A Remark on Bilinear Square Functions

Abstract

We provide some remarks concerning a bilinear square function formed by products of Littlewood–Paley operators over arbitrary intervals. For 1 < p ₁, p ₂ < ∞ with 1∕p = 1∕p ₁ + 1∕p ₂, we show that this square function is bounded from $L^{p_{1}}(\mathbf{R}) \times L^{p_{2}}(\mathbf{R})$ to L ^p(R) when p > 2∕3 and unbounded when p < 2∕3.

Little work is known in the area of bilinear Littlewood–Paley square functions besides the articles of Lacey [6], Diestel [3], and Bernicot [1]. In this note, we study a bilinear square function formed by products of Littlewood–Paley operators over arbitrary intervals.

Given an interval I = [a, b) on R, let $\Delta _{I}$ be the Littlewood–Paley operator defined by multiplication by the characteristic function of I on the Fourier transform side. The Fourier transform of an integrable function g on R is defined by

$$\displaystyle{\hat{g}(\xi ) =\int _{\mathbf{R}}g(x)e^{-2\pi ix\xi }\,dx}$$

and its inverse Fourier transform is defined by $g^{\vee }(\xi ) =\hat{ g}(-\xi )$. In terms of these operators we have $\Delta _{I}(g) =\big (\hat{g}\chi _{I}\big)^{\vee }$.

The Littlewood–Paley square function associated with the function f on R is given by

$$\displaystyle{ S(f) =\Big (\sum _{j\in \mathbf{Z}}\vert \Delta _{I_{j}}(f)\vert ^{2}\Big)^{\frac{1} {2} }\,, }$$

(1)

where I _j  = [−2^j+1, −2^j) ∪ [2^j, 2^j+1) and the classical Littlewood–Paley theorem says that

$$\displaystyle{\big\|S(f)\big\|_{L^{p}(\mathbf{R})} \leq C_{p}\|f\|_{L^{p}(\mathbf{R})}}$$

where 1 < p < ∞ and C _p is a constant independent of the function f in L ^p(R) (but depends on p).

In this note, we are interested in estimates for Littlewood–Paley square functions formed by products of Littlewood–Paley operators acting on two functions. To be precise, let a _j and b _j be strictly increasing sequences on the real line with the properties lim _{j → ∞} a _j  = lim _{j → ∞} b _j  = ∞ and lim _{j → −∞} a _j  = lim _{j → −∞} b _j  = −∞ and consider the bilinear square function

$$\displaystyle{S_{2}(f,g) =\Big (\sum _{j\in \mathbf{Z}}\vert \Delta _{[a_{j},a_{j+1})}(f)\Delta _{[b_{j},b_{j+1})}(g)\vert ^{2}\Big)^{\frac{1} {2} }}$$

defined for suitable functions f, g on the line. We consider the question whether S ₂ satisfies the inequality

$$\displaystyle{ \big\|S_{2}(f,g)\big\|_{L^{p}(\mathbf{R})} \leq C_{p_{1},p_{2}}\|f\|_{L^{p_{1}}(\mathbf{R})}\|g\|_{L^{p_{2}}(\mathbf{R})} }$$

(2)

for some constant $C_{p_{1},p_{2}}$ independent of f, g where 1 < p ₁, p ₂ < ∞ and 1∕p = 1∕p ₁ + 1∕p ₂. We have the following result concerning this operator:

FormalPara Theorem 1.

Let 1 < p ₁ ,p ₂ < ∞ be given and define p by setting 1∕p = 1∕p ₁ + 1∕p ₂ . Then if p > 2∕3, there is a constant $C_{p_{1},p_{2}}$ such that (2) holds for all functions f,g on the line. Conversely, if (2) holds, then we must have p ≥ 2∕3.

FormalPara Proof.

Introduce the maximal function

$$\displaystyle{\mathcal{M}(f) =\sup _{-\infty <a<b<\infty }\vert \Delta _{[a,b)}(f)\vert }$$

and notice that is pointwise controlled by

$$\displaystyle{2\sup _{a\in \mathbf{R}}\vert \Delta _{(-\infty,a)}(f)\vert }$$

and thus is controlled by the following version of the Carleson operator

$$\displaystyle{\mathcal{C}(f)(x) =\sup _{N>0}\bigg\vert \int _{-\infty }^{N}\hat{f}(\xi )e^{2\pi ix\xi }d\xi \bigg\vert \,.}$$

In view of the Carleson–Hunt theorem [2, 5] we have that $\mathcal{C}$ is bounded on L ^r(R) for 1 < r < ∞.

