Sharp local smoothing estimates for Fourier integral operators. (English) Zbl 1494.42013
Ciatti, Paolo (ed.) et al., Geometric aspects of harmonic analysis. Proceedings of the INdAM meeting, Cortona, Italy, June 25–29, 2018. Cham: Springer. Springer INdAM Ser. 45, 29-105 (2021).
The paper under review is a survey on the local smoothing conjecture for Fourier integral operators.
For the purposes of the paper under review, the investigation of FIOs dates back to the groundbreaking treaties [L. Hörmander, Acta Math. 127, 79–183 (1971; Zbl 0212.46601)] and [J. J. Duistermaat and L. Hörmander, ibid. 128, 183–269 (1972; Zbl 0232.47055)]. An FIO of order \(\mu\) is an operator that one originally defines on \(\mathcal{S}(\mathbb{R}^d)\) as \[ \mathcal{F}f(x):=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{i\phi(x,\xi)}a(x,\xi)\widehat{f}(\xi)\,\mbox{d}\xi \] where \(\widehat{f}\) is the Fourier integral of \(f\), \(\Phi\), the phase factor, is homogeneous of degree 1 in \(\xi\) and smooth away from \(\xi=0\) on the support of \(a\) while the amplitude \(a\) is in a smoothness class \(S^\mu\).
For the sake of this review, one might concentrate on the half-wave propagators given by \[ e^{\mp it\sqrt{-\Delta}}f(x):=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{i(\langle x,\xi\rangle\pm t|\xi|)} \widehat{f}(\xi) \,\mbox{d}\xi \] which are of order \(0\) and allow to solve the wave equation. Note that this operator depends on a parameter \(t\) that will play a key role in the future. The first aim of this survey is to provide a gentle introduction to FIOs in which the main concepts and results are introduced. Under a reasonable conjecture on the phase function (the so-called Mixed Hessian Condition), the boundedness of FIOs on \(L^p\) has been established in [A. Seeger et al., Ann. Math. (2) 134, No. 2, 231–251 (1991; Zbl 0754.58037)]. More precisely, if \(\mathcal{F}\) is an FIO of order \(\mu\) and \(1<p<+\infty\) then \[ \|\mathcal{F}f\|_{L^p_{-\mu-\tilde s_p}(\mathbb{R}^d)}\lesssim\|f\|_{L^p(\mathbb{R}^d)}\tag{1} \] where \(L^p_s\) stands for an \(L^p\) (Sobolev space of order \(s\)) and \(\bar s_p=(d-1)\left|\dfrac{1}{p}-\dfrac{1}{2}\right|\). The local smoothing estimate deals with a \(1\)-parameter family of FIOs \((\mathcal{F}_t)_t\) like the half-wave propagators \(e^{\pm i\sqrt{-\Delta}}\). Now, the FIOs are given by a phase factor \(\phi(x,\xi,t)\) and an amplitude \(a(x,\xi,t)\) depend on a “time” parameter \(t\). To explain the conjecture, note that (1) would directly imply an estimate of the form \[ \left(\int_{t\in[0,1]}\|\mathcal{F}_tf\|_{L^p_{-\mu-\bar s_p}(\mathbb{R}^d)}^p\,\mbox{d}t\right)^{1/p} \lesssim\|f\|_{L^p(\mathbb{R}^d)}. \] The local smoothing conjecture asserts that under some conditions on the amplitude and phase (still a “mixed Hessian condition” for \(\phi\) and a “curvature condition”), averaging in time allows for improving the smoothness of \(\mathcal{F}_t\) i.e. that one might replace \(-\mu-\bar s_p\) by a large smoothness index. \[ S_R^\delta f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{\langle x,\xi\rangle}(1-|t/R|)_+^\delta\widehat{f}(\xi)\,\mbox{d}\xi. \] Conjecture (Bochner Riesz): Let \(1\leq p\leq +\infty\), \(\delta(p)=\max\left(d\left|\dfrac{1}{2}-\dfrac{1}{p}\right|,0\right)\). Then, if \(f\in L^p(\mathbb{R}^d)\) and \(\delta<\delta(p)\), \(S_R^\delta f\to f\) in \(L^p\) when \(R\to\infty\). The Bochner-Riesz conjecture is known to imply the Restriction Conjecture (which roughly asks for which \(p\)’s can the Fourier transform of an \(f\in L^p(\mathbb{R}^d)\) be meaningfully be restricted to the unit sphere of \(\mathbb{R}^d\)). In turn, the restriction conjecture implies the Kakeya conjecture (which roughly speaking concerns how tubes that point in different directions can overlap). In the simplest case of the half-wave propagator (for which \(\mu=0\)), the conjecture is the following [C. D. Sogge, Invent. Math. 104, No. 2, 349–376 (1991; Zbl 0754.35004)]: Conjecture (Local Smoothing for the half-wave propagator on \(\mathbb{R}^d\)). For \(d\geq 2\), the inequality \[ \left(\int_{t\in[0,1]}\|e^{it\sqrt{-\Delta}}f\|_{L^p_{\sigma-\bar s_p}(\mathbb{R}^d)}^p\,\mbox{d}t\right)^{1/p} \lesssim\|f\|_{L^p(\mathbb{R}^d)}. \] holds for all \(f\in L^p(\mathbb{R}^d)\) when \(\sigma<1/p\) if \(\frac{2d}{d-1}\leq p<+\infty\) and \(\sigma<\bar s_p\) if \(2<p\leq \frac{2d}{d-1}\). Even in the case \(d=2\), this conjecture is still open, despite numerous partial results. The survey gives the state of the art on this conjecture. The importance of this conjecture is explained in Section 3 of the survey as its relations to other important results and conjectures in harmonic analysis are explained. First, it is shown that the Local Smoothing Conjecture for the half-wave propagator on \(\mathbb{R}^d\) implies the Bochner-Riesz Conjecture about summability of classical Fourier integrals: define the Bochner-Riesz Means as \[ S_R^\delta f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{\langle x,\xi\rangle}(1-|t/R|)_+^\delta\widehat{f}(\xi)\,\mbox{d}\xi. \]
The connection between the local smoothing conjecture and estimates of the spherical maximal function is also explained. In particular, it is shown how some known cases of the local smoothing conjecture imply the Circular Maximal Theorem, [J. Bourgain, J. Anal. Math. 47, 69–85 (1986; Zbl 0626.42012)]. For general \(1\)-parameter families of FIOs, it turns out that the range of smoothness has to be restricted. Indeed, as surveyed in Section 4, there exist, families of FIOs \(\mathcal{F},_t\) such that one may not replace \(e^{-i\sqrt{-\Delta}}\) by \(\mathcal{F}_t\) in the local smoothing conjecture and a new conjecture needs to be made: Conjecture (Local smoothing for general families of FIOs). For \(d\geq 2\), \[ \bar p_d=\begin{cases}\dfrac{2(d+1)}{d-1}&\mbox{ when }d\text{ is odd}\\ \dfrac{2(d+2)}{d}& \mbox{ when }d\text{ is even}\end{cases}. \] Let \(\bar p_d\leq p<+\infty\) and \(\sigma<1/p\). Let \(\mathcal{F}_t\) be a family of FIOs of order \(\mu\) satisfying a mixed Hessian condition and a curvature condition. Then the inequality \[ \left(\int_{t\in[0,1]}\|\mathcal{F}_tf\|_{L^p_{\sigma-\bar s_p}(\mathbb{R}^d)}^p\,\mbox{d}t\right)^{1/p} \lesssim\|f\|_{L^p(\mathbb{R}^d)}. \] holds for all \(f\in L^p(\mathbb{R}^d)\).
