on a factor Tsχp2\|T_{s}\chi\|_{p_{2}} to estimate (2.7) by

Cχ,p22hTsχp3TsχB(fTs+tχ,g)p1𝑑s𝑑tCχ,ρ,ϕ,p1,p22hTsχp3(1+t2)1fTs+tχ~p1(AgV2+ϵ)(s)𝑑s𝑑tCχ,ρ,ϕ,p1,p2hp3(1+t2)1fp1(AgV2+ϵ)p2𝑑t.C_{\chi,p_{2}}\int_{{\mathbb{R}}^{2}}\|hT_{s}\chi\|_{p_{3}}\|T_{s}\chi B(fT_{s+t}\chi,g)\|_{p_{1}}\,dsdt\\ \leq C_{\chi,\rho,\phi,p_{1},p_{2}}\int_{{\mathbb{R}}^{2}}\|hT_{s}\chi\|_{p_{3}}(1+t^{2})^{-1}\|fT_{s+t}\tilde{\chi}\|_{p_{1}}{\mathcal{M}}(\|Ag\|_{V^{2+\epsilon}})(s)\,dsdt\\ \leq C_{\chi,\rho,\phi,p_{1},p_{2}}\int_{{\mathbb{R}}}\|h\|_{p_{3}}(1+t^{2})^{-1}\|f\|_{p_{1}}\|{\mathcal{M}}(\|Ag\|_{V^{2+\epsilon}})\|_{p_{2}}\,dt.

In the last line, we have applied Hölder’s inequality and noted

hTsχp3p3𝑑s=2|h(x)Tsχ(x)|p3𝑑x𝑑s=Cχ,p3hp3p3.\int_{\mathbb{R}}\|hT_{s}\chi\|_{p_{3}}^{p_{3}}\,ds=\int_{{\mathbb{R}}^{2}}|h(x)T_{s}\chi(x)|^{p_{3}}dxds=C_{\chi,p_{3}}\|h\|_{p_{3}}^{p_{3}}.

The same computation also applies to the factor with Lp1L^{p_{1}} norm. The integral in tt is trivial, and hence we obtain for (2.7) the upper bound

Cχ,ρ,ϕ,p1,p2hp3fp1(AgV2+ϵ)p2.C_{\chi,\rho,\phi,p_{1},p_{2}}\|h\|_{p_{3}}\|f\|_{p_{1}}\|\mathcal{M}(\|Ag\|_{V^{2+\epsilon}})\|_{p_{2}}.

It remains to observe

(AgV2+ϵ)p2Cϕ,p2gp2.\|\mathcal{M}(\|Ag\|_{V^{2+\epsilon}})\|_{p_{2}}\leq C_{\phi,p_{2}}\|g\|_{p_{2}}.

This follows from the Hardy–Littlewood maximal theorem and a well-known variational bound stated in the following theorem, whose general formulation we quote from Jones, Seeger and Wright [12].

Theorem 2.5.

Let 1<p<1<p<\infty and 2<r<2<r<\infty. Let ϕ\phi be a Schwartz function and define ϕt(x)=2tϕ(2tx)\phi_{t}(x)=2^{-t}\phi(2^{-t}x). Define the averaging operators

Ah(x)(t)=h(xy)ϕt(y)𝑑y.Ah(x)(t)=\int_{\mathbb{R}}h(x-y)\phi_{t}(y)\,dy.

Then we have the bound

AhVrLpCp,rhp.\|\|Ah\|_{V^{r}}\|_{L^{p}}\leq C_{p,r}\|h\|_{p}.

The theorem goes back to Lépingle in the martingale setting [17], and it was introduced for applications in harmonic analysis by Bourgain [2]. However, the precise formulation we use is from [12]. Note that the convolution operator appearing in this Theorem can be written equivalently as a multiplier corresponding to our definition of AA. This completes the proof of Lemma 2.4. ∎

We are now ready to prove Theorem 1.1. What remains is to pass from the smooth model in Lemma 2.4 to the rough paraproduct defined using characteristic functions of intervals instead of smooth Schwartz bumps.

Proof of Theorem 1.1.

We regroup the multiplier operator of Theorem 1.1 as

Bm(f,g)=jM(,j1)fM[2j1,2j)g.B_{m}(f,g)=\sum_{j\in{\mathbb{N}}}M_{(-\infty,-j-1)}fM_{[2^{-j-1},2^{-j})}g.

Pick a smooth function ρ\rho supported in [1,1)[-1,1) such that

kTkρ(k)=kρ(ξ+k)=1.\sum_{k\in{\mathbb{Z}}}T_{-k}\rho(k)=\sum_{k\in{\mathbb{Z}}}\rho(\xi+k)=1.

We compare BmB_{m} with

Bm~(f,g)=jk>jMTkρfM[2j1,2j)g.B_{\tilde{m}}(f,g)=\sum_{j\in{\mathbb{N}}}\sum_{k>j}M_{T_{-k}\rho}fM_{[2^{-j-1},2^{-j})}g.

The difference satisfies

(Bm~Bm)(f,g)=jMTj1ρM[j1,j]fM[2j1,2j)g,(B_{\tilde{m}}-B_{m})(f,g)=\sum_{j\in{\mathbb{N}}}M_{T_{-j-1}\rho}M_{[-j-1,-j]}fM_{[2^{-j-1},2^{-j})}g,

and we have the estimate

(Bm~Bm)(f,g)p3(j|MTj1ρM[j1,j]f|2)1/2p1(j|M[2j1,2j)g|2)1/2p2.\|(B_{\tilde{m}}-B_{{m}})(f,g)\|_{p_{3}^{\prime}}\\ \leq\|(\sum_{j\in{\mathbb{N}}}|M_{T_{-j-1}\rho}M_{[-j-1,-j]}f|^{2})^{1/2}\|_{p_{1}}\|(\sum_{j\in{\mathbb{N}}}|M_{[2^{-j-1},2^{-j})}g|^{2})^{1/2}\|_{p_{2}}.

