A maximal function for families of Hilbert transforms along homogeneous curves

Abstract

Let \(H^{(u)}\) be the Hilbert transform along the parabola \((t, ut^2)\) where \(u\in \mathbb {R}\). For a set U of positive numbers consider the maximal function \({\mathcal {H}}^U \,f= \sup \{|H^{(u)}\, f|: u\in U\}\). We obtain an (essentially) optimal result for the \(L^p\) operator norm of \({\mathcal {H}}^U\) when \(2<p<\infty \). The results are proved for families of Hilbert transforms along more general nonflat homogeneous curves.

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Notes

  1. 1.

    Added in September 2019: After the submission of this paper the authors showed the bound of Proposition 7.1 for general lacunary sets U, in the full range \(1<p<\infty \). This result can be found in the paper [17] which also contains \(L^p\) results, \(p<2\), for more general sets U, under suitable dimension assumptions.

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Acknowledgements

S.G. was supported in part by a direct grant for research from the Chinese University of Hong Kong (4053295). A.S. was supported in part by National Science Foundation grants DMS 1500162 and 1764295. He would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program Approximation, Sampling and Compression in Data Science where some work on this paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1. P.Y. was supported in part by a General Research Fund CUHK14303817 from the Hong Kong Research Grant Council, and direct grants for research from the Chinese University of Hong Kong (4441563 and 4441651).

The authors thank Rajula Srivastava for reading a draft of this paper and for providing useful comments.

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Correspondence to Shaoming Guo.

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Appendix A: Proof of Proposition 3.4

Appendix A: Proof of Proposition 3.4

The proof is a modification of the argument for the standard Cotlar inequality regarding truncations of singular integrals, cf. [26, §I.7].

Let \(m_j(\xi ) =\eta (2^{-j}\xi )m(\xi )\) and let \(a_j(\xi ) = m_j(2^j\xi )\). We pick \(0<\varepsilon <\min \{\alpha -d,1\}\). Then by assumption

$$\begin{aligned} \sup _{j\in {\mathbb {Z}}}\Vert a_j\Vert _{\mathscr {L}^1_\alpha }\le B<\infty \end{aligned}$$
(A.1)

which implies that \(|{\mathcal {F}}^{-1} [a_j](x)|\le CB(1+|x|)^{-d-\varepsilon }\), and thus, with \(K_j= {\mathcal {F}}^{-1} [m_j]\),

$$\begin{aligned} |K_j(x)|+ 2^{-j} |\nabla K_j(x)| \le C B 2^{jd} (1+2^j|x|)^{-d-\varepsilon }. \end{aligned}$$

For Schwartz functions f we have \(S f=\sum _{j\in {\mathbb {Z}}} K_j* f\) and \(S_nf=\sum _{j\le n} K_j*f\).

Lemma A.1

Fix \(\tilde{x}\in {\mathbb {R}}^d\) and \(n\in {\mathbb {Z}}\), and let \(g(y)= f(y){\mathbb {1}}_{B(\tilde{x}, 2^{-n})}(y)\) and \(h=f-g\). Then

  1. (i)

    \(|S_n g(\tilde{x})|\lesssim B \, M[f](\tilde{x})\).

  2. (ii)

    \(|S_nh(\tilde{x})-S h(\tilde{x})|\lesssim B\, M[f](\tilde{x})\).

  3. (iii)

    For \(|w-\tilde{x}|\le 2^{-n-1}\) we have \(|Sh(\tilde{x})-Sh(w)|\lesssim B \, M[f](\tilde{x}).\)

Proof

By appropriate normalization of the multiplier we may assume \(B=1\).

  1. (i)

    is immediate since for \(j\le n\)

    $$\begin{aligned} |K_j*g(\tilde{x})|\lesssim 2^{jd}\int _{|\tilde{x}-y|\le 2^{-n}}|g(y)| dy \lesssim 2^{(j-n)d} M[g](\tilde{x}) \end{aligned}$$

    and the assertion follows since \(|g|\le |f|\).

For (ii) notice that \(|S_n h(\tilde{x})-S h(\tilde{x})|\le \sum _{j>n}|K_j* h(\tilde{x})|\). For \(j>n\) we estimate

where the slashed integral denotes the average. Thus we get

$$\begin{aligned} \sum _{j\ge n} |K_j*h(\tilde{x})|\lesssim M[h](\tilde{x}) \end{aligned}$$

and, since \(|h|\le |f|\), the assertion follows.

Concerning (iii) we consider the terms \(K_j*h(\tilde{x})-K_j*h(w)\) separately for \(j\le n\) and \(j>n\). The term \(\sum _{j>n} |K_j*h(\tilde{x})|\) was already dealt with in (ii). Since \(|w-\tilde{x}|\le 2^{-n-1}\) we have \(|w-y|\approx |\tilde{x}-y| \) for \(|\tilde{x}-y|\ge 2^{-n}\) and thus the previous calculation also yields

$$\begin{aligned} \sum _{j> n} |K_j*h(w)|\lesssim M[h](\tilde{x}) \lesssim Mf(\tilde{x}). \end{aligned}$$

It remains to consider the terms for \(j\le n\). In that range we write

$$\begin{aligned} K_j*h(\tilde{x})-K_j*h(w) = \int _0^1\int \limits _{|\tilde{x}-y|\ge 2^{-n} }\langle \tilde{x}-w,\nabla K_j (w+s(\tilde{x}-w)-y)\rangle h(y) dy\, ds. \end{aligned}$$