The main aim of this survey is to present an outline proof of this conjecture when the dimension \(d\) is odd, following the author’s work [D. Beltran et al., Anal. PDE 13, No. 2, 403–433 (2020; Zbl 1436.35340)]. The main ingredient here is a decoupling inequality and is based on T. Wolff’s approach to the local smoothing conjecture [T. Wolff, Geom. Funct. Anal. 10, No. 5, 1237–1288 (2000; Zbl 0972.42005)]. The key ingredient (to which Section 5 is devoted) is an extension of the celebrated Bourgain-Demeter Decoupling Inequality [J. Bourgain and C. Demeter, Ann. Math. (2) 182, No. 1, 351–389 (2015; Zbl 1322.42014)] that was extended by the authors to the variable coefficient setting.
This survey covers many topics related to several conjectures central to modern harmonic analysis and presents some of the most advanced techniques in the field. It can be considered a must-read for people willing to keep up to date with those topics.
For the entire collection see [Zbl 1470.42001].
For the purposes of the paper under review, the investigation of FIOs dates back to the groundbreaking treaties [L. Hörmander, Acta Math. 127, 79–183 (1971; Zbl 0212.46601)] and [J. J. Duistermaat and L. Hörmander, ibid. 128, 183–269 (1972; Zbl 0232.47055)]. An FIO of order \(\mu\) is an operator that one originally defines on \(\mathcal{S}(\mathbb{R}^d)\) as \[ \mathcal{F}f(x):=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{i\phi(x,\xi)}a(x,\xi)\widehat{f}(\xi)\,\mbox{d}\xi \] where \(\widehat{f}\) is the Fourier integral of \(f\), \(\Phi\), the phase factor, is homogeneous of degree 1 in \(\xi\) and smooth away from \(\xi=0\) on the support of \(a\) while the amplitude \(a\) is in a smoothness class \(S^\mu\).
For the sake of this review, one might concentrate on the half-wave propagators given by \[ e^{\mp it\sqrt{-\Delta}}f(x):=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{i(\langle x,\xi\rangle\pm t|\xi|)} \widehat{f}(\xi) \,\mbox{d}\xi \] which are of order \(0\) and allow to solve the wave equation. Note that this operator depends on a parameter \(t\) that will play a key role in the future. The first aim of this survey is to provide a gentle introduction to FIOs in which the main concepts and results are introduced. Under a reasonable conjecture on the phase function (the so-called Mixed Hessian Condition), the boundedness of FIOs on \(L^p\) has been established in [A. Seeger et al., Ann. Math. (2) 134, No. 2, 231–251 (1991; Zbl 0754.58037)]. More precisely, if \(\mathcal{F}\) is an FIO of order \(\mu\) and \(1<p<+\infty\) then \[ \|\mathcal{F}f\|_{L^p_{-\mu-\tilde s_p}(\mathbb{R}^d)}\lesssim\|f\|_{L^p(\mathbb{R}^d)}\tag{1} \] where \(L^p_s\) stands for an \(L^p\) (Sobolev space of order \(s\)) and \(\bar s_p=(d-1)\left|\dfrac{1}{p}-\dfrac{1}{2}\right|\). The local smoothing estimate deals with a \(1\)-parameter family of FIOs \((\mathcal{F}_t)_t\) like the half-wave propagators \(e^{\pm i\sqrt{-\Delta}}\). Now, the FIOs are given by a phase factor \(\phi(x,\xi,t)\) and an amplitude \(a(x,\xi,t)\) depend on a “time” parameter \(t\). To explain the conjecture, note that (1) would directly imply an estimate of the form \[ \left(\int_{t\in[0,1]}\|\mathcal{F}_tf\|_{L^p_{-\mu-\bar s_p}(\mathbb{R}^d)}^p\,\mbox{d}t\right)^{1/p} \lesssim\|f\|_{L^p(\mathbb{R}^d)}. \] The local smoothing conjecture asserts that under some conditions on the amplitude and phase (still a “mixed Hessian condition” for \(\phi\) and a “curvature condition”), averaging in time allows for improving the smoothness of \(\mathcal{F}_t\) i.e. that