The second factor is bounded by Cp2gp2C_{p_{2}}\|g\|_{p_{2}} by the following well-known square function estimate due to Rubio de Francia [27].

Theorem 2.6.

Let \mathcal{I} be a collection of pairwise disjoint intervals. Then, for 2<p<2<p<\infty, there is a constant CpC_{p} such that for all Schwartz functions ff we have

(I|MIf(x)|2)1/2pCpfp.\|(\sum_{I\in{\mathcal{I}}}|M_{I}f(x)|^{2})^{1/2}\|_{p}\leq C_{p}\|f\|_{p}.

To estimate the first factor, we note that the operator MTj1ρM_{T_{-j-1}\rho} is dominated pointwise by a constant multiple of the Hardy–Littlewood maximal function. Using the Fefferman–Stein maximal inequality [9] as well as Theorem 2.6, we can estimate

(j(M(Tj1ρM[j1,j]f)2)1/2p1Cρ,p1(j(M[(j+1),j]f)2)1/2p1Cρ,p1(j(M[(j+1),j]f)2)1/2p1Cρ,p1fp1.\|(\sum_{j}(M_{(T_{-j-1}\rho}M_{[-j-1,-j]}f)^{2})^{1/2}\|_{p_{1}}\\ \leq C_{\rho,p_{1}}\|(\sum_{j}(\mathcal{M}M_{[-(j+1),-j]}f)^{2})^{1/2}\|_{p_{1}}\\ \leq C_{\rho,p_{1}}\|(\sum_{j}(M_{[-(j+1),-j]}f)^{2})^{1/2}\|_{p_{1}}\leq C_{\rho,p_{1}}\|f\|_{p_{1}}.

It then remains to estimate Bm~B_{\tilde{m}}. We cut the intervals [2j1,2j)[2^{-j-1},2^{-j}) into two equally long halves. For β{0,1}\beta\in\{0,1\}, define

mβ(ξ,η)\displaystyle m_{\beta}(\xi,\eta) =kTkρ(ξ)j,j<k1[2j1,2j1+2j2)+β2j2(η)\displaystyle=\sum_{k\in{\mathbb{N}}}T_{-k}\rho(\xi)\sum_{j\in{\mathbb{N}},j<k}1_{[2^{-j-1},2^{-j-1}+2^{-j-2})+\beta 2^{-j-2}}(\eta) (2.10)

so that m~=m0+m1\tilde{m}=m_{0}+m_{1}. We estimate the corresponding two multiplier operators separately. Set

g~β=jM[2j1,2j1+2j2)+β2j2g,\tilde{g}_{\beta}=\sum_{j\in{\mathbb{N}}}M_{[2^{-j-1},2^{-j-1}+2^{-j-2})+\beta 2^{-j-2}}g, (2.11)

and observe

j,j<kM[2j1,2j1+2j2)+β2j2g=g~βϕβ,kg~β,\sum_{j\in{\mathbb{N}},j<k}M_{[2^{-j-1},2^{-j-1}+2^{-j-2})+\beta 2^{-j-2}}g=\tilde{g}_{\beta}-\phi_{\beta,k}*\tilde{g}_{\beta},

where ϕ^β,k=ϕ^β(2kx)\widehat{\phi}_{\beta,k}=\widehat{\phi}_{\beta}(2^{k}x) and ϕ^β\widehat{\phi}_{\beta} is a smooth function supported in

[1,1]+[β/4,β/4][-1,1]+[-\beta/4,\beta/4]

and constant one on

[3/4,3/4]+[β/4,β/4].[-3/4,3/4]+[-\beta/4,\beta/4].

Hence the multiplier operator corresponding to (2.10) becomes

k(MTkρf)g~βk(MTkρf)(ϕβ,kg~β).\sum_{k\in{\mathbb{N}}}(M_{T_{-k}\rho}f)\tilde{g}_{\beta}-\sum_{k\in{\mathbb{N}}}(M_{T_{-k}\rho}f)(\phi_{\beta,k}*\tilde{g}_{\beta}). (2.12)

The second term in (2.12) is bounded by

k(MTkρf)(ϕβ,kg~β)p3Cp1,p2fp1g~βp2Cp1,p2fp1gp2,\|\sum_{k\in{\mathbb{N}}}(M_{T_{-k}\rho}f)(\phi_{\beta,k}*\tilde{g}_{\beta})\|_{p_{3}^{\prime}}\leq C_{p_{1},p_{2}}\|f\|_{p_{1}}\|\tilde{g}_{\beta}\|_{p_{2}}\leq C_{p_{1},p_{2}}\|f\|_{p_{1}}\|g\|_{p_{2}},

the first inequality following from Lemma 2.4 and the second inequality being a consequence of the following Theorem 2.7 below by Coifman, Rubio de Francia and Semmes [4].

Theorem 2.7.

Let nn be a bounded function on the real line. If

supjn1[2j,2j+2)[2j,2j+2)V1A,\sup_{j\in{\mathbb{Z}}}\left\lVert n1_{[2^{j},2^{j+2})\cup[-2^{j},-2^{j+2})}\right\rVert_{V^{1}}\leq A,

then for each 1<p<1<p<\infty there is a constant CpC_{p} such that

nMpCpA.\|n\|_{M^{p}}\leq C_{p}A.