Consider the case where 2 ≤ p ₁ < ∞ and 1 < p ₂ < ∞. Then we have that

$$\displaystyle{S_{2}(f,g) \leq \Big (\sum _{j\in \mathbf{Z}}\vert \Delta _{[a_{j},a_{j+1})}(f)\vert ^{2}\Big)^{\frac{1} {2} }\sup _{j\in \mathbf{Z}}\vert \Delta _{[b_{ j},b_{j+1})}(g)\vert = S(f)\mathcal{M}(g)}$$

where S is defined as in (1) with [a _j , a _j+1) in place of I _j . In view of the Rubio de Francia inequality [7] we have that S is bounded on L ^r(R) for 2 ≤ r < ∞. An application of Hölder’s inequality yields the inequality

$$\displaystyle{ \big\|S_{2}(f,g)\big\|_{L^{p}(\mathbf{R})} \leq \| S(f)\|_{L^{p_{1}}(\mathbf{R})}\|\mathcal{M}(g)\|_{L^{p_{2}}(\mathbf{R})} }$$

(3)

and this (2) follows from the preceding inequality combined with the boundedness of S on $L^{p_{1}}(\mathbf{R})$ and $\mathcal{M}$ on $L^{p_{2}}(\mathbf{R})$.

An analogous argument holds with the roles of p ₁ and p ₂ are reversed, i.e., when we have 1 < p ₁ < ∞ and 2 ≤ p ₂ < ∞. Thus boundedness holds for all pairs (p ₁, p ₂) for which either p ₁ ≥ 2 or p ₂ ≥ 2. But there exist points (p ₁, p ₂) with p = (1∕p ₁ + 1∕p ₂)⁻¹ > 2∕3 for which neither p ₁ nor p ₂ is at least 2. (For instance, p ₁ = p ₂ = 7∕5). To deal with these intermediate points we use interpolation.

Given a pair of points (p ₁, p ₂) with p = (1∕p ₁ + 1∕p ₂)⁻¹ > 2∕3 and 1 < p ₁, p ₂ < 2, we pick two pairs of points (p ₁ ¹, p ₂ ¹) and (p ₁ ², p ₂ ²) with

$$\displaystyle{p> p^{1} = (1/p_{ 1}^{1} + 1/p_{ 2}^{1})^{-1} = p^{2} = (1/p_{ 1}^{2} + 1/p_{ 2}^{2})^{-1}> 2/3}$$

and 1 < p ₂ ¹ < 2 < p ₁ ¹ < ∞, < 2 and 1 < p ₂ ² < 2 < p ₁ ² < ∞. For instance, we take (p ₁ ¹, p ₂ ¹, p ¹) near (1, 2, 2∕3) and (p ₁ ², p ₂ ², p ²) near (2, 1, 2∕3). Then consider the three points W ₁ = (1∕p ₁ ¹, 1∕p ₂ ¹, 1∕p ¹), W ₂ = (1∕p ₁ ², 1∕p ₂ ², 1∕p ²), and W ₃ = (1∕2, 1∕2, 1) and notice that the point (1∕p ₁, 1∕p ₂, 1∕p) lies in the interior of the convex hull of W ₁, W ₂, and W ₃. We consider the bi-sublinear operator

$$\displaystyle{(f,g)\mapsto S_{2}(f,g)}$$

which is bounded at the points W ₁, W ₂, and W ₃. Using Corollary 7.2.4 in [4] we obtain that S ₂ is bounded from $L^{p_{1}}(\mathbf{R}) \times L^{p_{2}}(\mathbf{R})$ to L ^p(R). This completes the proof in the remaining case.

Next, we turn to the converse assertion of the theorem. Suppose that for some 1 < p ₁, p ₂ < ∞ with 1∕p = 1∕p ₁ + 1∕p ₂ estimate (2) holds for some constant $C_{p_{1},p_{2}}$ and all suitable functions f, g on the line. Now consider the sequences a _j  = b _j  = j and the functions

$$\displaystyle{f_{N} = g_{N} =\chi _{[0,N)}^{\vee }\,.}$$

Then we have

$$\displaystyle{f_{N}(x) =\chi _{[0,N]}^{\vee }(x) =\int _{ 0}^{N}e^{2\pi ix\xi }d\xi = \frac{e^{2\pi iNx} - 1} {2\pi ix} }$$

and for j = 0, 1, …, N − 1 we have

$$\displaystyle{\Delta _{[j,j+1)}(f_{N})(x) =\int _{ j}^{j+1}\!\!e^{2\pi ix\xi }d\xi = e^{2\pi ixj}\int _{ 0}^{1}\!\!e^{2\pi ix\xi }d\xi = \frac{e^{2\pi ixj}(e^{2\pi ix} - 1)} {2\pi ix} \,.}$$

