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The Boundedness of the (Sub)bilinear Maximal Function Along “Non-flat” Smooth Curves

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Abstract

Let \(\mathcal {NF}\) be the class of smooth non-flat curves near the origin and near infinity introduced in Lie (Am J Math 137(2):313–363, 2015) and let \(\gamma \in \mathcal {NF}\). We show—via a unifying approach relative to the corresponding bilinear Hilbert transform \(H_{\Gamma }\)—that the (sub)bilinear maximal function along curves \(\Gamma =(t,-\gamma (t))\) defined as

$$\begin{aligned} M_\Gamma (f,g)(x):=\sup \limits _{\varepsilon >0} \frac{1}{2\varepsilon } \int _{-\varepsilon }^\varepsilon |f(x-t)g(x+\gamma (t))|dt \end{aligned}$$

is bounded from \(L^p(\mathbb {R})\times L^{q}(\mathbb {R})\rightarrow L^r(\mathbb {R})\) for all p, q and r Hölder indices, i.e. \(\frac{1}{p}+\frac{1}{q}=\frac{1}{r}\), with \(1<p,\,q\le \infty \) and \(1\le r\le \infty \). This is the maximal boundedness range for \(M_{\Gamma }\), that is, our result is sharp.

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Notes

  1. Throughout the paper we allow this slight abuse by referring to our maximal operator as “bilinear”, though of course, strictly speaking, this is a sub-linear operator in each of the two inputs.

  2. Our bibliography here is not exhaustive, listing only the references that are closest and most relevant to our concise study.

  3. It is still an open problem if this range is optimal.

  4. I.e., any expression of the form \(\gamma (t):=\sum _{j=1}^d a_j t^{\alpha _j}\) with \(\{a_j\}_{j=1}^{d}\subset \mathbb {R}\), \(\{\alpha _j\}_{j=1}^{d}\subset \mathbb {R}\setminus \{1\}\) and \(d\in \mathbb {N}\). Here, we use the following convention: if \(\alpha ,\,t\in \mathbb {R}\) we let \(t^{\alpha }\) stand for either \(|t|^{\alpha }\) or \(\text {sgn}\, (t)\, |t|^{\alpha }\). In a follow up paper [9], the first author extends the present result (and its singular integral analogue in [17] and [18]) to the case in which one allows the curve \(\gamma \) to be a generalized polynomial but with the linear term included.

  5. It is easy to notice that our main result extends to the class of curves \(\mathcal {NF}^{C}\) that is defined to be the set of all curves \(\gamma \,+\,C\) with \(\gamma \in \mathcal {NF}\) and \(C\in \mathbb {R}\), i.e., our result is invariant under translation by constants of \(\gamma \) with \(\gamma \in \mathcal {NF}\).

  6. Observe that one gets trivially the desired bound for the triple of indices \((p,q,r)=(\infty ,\infty ,\infty )\) corresponding to the point C in Fig. 1. That is why we will exclude this case from all our future reasonings.

  7. Same type of decomposition holds for \(H_j\).

  8. Here M stands for the standard Hardy–Littlewood maximal function.

  9. In our later reasonings we will often ignore the constant \(\frac{1}{\sqrt{2\pi }}\).

  10. As expected, in the maximal function case, the mean zero condition of \(\varphi \) becomes irrelevant and one can replace the original conditions imposed on \(\varphi \) by merely \(\check{\varphi }\in \mathcal {S}(\mathbb {R})\) any (normalized) function with \(\check{\varphi }\ge 0\) and \(\Vert \check{\varphi }\Vert _{L^1(\mathbb {R})}=1\). Given this, in what follows we no longer specify the dependence of our maximal function on \(\varphi \).

  11. The condition discussed below will be needed when discussing the boundedness of our operator \(M_{\Gamma }(f,g)\) in the regime \(L^{\infty }\times L^q\) for \(1<q\le \infty \)—see Sect. 5.2.

  12. The regularity index N here can be lowered but we will not detail this fact here.

  13. Throughout this subsection \(q=p'\).

  14. It is important to notice here that as opposed to the similar object defined in [18], in our context, in definition (4.9) below, the operator \(T_{j,m}(f,g)\) is taken with absolute values thus making \(\Lambda _{j,m}(f,g,h)\) a sublinear “form”.

  15. We will only recall here that \(\phi _{j,p_0}(\eta ):= \phi (\frac{\gamma '(2^{-j})\,\eta }{2^j}-p_0) \) and \(\psi _{m,p_0,j}(\xi ):= 2^{-\frac{m}{2}}\,e^{- i p_0 R(\frac{\xi }{2^j\,p_0})}\,\phi (\frac{\xi }{2^{m+j}})\) where \(j\in \mathbb {N}\) and \(p_0\in [2^m,\, 2^{m+1})\cap \mathbb {N}\) with \(m\in \mathbb {N}\).

  16. We remind the reader that the boundedness along the edge (AB) was proved in Sect. 4.1.

  17. Throughout this section we return to the original definition of our operator \(M_\Gamma (f,g)\) in (1.1).

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47906, USA

    Alejandra Gaitan & Victor Lie

  2. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700, Bucharest, RO, Romania

    Victor Lie

Authors
  1. Alejandra Gaitan
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  2. Victor Lie
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Correspondence to Victor Lie.

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Communicated by Loukas Grafakos.

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The second author was supported by the National Science Foundation under Grant No. DMS-1500958. The final revision of the paper before publication was performed while the second author was supported by the National Science Foundation under Grant No. DMS-1900801.

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Gaitan, A., Lie, V. The Boundedness of the (Sub)bilinear Maximal Function Along “Non-flat” Smooth Curves. J Fourier Anal Appl 26, 69 (2020). https://doi.org/10.1007/s00041-020-09770-6

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