This is the sharp version of the Hörmander–Mikhlin multiplier theorem, a stronger version of Theorem 2.2 above. The assumption on the total variation is obviously satisfied by the multiplier in (2.11). It only jumps a uniformly bounded number of times in each of the test intervals, each of the jumps being of height one. Hence we have the desired bound for the second term in (2.12).

For the first term in (2.12), we have the bound

k(MTkρf)g~βp3k(MTkρf)p1g~p2Cρ,p1fp1gp2\|\sum_{k\in{\mathbb{N}}}(M_{T_{-k}\rho}f)\tilde{g}_{\beta}\|_{p_{3}^{\prime}}\leq\|\sum_{k\in{\mathbb{N}}}(M_{T_{-k}\rho}f)\|_{p_{1}}\|\tilde{g}\|_{p_{2}}\leq C_{\rho,p_{1}}\|f\|_{p_{1}}\|g\|_{p_{2}}

where we again used Theorem 2.7 for the second factor. The first factor is estimated by Theorem 2.2 as the corresponding multiplier is locally constant except for in [1,1][-1,1] where it is a smooth function of total variation one. ∎

Proof of Corollary 1.2.

We write the multiplier in the corollary as

m=j1[(j+1),j)(ξ)1[2j,1)(η)+jmj,m=\sum_{j\in{\mathbb{N}}}1_{[-(j+1),-j)}(\xi)1_{[2^{-j},1)}(\eta)+\sum_{j\in{\mathbb{N}}}m_{j}, (2.13)

where mjm_{j} is the characteristic function of the set

{(ξ,η):(j+1)ξj,2ξη<2j}.\{(\xi,\eta):-(j+1)\leq\xi\leq-j,2^{\xi}\leq\eta<2^{-j}\}. (2.14)

The bilinear multiplier corresponding to the first summand in (2.13) is bounded by Theorem 1.1.

To estimate the second summand, we pair the bilinear multiplier with an arbitrary function hLp3h\in L^{p_{3}} and write

jBmj(f,g)(x)h(x)dx=jBmj(M[(j+1),j)f,M[2(j+1),2j)g)(x)×M[j,(j+1))+[2j,2(j+1))h(x)dx.\int_{\mathbb{R}}\sum_{j\in{\mathbb{N}}}B_{m_{j}}(f,g)(x)h(x)\,dx\\ =\sum_{j\in{\mathbb{N}}}\int_{\mathbb{R}}B_{m_{j}}(M_{[-(j+1),-j)}f,M_{[2^{-(j+1)},2^{-j})}g)(x)\\ \times M_{[j,(j+1))+[-2^{-j},-2^{-(j+1)})}h(x)\,dx.

Here we have inserted an additional multiplier to restrict the frequency support of hh. This is possible as a bilinear multiplier applied to a pair of functions with frequency supports in the intervals II and JJ, results in a function that has frequency support in IJ-I-J.

We identify each piece BmjB_{m_{j}} with a multiplier of the form corresponding to the second case in the list of examples of convex sets in the introduction, that is, we notice the slope of the curved boundary line of (2.14) is bounded above and below by comparable numbers. Each BmjB_{m_{j}} is hence individually bounded, and we can estimate the display above by

jM[(j+1),j)fp1M[2(j+1),2j)gp2M[j,(j+1))+[2j,2(j+1))hp3\sum_{j\in{\mathbb{N}}}\|M_{[-(j+1),-j)}f\|_{p_{1}}\|M_{[2^{-(j+1)},2^{-j})}g\|_{p_{2}}\|M_{[j,(j+1))+[-2^{-j},-2^{-(j+1)})}h\|_{p_{3}}

Applying Hölder’s inequality, we estimate this by

(j|M[(j+1),j)f|p1)1/p1p1×(j|M[2(j+1),2j)g|p2)1/p2p2×(j|M[j,(j+1))+[2j,2(j+1))h|p3)1/p3p3.\|(\sum_{j\in{\mathbb{N}}}|M_{[-(j+1),-j)}f|^{p_{1}})^{1/p_{1}}\|_{p_{1}}\\ \times\|(\sum_{j\in{\mathbb{N}}}|M_{[2^{-(j+1)},2^{-j})}g|^{p_{2}})^{1/p_{2}}\|_{p_{2}}\\ \times\|(\sum_{j\in{\mathbb{N}}}|M_{[j,(j+1))+[-2^{-j},-2^{-(j+1)})}h|^{p_{3}})^{1/p_{3}}\|_{p_{3}}.

Because p1>2p_{1}>2 and the intervals [(j+1),j)[-(j+1),-j) are disjoint, we can use Theorem 2.6 and estimate the first factor by

(j|M[(j+1),j)f|p1)1/p1p1(j|M[(j+1),j)f|2)1/2p1Cp1fp1.\|(\sum_{j\in{\mathbb{N}}}|M_{[-(j+1),-j)}f|^{p_{1}})^{1/p_{1}}\|_{p_{1}}\leq\|(\sum_{j\in{\mathbb{N}}}|M_{[-(j+1),-j)}f|^{2})^{1/2}\|_{p_{1}}\\ \leq C_{p_{1}}\|f\|_{p_{1}}.

Similarly, we can estimate the factors corresponding to gg and hh. The intervals [2(j+1),2j)[2^{-(j+1)},2^{-j}) relevant to gg are also disjoint, and the intervals [j,(j+1))+[2j,2(j+1))[j,(j+1))+[-2^{-j},-2^{-(j+1)}) are disjoint for jj even or jj odd separately. This concludes the the desired bound for the second term in (2.13) and hence the proof of Corollary 1.2. ∎

3. Multi-lacunary paraproducts

In this section, we prove Theorem 1.4. We begin by noting the natural decomposition of multi-lacunary sets.