Consequently,

$$\displaystyle{\Big(\sum _{j=0}^{N-1}\big\vert \Delta _{ [j,j+1)}(f_{N})(x)\Delta _{[j,j+1)}(g_{N})(x)\big\vert ^{2}\Big)^{\frac{1} {2} } = \sqrt{N}\,\,\bigg\vert \frac{e^{2\pi ix} - 1} {2\pi ix} \bigg\vert ^{2}}$$

and thus

$$\displaystyle{\big\|S_{2}(f_{N},g_{N})\big\|_{L^{p}} \geq \sqrt{N}\,\bigg\|\frac{(e^{2\pi ix} - 1)^{2}} {4\pi ^{2}x^{2}} \bigg\|_{L^{p}} = c\,\sqrt{N}}$$

as long as p > 1∕2. On the other hand we have

$$\displaystyle{\|f_{N}\|_{L^{p_{1}}} = N^{1- \frac{1} {p_{1}} }\bigg\| \frac{e^{2\pi ix} - 1} {2\pi ix} \bigg\|_{L^{p_{1}}} = c_{p_{1}}\,N^{1- \frac{1} {p_{1}} }}$$

whenever 1 < p ₁ < ∞.

Now suppose that (2) holds. Then we must have

$$\displaystyle{ \big\|S_{2}(f_{N},g_{N})\big\|_{L^{p}(\mathbf{R})} \leq C_{p_{1},p_{2}}\|f_{N}\|_{L^{p_{1}}(\mathbf{R})}\|g_{N}\|_{L^{p_{2}}(\mathbf{R})} }$$

(4)

and this implies that

$$\displaystyle{c\,\sqrt{N} \leq C_{p_{1},p_{2}}c_{p_{1}}N^{1- \frac{1} {p_{1}} }c_{p_{ 2}}N^{1- \frac{1} {p_{2}} } = C_{p_{ 1},p_{2}}c_{p_{1}}c_{p_{2}}N^{2-\frac{1} {p} }}$$

which forces p ≥ 2∕3 by letting N → ∞. □ 

It is unclear to us at the moment as to what happens when p = 2∕3.

We now discuss a related larger square function. Let 1 < p ₁, p ₂ < ∞ with 1∕p ₁ + 1∕p ₂ = 1∕p. It is not hard to see that the square function

$$\displaystyle{S_{22}(f,g) =\Big (\sum _{j\in \mathbf{Z}}\sum _{k\in \mathbf{Z}}\vert \Delta _{[a_{j},a_{j+1})}(f)\Delta _{[b_{k},b_{k+1})}(g)\vert ^{2}\Big)^{\frac{1} {2} }}$$

is bounded from $L^{p_{1}}(\mathbf{R}) \times L^{p_{2}}(\mathbf{R})$ to L ^p(R) if and only if p ₁, p ₂ ≥ 2. Indeed, one direction is a trivial consequence of Hölder’s inequality; for the other direction, let

$$\displaystyle{f_{N}(x) = g_{N}(x) =\chi _{[0,N]}^{\vee }(x) =\int _{ 0}^{N}e^{2\pi ix\xi }\,d\xi = \frac{e^{2\pi iNx} - 1} {2\pi ix} \,.}$$

The preceding argument shows that

$$\displaystyle{\big\|S_{22}(f_{M},g_{N})\big\|_{L^{p}} \geq c^{2}\sqrt{M}\,\sqrt{N}}$$

and we also have

$$\displaystyle{\|f_{M}\|_{L^{p_{1}}(\mathbf{R})}\|g_{N}\|_{L^{p_{2}}(\mathbf{R})} = c_{p_{1}}c_{p_{2}}M^{1- \frac{1} {p_{1}} }N^{1- \frac{1} {p_{2}} }\,.}$$

Hence, letting M → ∞ with N fixed or N → ∞ with M fixed, we obtain that both p ₁ and p ₂ satisfy p ₁, p ₂ ≥ 2.

I would like to end this note by expressing a few feelings about Cora Sadosky. Although, I have not had a very close personal relationship with her, I have always admired the great dedication and enthusiasm Cora has displayed in mathematics and the sincere love and support she has provided to young people who wished to pursue a research career in harmonic analysis. I warmly recall the personal interest she showed in my search for a permanent position in the USA. Cora’s untimely passing away was a big loss for our harmonic analysis community and we are all proud of the strong legacy she has left behind.