Since \(|w-\tilde{x}| \le 2^{-n-1}\) we can replace \(|w+s(\tilde{x}-w)-y| \) in the integrand with \(|\tilde{x}-y|\) and estimate the displayed expression by \(C\sum _{l\ge 0} A_{l,j,n}\) where

Summing in \(l > 0\) and then \(j\le n\) yields

$$\begin{aligned} \sum _{j\le n} |K_j*h(\tilde{x})-K_j*h(w)| \lesssim Mh(\tilde{x}) \lesssim Mf(\tilde{x}). \end{aligned}$$
(A.2)

Proof of (3.7)

We proceed arguing as in [26, §I.7]. Fix \(\tilde{x}\in {\mathbb {R}}^d\) and \(n\in {\mathbb {Z}}\) and define g and h as in the lemma. For (suitable) w with \(|w-\tilde{x}|\le 2^{-n-1}\) we write

$$\begin{aligned} S_n f(\tilde{x})&= S_n g(\tilde{x})+ (S_n-S)h(\tilde{x}) + Sh(\tilde{x})\nonumber \\&=S_n g(\tilde{x})+ (S_n-S)h(\tilde{x}) + Sh(\tilde{x})- Sh(w) + Sf(w)- Sg(w). \end{aligned}$$
(A.3)

By Lemma A.1

$$\begin{aligned} |S_n g(\tilde{x})|+ |(S_n-S)h(\tilde{x})| + |Sh(\tilde{x})- Sh(w)| \lesssim B \,M[f](\tilde{x}) \end{aligned}$$

and it remains to consider the term \(Sf(w)-Sg(w)\) for a substantial set of w with \(|w-\tilde{x}|\le 2^{-n-1}\).

By the Mikhlin-Hörmander theorem we have for all \(f\in L^1({\mathbb {R}}^d)\) and all \(\lambda >0\)

$$\begin{aligned} {\mathrm{meas}}( \{x: |Sf(x)|>\lambda \}) \le A \lambda ^{-1} \Vert f\Vert _1 \end{aligned}$$

where \(A \le C_{\alpha ,d}B\).

Now let \(\delta \in (0,1/2)\) and consider the set

$$\begin{aligned} \Omega _n(\tilde{x}, \delta )=\big \{ w: |w-\tilde{x}|< 2^{-n-1}, \quad |Sg(w)|> 2^d\delta ^{-1} A \,M[f](\tilde{x})\big \}. \end{aligned}$$

In (A.3) we can estimate the term |Sg(w)| by \(2^d\delta ^{-1} A \,M[f](\tilde{x})\) when \(w\in B(\tilde{x}, 2^{-n-1}) {\setminus } \Omega _n(\tilde{x},\delta )\). Hence we obtain

$$\begin{aligned} |S_n f(\tilde{x})| \le \inf _{w\in B(\tilde{x}, 2^{-n-1}) {\setminus } \Omega _n(\tilde{x},\delta )} |Sf(w)| + C(\alpha , d) B (1+\delta ^{-1}) M[f](\tilde{x}). \end{aligned}$$
(A.4)

By the weak type inequality for S we have

$$\begin{aligned} {\mathrm{meas}}(\Omega _n(\tilde{x},\delta ))&\le \frac{A\Vert g\Vert _1}{2^d\delta ^{-1} A M[f](\tilde{x})} =\frac{\delta }{2^d M[f](\tilde{x})}\int _{|\tilde{x}-y|\le 2^{-n}}|f(y)| dy\\&\le \delta \, 2^{-d}\,{\mathrm{meas}}(B(\tilde{x}, 2^{-n})) =\delta \, {\mathrm{meas}}(B(\tilde{x}, 2^{-n-1})). \end{aligned}$$

Hence \({\mathrm{meas}}( B(\tilde{x}, 2^{-n-1}) {\setminus } \Omega _n(\tilde{x},\delta ))\ge (1-\delta ) {\mathrm{meas}}(B(\tilde{x}, 2^{-n-1}))\) and thus for all \(r>0\)

$$\begin{aligned}&\inf _{w\in B(\tilde{x}, 2^{-n-1}) {\setminus } \Omega _n(\tilde{x},\delta )} |Sf(w)| \\&\quad \le \left( \frac{1}{{\mathrm{meas}}(B(\tilde{x}, 2^{-n-1}) {\setminus } \Omega _n(\tilde{x},\delta ))} \int _{B(\tilde{x}, 2^{-n-1})} |Sf (w)|^r dw\right) ^{1/r}\\&\quad \le \left( \frac{1}{(1-\delta )|B(\tilde{x}, 2^{-n-1})|} \int _{B(\tilde{x}, 2^{-n-1})} |Sf (w)|^r dw\right) ^{1/r}. \end{aligned}$$

We obtain

$$\begin{aligned} |S_n f(\tilde{x})| \le (1-\delta )^{-1/r} (M[|Sw|^r](\tilde{x}))^{1/r} + C(\alpha ,d) (1+\delta ^{-1}) B\,M[f](\tilde{x}) \end{aligned}$$

uniformly in n. This implies (3.7). \(\square \)

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Guo, S., Roos, J., Seeger, A. et al. A maximal function for families of Hilbert transforms along homogeneous curves. Math. Ann. 377, 69–114 (2020). https://doi.org/10.1007/s00208-019-01915-3

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