Lemma 3.1.

If XX is (d,b)(d,b)-lacunary, then there exists a partition

X=O0OdX=O_{0}\cup\dots\cup O_{d}

such that for every i<di<d the set

Xi:=O0OiX_{i}:=O_{0}\cup\dots\cup O_{i}

is (i,b)(i,b)-lacunary and any two points ξ,ξ\xi,\xi^{\prime} in Oi+1O_{i+1} satisfy

dist(ξ,ξ)2bdist(ξ,Xi).\operatorname{dist}(\xi,\xi^{\prime})\geq 2^{-b}\operatorname{dist}(\xi,X_{i}).
Proof.

We successively decompose the limit sets in the definition of (d,b)(d,b)-lacunarity. ∎

Reductions

Consider the assumptions of Theorem 1.4. We first break up the multiplier by decomposing the interval [2j,22j)[2^{-j},2^{2-j}) into four equally long intervals and intersecting these intervals with [ξj,ζj)[\xi_{j},\zeta_{j}). In other words, for integers 0β30\leq\beta\leq 3, we choose numbers ηj(β)\eta^{(\beta)}_{j} and ζj(β)\zeta^{(\beta)}_{j} such that

1[ηj,ζj)=m=031[ηj(β),ζj(β))1_{[\eta_{j},\zeta_{j})}=\sum_{m=0}^{3}1_{[\eta_{j}^{(\beta)},\zeta_{j}^{(\beta)})}

and

2j+β2jηj(β)ζj(β)2j+(β+1)2j2^{-j}+\beta 2^{-j}\leq\eta^{(\beta)}_{j}\leq\zeta^{(\beta)}_{j}\leq 2^{-j}+(\beta+1)2^{-j} (3.1)

Then the multiplier (1.6) breaks up as a sum of four analogous expressions and it suffices to show the desired bound for each of the summands. We fix β\beta and suppress the dependency from the superscript for the rest of the proof.

We also modify the sequence ξj\xi_{j}. For each jj\in{\mathbb{N}}, let ξj\xi_{j}^{\prime} be the largest integer multiple of 24j2^{4-j} smaller than or equal to ξj\xi_{j}. Thanks to (1.5), the sequence ξ\xi^{\prime} satisfies

0ξj+25jξj10\leq\xi^{\prime}_{j}+2^{5-j}\leq\xi^{\prime}_{j-1}

Moreover, distance between ξj\xi_{j}^{\prime} and ξj\xi_{j^{\prime}}^{\prime} for two indices jj and jj^{\prime} is comparable to the distance of ξj\xi_{j} and ξj\xi_{j^{\prime}} with upper and lower factor at most 22. Hence the sequence ξj\xi_{j}^{\prime} is (d,b2)(d,b-2)-lacunary. Moreover, the intervals [ξj,ξj)[\xi_{j}^{\prime},\xi_{j}) are pairwise disjoint.

Lemma 3.2.

The multiplier BmB_{m^{\prime}} with

m(ξ,η)=n1[ξj,ξj)(ξ)1[ηj,ζj)(η)m^{\prime}(\xi,\eta)=\sum_{n}1_{[\xi_{j}^{\prime},\xi_{j})}(\xi)1_{[\eta_{j},\zeta_{j})}(\eta)

satisfies (1.2) with constant depending only on p1,p2p_{1},p_{2}.

Proof.

Recall that we denote by MIM_{I} the linear Fourier multiplier for the interval II. We write

Bm(f,g)p3=j(M[ξj,ξj)f)(M[ηj,ζj)g)p3(j|M[ξj,ξj)f|2)1/2p1(j|M[ηj,ζj)g|)1/2p2Cp1,p2fp1gp2,\|B_{m^{\prime}}(f,g)\|_{p_{3}^{\prime}}=\|\sum_{j}(M_{[\xi_{j}^{\prime},\xi_{j})}f)(M_{[\eta_{j},\zeta_{j})}g)\|_{p_{3}^{\prime}}\\ \leq\|(\sum_{j}|M_{[\xi_{j}^{\prime},\xi_{j})}f|^{2})^{1/2}\|_{p_{1}}\|(\sum_{j}|M_{[\eta_{j},\zeta_{j})}g|)^{1/2}\|_{p_{2}}\\ \leq C_{p_{1},p_{2}}\|f\|_{p_{1}}\|g\|_{p_{2}},

where we applied Cauchy-Schwarz and Hölder to pass to the second line and Theorem 2.6 to bound the individual factors. This proves the lemma. ∎

Because of the lemma, it suffices to estimate the multiplier (1.6) with ξj\xi_{j} replaced by ξj\xi_{j}^{\prime}. We shall do so and omit the prime in the forthcoming notation. Let XX be the multi-lacunary image of the sequence ξj\xi_{j}. Let YY be the collection of dyadic intervals II such that 3I3I contains a point of XX. Let ZZ be the collection of maximal dyadic intervals II such that 3I3I does not contain any point of XX. By maximality, the intervals of ZZ are pairwise disjoint. Let YjY_{j} be the collection of all dyadic intervals in YY that have length 24j2^{4-j} and are contained in [0,ξj)[0,\xi_{j}). Let ZjZ_{j} be the collection of intervals in ZZ which are contained in [0,ξj)[0,\xi_{j}) but not in any interval of YjY_{j}.

Lemma 3.3.

The intervals in YjZjY_{j}\cup Z_{j} have length at least 24j2^{4-j} and partition [0,ξj)[0,\xi_{j}).

Proof.

The intervals in YjY_{j} have length 24j2^{4-j} by definition. Let II be an interval in ZjZ_{j} of length at most 24j2^{4-j}. Let JJ be the dyadic interval of length 24j2^{4-j} containing II. Then JJ is contained in [0,ξj)[0,\xi_{j}), because ξj\xi_{j} is an integer multiple of 24j2^{4-j} and JJ contains I[0,ξj)I\subset[0,\xi_{j}). The interval JJ is not in YjY_{j}, since II is not contained in any interval of YjY_{j}. Hence JJ must be contained in an interval JZJ^{\prime}\in Z. As the intervals in ZZ are pairwise disjoint, II cannot be strictly contained in JJ^{\prime}, and hence II has to be equal to JJ^{\prime}. Consequently, |I|=24j|I|=2^{4-j}. This proves the statement about the length of the intervals.

We turn to the claim about partitioning. The intervals in YjY_{j} are pairwise disjoint, because they are dyadic intervals of equal length. The intervals in ZjZ_{j} are pairwise disjoint, because they are maximal. By construction, no interval of ZjZ_{j} can be contained in any interval of YjY_{j}. Conversely, no interval in YjY_{j} can be contained in any interval of ZjZ_{j}. Indeed, 3I3I with IZjI\in Z_{j} contains no point of XX, but 3J3J for JYjJ\in Y_{j} does contain a point of XX. Hence JIJ\nsubseteq I. Hence the intervals in YjZjY_{j}\cup Z_{j} are all pairwise disjoint.

To prove that the intervals form a cover, let ξ\xi be any point in [0,ξj)[0,\xi_{j}). Let II be the dyadic interval of length 24j2^{4-j} containing ξ\xi. It is contained in [0,ξj)[0,\xi_{j}). If II is in YjY_{j}, then ξ\xi is covered by intervals in YjZjY_{j}\cup Z_{j}. If it is not in YjY_{j}, then it is contained in an interval JJ in ZZ. The interval JJ does not contain ξj\xi_{j} by definition of ZZ. Hence it is contained in [0,ξj)[0,\xi_{j}). It is not contained in any interval of YjY_{j}, and hence it is in ZjZ_{j}. Again, ξ\xi is covered by YjZjY_{j}\cup Z_{j}. This proves the partition statement. ∎

Using Lemma 3.3, we may split Bm(f,g)B_{m}(f,g) as

jIZj(MIf)(M(ηj,ζj)g)+jIYj(MIf)(M(ηj,ζj)g).\sum_{j}\sum_{I\in Z_{j}}(M_{I}f)(M_{(\eta_{j},\zeta_{j})}g)+\sum_{j}\sum_{I\in Y_{j}}(M_{I}f)(M_{(\eta_{j},\zeta_{j})}g). (3.2)

It suffices to estimate the terms separately. Recall that we assume each ξj\xi_{j} to be an integer multiple of 24j2^{4-j} and each (ηj,ζj)(\eta_{j},\zeta_{j}) to actually be (ηj(β),ζj(β))(\eta_{j}^{(\beta)},\zeta_{j}^{(\beta)}) as defined in (3.1).

First term

Regrouping the sum in the first term and pairing with a dualizing function, the quantity to be estimated becomes

IZj:IZj(MIf)(M(ηj,ζj)g)(x)h(x)dx=IZ(MIf)[j:IZj(M(ηj,ζj)g)(x)]MI[0,22|I|]h(x)dx.\int\sum_{I\in Z}\sum_{j:I\in Z_{j}}(M_{I}f)(M_{(\eta_{j},\zeta_{j})}g)(x)h(x)\,dx\\ =\int\sum_{I\in Z}(M_{I}f)\left[\sum_{j:I\in Z_{j}}(M_{(\eta_{j},\zeta_{j})}g)(x)\right]M_{-I-[0,2^{-2}|I|]}h(x)\,dx. (3.3)

Here we used that the Fourier support of the product of MIfM_{I}f and M(ηj,ζj)gM_{(\eta_{j},\zeta_{j})}g is contained in

I+(ηj,ζj)I+[0,22j)I+[0,22|I|),I+(\eta_{j},\zeta_{j})\subset I+[0,2^{2-j})\subset I+[0,2^{-2}|I|),

the first inclusion following from (3.1) and the second one from Lemma 3.3. We may therefore apply the adjoint of the Fourier restriction to this interval to the dualizing function hh without changing the value of the duality pairing.

The intervals I+[0,22|I|)I+[0,2^{-2}|I|) have bounded overlap as II runs through ZZ. To see this, partition each such interval as disjoint union of II and JIJ_{I}, where JIJ_{I} is the dyadic interval to the right of II and has length 22|I|2^{-2}|I|. Then 3JI3J_{I} is contained in 3I3I and thus does not contain any point of XX. Hence JIJ_{I} is contained in an interval of ZZ with which it shares the left endpoint. As the intervals in ZZ are pairwise disjoint and each of them can only contain one interval JIJ_{I} as above, the intervals JIJ_{I} are pairwise disjoint.

Consider an interval II of ZZ and assume it has nonzero contribution to (3.3). Then there are integers jI<jIj_{I}<j^{I} such that II is contained in [0,ξj)[0,\xi_{j}) precisely if j<jIj<j^{I}, and it is not contained in any interval of YjY_{j} precisely if j>jIj>j_{I}. Hence we can notice

j:IZjM1(ηj,ζj)g=jI<j<jIM1(ηj,ζj)g=g~(ϕjIϕjI)\sum_{j:I\in Z_{j}}M_{1_{(\eta_{j},\zeta_{j})}}g=\sum_{j_{I}<j<j^{I}}M_{1_{(\eta_{j},\zeta_{j})}}g=\tilde{g}*(\phi_{j_{I}}-\phi_{j^{I}}) (3.4)

where

g~=jM1(ηj,ζj)g,ϕ^j(x)=ϕ^(2jx)\tilde{g}=\sum_{j\in{\mathbb{N}}}M_{1_{(\eta_{j},\zeta_{j})}}g,\quad\widehat{\phi}_{j}(x)=\widehat{\phi}(2^{j}x)

and ϕ\phi is a Schwartz function whose Fourier transform is supported on [β,β+3][\beta,\beta+3] and is constant one on [β+1,β+2][\beta+1,\beta+2]. Here we have used (3.1).

The expression in (3.4) is bounded pointwise by a constant times the Hardy–Littlewood maximal function g~{\mathcal{M}}\tilde{g} of g~\tilde{g}. Hence we estimate (3.3) by

C(IZ|M1If(x)|2)1/2(g~)(x)(IZ|MI[0,22|I|]h(x)|2)1/2𝑑x(IZ|MIf|2)1/2p1g~p2(IZ|MI[0,22|I|]h|2)1/2p3.C\int(\sum_{I\in Z}|M_{1_{I}}f(x)|^{2})^{1/2}({\mathcal{M}}\tilde{g})(x)(\sum_{I\in Z}|M_{-I-[0,2^{-2}|I|]}h(x)|^{2})^{1/2}\,dx\\ \leq\|(\sum_{I\in Z}|M_{I}f|^{2})^{1/2}\|_{p_{1}}\|\mathcal{M}\tilde{g}\|_{p_{2}}\|(\sum_{I\in Z}|M_{-I-[0,2^{-2}|I|]}h|^{2})^{1/2}\|_{p_{3}}.

With Rubio de Francia’s square function inequality from Theorem 2.6, using the bounded overlap of I+[0,22|I|]I+[0,2^{-2}|I|], we estimate the last display by

Cp1,p2fp1g~p2hp3Cp1,p2fp1gp2hp3C_{p_{1},p_{2}}\|f\|_{p_{1}}\|\tilde{g}\|_{p_{2}}\|h\|_{p_{3}}\leq C_{p_{1},p_{2}}\|f\|_{p_{1}}\|{g}\|_{p_{2}}\|h\|_{p_{3}}

In the last inequality, we have also used Theorem 2.7. The assumption on the total variation is obviously satisfied as the multiplier only jumps a bounded number of times by one in each of the test intervals. This completes the bound for the first term in (3.2).

Second term

We turn to the second term in (3.2). We decompose YY as the union

Y=iY(i)Y=\bigcup_{i}Y^{(i)}

where Y(i)Y^{(i)} contains the intervals II of YY such that 3I3I contains a point of O(i)O^{(i)} but no point of any O(i)O^{(i^{\prime})} with i<ii^{\prime}<i. As the parameter ii only ranges form 0 to dd, it suffices to consider the Y(i)Y^{(i)} separately and prove a bound for

jIYjY(i)(MIf)(M(ηj,ζj)g).\sum_{j}\sum_{I\in Y_{j}\cap Y^{(i)}}(M_{I}f)(M_{(\eta_{j},\zeta_{j})}g). (3.5)

Let W(i)W^{(i)} be the maximal intervals in Y(i)Y^{(i)}. In the next lemma, we single out two facts that we will need later. The proof is similar, but not identical, to the argument that proved an analogous statement for ZZ.

Lemma 3.4.

Let IW(i)I\in W^{(i)}. If there is JW(i)J\in W^{(i)} with

J(I+[0,22|I|)),J\cap(I+[0,2^{-2}|I|))\neq\varnothing,

then |J|22|I||J|\geq 2^{-2}|I|. In particular, the intervals I+[0,22|I|)I+[0,2^{-2}|I|) with IW(i)I\in W^{(i)} have bounded overlap.

Proof.

Take IW(i)I\in W^{(i)} and write I+[0,22|I|]=IJII+[0,2^{-2}|I|]=I\cup J_{I} as a disjoint union. Assume JIJ_{I} intersects another interval JW(i)J\in W^{(i)}. It suffices to show that JIJ_{I} is contained in JJ. Suppose JJ is contained in JIJ_{I}. Then 3JI3J_{I} contains 3J3J and thus a point from O(i)O^{(i)}. On the other hand, 3JI3J_{I} is contained in 3I3I and thus does not contain any point from any O(i)O^{(i^{\prime})} with i<ii^{\prime}<i. Hence JIJ_{I} is contained in an interval of W(i)W^{(i)}. As the intervals of W(i)W^{(i)} are pairwise disjoint, this interval must be JJ and hence JJ is equal to JIJ_{I}. In particular, a right neighbor of IW(i)I\in W^{(i)} has at least one quarter of the length of II. ∎

To estimate (3.5), we sort the intervals by their containment in maximal intervals, pair with a dualizing function, and realize the restriction of the Fourier support of the dualizing function with a multiplier as

jJW(i)IJIYjY(i)MIf(x)M[ηj,ζj)g(x)MJ[0,22|J|)(h)(x)dx.\int\sum_{j}\sum_{J\in W^{(i)}}\sum_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\end{subarray}}M_{I}f(x)M_{[\eta_{j},\zeta_{j})}g(x)M_{-J-[0,2^{-2}|J|)}(h)(x)dx. (3.6)

We break the innermost sum up by considering separately the cases |I|>2b4|J||I|>2^{-b-4}|J| and |I|2b4|J||I|\leq 2^{-b-4}|J|.

The sum with |I|>2b4|J||I|>2^{-b-4}|J| is estimated by

(jJW(i)(IJIYjY(i)|I|>2b4|J|MIf)2)1/2p1×(j|M(ηj,ζj)g|2)1/2p2(JW(i)|MJ+21|J|h|2)1/2p3.\|(\sum_{j}\sum_{J\in W^{(i)}}(\sum_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\\ |I|>2^{-b-4}|J|\end{subarray}}M_{I}f)^{2})^{1/2}\|_{p_{1}}\\ \times\|(\sum_{j}|M_{(\eta_{j},\zeta_{j})}g|^{2})^{1/2}\|_{p_{2}}\|(\sum_{J\in W^{(i)}}|M_{J+2^{-1}|J|}h|^{2})^{1/2}\|_{p_{3}}.

The three factors are estimated by Theorem 2.6. It is clear that the intervals (ηj,ζj)(\eta_{j},\zeta_{j}) are pairwise disjoint. By Lemma 3.4, J+21|J|J+2^{-1}|J| have bounded overlap. For the first factor, we use the disjointness of the various intervals in YjY_{j} to write it as

(JW(i)k=0b+3IJIY(i)|I|=2k|J||M1If|2)1/2p1\|(\sum_{J\in W^{(i)}}\sum_{k=0}^{b+3}\sum_{\begin{subarray}{c}I\subset J\\ I\in Y^{(i)}\\ |I|=2^{-k}|J|\end{subarray}}|M_{1_{I}}f|^{2})^{1/2}\|_{p_{1}}

As the various intervals JJ are disjoint, we obtain the estimate

Cp1(b+4)fp1C_{p_{1}}(b+4)\|f\|_{p_{1}}

by Theorem 2.6. This completes the bound of the sum over |I|>2b4|J||I|>2^{-b-4}|J| in (3.6).

We turn to the sum in (3.6) over |I|2b4|J||I|\leq 2^{-b-4}|J|. Let V(i)V^{(i)} be the collection of dyadic intervals II such that there exists JW(i)J\in W^{(i)} with IJI\subset J and 2b+4|I|=|J|2^{b+4}|I|=|J| and at least one III^{\prime}\subset I with IYjY(i)I^{\prime}\in Y_{j}\cap Y^{(i)}. Given a pair of neighboring intervals in W(i)W^{(i)}, the right interval is at least one quarter as wide as left interval by Lemma 3.4. The same property is inherited by the refined family V(i)V^{(i)}. Accordingly, it suffices to estimate

jJV(i)IJIYjY(i)MIf(x)M[ηj,ζj)g(x)MJ[0,22|J|)h(x)dx,\int\sum_{j}\sum_{J\in V^{(i)}}\sum_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\end{subarray}}M_{I}f(x)M_{[\eta_{j},\zeta_{j})}g(x)M_{-J-[0,2^{-2}|J|)}h(x)dx, (3.7)

where the Fourier support of hh is now given in terms of an interval in V(i)V^{(i)}.

For each JV(i)J\in V^{(i)}, there is exactly one point of O(i)O^{(i)} contained in 7J7J. Existence of at least one such a point follows by our requirement that JJ contains an interval from YjY(i)Y_{j}\cap Y^{(i)}. To show that there cannot be more than one, suppose there were two such points ξ,ξ\xi,\xi^{\prime}. By the multi-lacunarity, there would then exist a point ξ′′\xi^{\prime\prime} of O(i)O^{(i^{\prime})} with i<ii^{\prime}<i and

|ξ′′ξ|2b|ξξ|72b|J|716|I||\xi^{\prime\prime}-\xi|\leq 2^{b}|\xi-\xi^{\prime}|\leq 7\cdot 2^{b}|J|\leq\frac{7}{16}|I^{\prime}|

where II^{\prime} is the interval in W(i)W^{(i)} with IJI^{\prime}\supset J. This would imply ξ′′ 3I\xi^{\prime\prime} \in 3I^{\prime}, which contradicts the fact IY(i)I^{\prime}\in Y^{(i)}. Hence there is at most one point ξJ3JO(i)\xi_{J}\in 3J\cap O^{(i)}, and this point is the unique point from O(i)O^{(i)} withing the whole 7J7J.

Let ϕ\phi be a Schwartz function whose Fourier transform is supported on [28,28][-2^{-8},2^{-8}] and equal to one in [29,29][-2^{-9},2^{-9}]. Write ϕ^ξ,j(η)=ϕ^(2j(ηξ))\widehat{\phi}_{\xi,j}(\eta)=\widehat{\phi}(2^{j}(\eta-\xi)). We compare (3.7) with

jJV(i)(ϕξJ,jf)(x)M(ηj,ζj)g(x)MJ[0,22|J|)(h)(x)dx,\int\sum_{j}\sum_{J\in V^{(i)}}(\phi_{\xi_{J},j}*f)(x)M_{(\eta_{j},\zeta_{j})}g(x)M_{-J-[0,2^{-2}|J|)}(h)(x)dx, (3.8)

where we have replaced MIM_{I} by a convolution with ϕξJ,j\phi_{\xi_{J},j}, and the sum over II has been removed. We first prove bounds on (3.8), and then we show how to pass from the original expression to (3.8).

We note that we may now further restrict the Fourier transform of hh to the interval

IJ,j=[ξJ28j,ξJ+28j][ηj,ζj).I_{J,j}=-[\xi_{J}-2^{-8-j},\xi_{J}+2^{-8-j}]-[\eta_{j},\zeta_{j}).

Because ξJ\xi_{J} is a point unique to 7J7J, because the intervals JJ are disjoint, and because the sequences [ηj,ζj)[\eta_{j},\zeta_{j}) are subject to the condition (1.4), we see that the intervals IJ,jI_{J,j} form a disjoint family as JJ varies and disjoint and lacunary family as jj varies. We thus estimate (3.8) by

(JV(i)supj|ϕξJ,j(MJf)|2)1/2p1(j|M(ηj,ζj)g|2)1/2p2×(JV(i)j|MIJ,jh|2)1/2p3\|(\sum_{J\in V^{(i)}}\sup_{j}|\phi_{\xi_{J},j}*(M_{J}f)|^{2})^{1/2}\|_{p_{1}}\|(\sum_{j}|M_{(\eta_{j},\zeta_{j})}g|^{2})^{1/2}\|_{p_{2}}\\ \times\|(\sum_{J\in V^{(i)}}\sum_{j}|M_{I_{J,j}}h|^{2})^{1/2}\|_{p_{3}}

The last two factors we estimate with Theorem 2.6. In the first factor, we bound the convolution product by the Hardy–Littlewood maximal function and apply the Fefferman–Stein maximal inequality as well as Theorem 2.6. This concludes the proof of the bound for (3.8).

It remains to bound the difference of (3.7) and (3.8). Define

wj,J=(IJIYjY(i)1I)ϕξJ,j^1J.w_{j,J}=\left(\sum_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\end{subarray}}1_{I}\right)-\widehat{\phi_{\xi_{J},j}}1_{J}.

Let ψ~j,J\tilde{\psi}_{j,J} be a Schwartz function equal to one in [ξJ25j,ξJ+25j][\xi_{J}-2^{5-j},\xi_{J}+2^{5-j}] and zero outside [ξJ26j,ξJ+26j][\xi_{J}-2^{6-j},\xi_{J}+2^{6-j}]. Defining ψj,J=ψ~j,JϕξJ,j^\psi_{j,J}=\tilde{\psi}_{j,J}-\widehat{\phi_{\xi_{J},j}}, we then see that ψj,J\psi_{j,J} is a Schwartz function supported in [ξJ26j,ξJ+26j][\xi_{J}-2^{6-j},\xi_{J}+2^{6-j}], vanishing in [29j,29j][-2^{-9-j},2^{-9-j}] and satisfying

wj,J=1Jψj,J(IJIYjY(i)1I(ξJ29j,ξJ+29j)).w_{j,J}=1_{J}\psi_{j,J}\left(\sum_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\end{subarray}}1_{I\setminus(\xi_{J}-2^{-9-j},\xi_{J}+2^{-9-j})}\right). (3.9)

Indeed, ξJ\xi_{J} is the unique point of iiO(i)\bigcup_{i^{\prime}\leq i}O^{(i)} in 7J7J. If there is IYjY(i)I\in Y_{j}\cap Y^{(i)} with IJI\subset J and IJξJ\partial I\setminus\partial J\ni\xi_{J}, then II^{\prime} with |I|=|I||I^{\prime}|=|I| and II={ξJ}I^{\prime}\cap I=\{\xi_{J}\} is also in YjY(i)Y_{j}\cap Y^{(i)} and contained in JJ. Hence

ξJ(IJIYjY(i)I)J.\xi_{J}\notin\partial\left(\bigcup_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\end{subarray}}I\right)\setminus\partial J.

On the other hand, the union above is contained in [ξJ25j,ξJ+25j][\xi_{J}-2^{5-j},\xi_{J}+2^{5-j}], and so the equation (3.9) is justified.

As we know the bounds for (3.8), it suffices to bound

(JV(i)j|IJIYjY(i)MIMJfϕξJ,j(MJf)|2)1/2p1×(j|M(ηj,ζj)g|2)1/2p2(JV(i)|M(J[0,22|J|))h|2)1/2p3.\|(\sum_{J\in V^{(i)}}\sum_{j}|\sum_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\end{subarray}}M_{I}M_{J}f-\phi_{\xi_{J},j}*(M_{J}f)|^{2})^{1/2}\|_{p_{1}}\\ \times\|(\sum_{j}|M_{(\eta_{j},\zeta_{j})}g|^{2})^{1/2}\|_{p_{2}}\|(\sum_{J\in V^{(i)}}|M_{(-J-[0,2^{-2}|J|))}h|^{2})^{1/2}\|_{p_{3}}.

Here the last two factors are readily estimated by Theorem 2.6. We focus on the first factor. By the observation (3.9), we can bound the multiplier operator associated with frequency symbol ψj,J\psi_{j,J} by the Hardy–Littlewood maximal function and apply the Fefferman–Stein inequality to control the first factor by

(JV(i)j|IJIYjY(i)MI(ξJ29j,ξJ+29j)MJf|2)1/2p1.\|(\sum_{J\in V^{(i)}}\sum_{j}|\sum_{\begin{subarray}{c}I\subset J\\ I\in Y_{j}\cap Y^{(i)}\end{subarray}}M_{I\setminus(\xi_{J}-2^{-9-j},\xi_{J}+2^{-9-j})}M_{J}f|^{2})^{1/2}\|_{p_{1}}.

However, the intervals above have bounded overlap, and hence we are in a position to apply Theorem 2.6 for one last time. This completes the proof of Theorem 1.4.

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