one might replace \(-\mu-\bar s_p\) by a large smoothness index. \[ S_R^\delta f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{\langle x,\xi\rangle}(1-|t/R|)_+^\delta\widehat{f}(\xi)\,\mbox{d}\xi. \] Conjecture (Bochner Riesz): Let \(1\leq p\leq +\infty\), \(\delta(p)=\max\left(d\left|\dfrac{1}{2}-\dfrac{1}{p}\right|,0\right)\). Then, if \(f\in L^p(\mathbb{R}^d)\) and \(\delta<\delta(p)\), \(S_R^\delta f\to f\) in \(L^p\) when \(R\to\infty\). The Bochner-Riesz conjecture is known to imply the Restriction Conjecture (which roughly asks for which \(p\)’s can the Fourier transform of an \(f\in L^p(\mathbb{R}^d)\) be meaningfully be restricted to the unit sphere of \(\mathbb{R}^d\)). In turn, the restriction conjecture implies the Kakeya conjecture (which roughly speaking concerns how tubes that point in different directions can overlap). In the simplest case of the half-wave propagator (for which \(\mu=0\)), the conjecture is the following [C. D. Sogge, Invent. Math. 104, No. 2, 349–376 (1991; Zbl 0754.35004)]: Conjecture (Local Smoothing for the half-wave propagator on \(\mathbb{R}^d\)). For \(d\geq 2\), the inequality \[ \left(\int_{t\in[0,1]}\|e^{it\sqrt{-\Delta}}f\|_{L^p_{\sigma-\bar s_p}(\mathbb{R}^d)}^p\,\mbox{d}t\right)^{1/p} \lesssim\|f\|_{L^p(\mathbb{R}^d)}. \] holds for all \(f\in L^p(\mathbb{R}^d)\) when \(\sigma<1/p\) if \(\frac{2d}{d-1}\leq p<+\infty\) and \(\sigma<\bar s_p\) if \(2<p\leq \frac{2d}{d-1}\). Even in the case \(d=2\), this conjecture is still open, despite numerous partial results. The survey gives the state of the art on this conjecture. The importance of this conjecture is explained in Section 3 of the survey as its relations to other important results and conjectures in harmonic analysis are explained. First, it is shown that the Local Smoothing Conjecture for the half-wave propagator on \(\mathbb{R}^d\) implies the Bochner-Riesz Conjecture about summability of classical Fourier integrals: define the Bochner-Riesz Means as \[ S_R^\delta f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}e^{\langle x,\xi\rangle}(1-|t/R|)_+^\delta\widehat{f}(\xi)\,\mbox{d}\xi. \]
The connection between the local smoothing conjecture and estimates of the spherical maximal function is also explained. In particular, it is shown how some known cases of the local smoothing conjecture imply the Circular Maximal Theorem, [J. Bourgain, J. Anal. Math. 47, 69–85 (1986; Zbl 0626.42012)]. For general \(1\)-parameter families of FIOs, it turns out that the range of smoothness has to be restricted. Indeed, as surveyed in Section 4, there exist, families of FIOs \(\mathcal{F},_t\) such that one may not replace \(e^{-i\sqrt{-\Delta}}\) by \(\mathcal{F}_t\) in the local smoothing conjecture and a new conjecture needs to be made: Conjecture (Local smoothing for general families of FIOs). For \(d\geq 2\), \[ \bar p_d=\begin{cases}\dfrac{2(d+1)}{d-1}&\mbox{ when }d\text{ is odd}\\ \dfrac{2(d+2)}{d}& \mbox{ when }d\text{ is even}\end{cases}. \] Let \(\bar p_d\leq p<+\infty\) and \(\sigma<1/p\). Let \(\mathcal{F}_t\) be a family of FIOs of order \(\mu\) satisfying a mixed Hessian condition and a curvature condition. Then the inequality \[ \left(\int_{t\in[0,1]}\|\mathcal{F}_tf\|_{L^p_{\sigma-\bar s_p}(\mathbb{R}^d)}^p\,\mbox{d}t\right)^{1/p} \lesssim\|f\|_{L^p(\mathbb{R}^d)}. \] holds for all \(f\in L^p(\mathbb{R}^d)\).
The main aim of this survey is to present an outline proof of this conjecture when the dimension \(d\) is odd, following the author’s work [D. Beltran et al., Anal. PDE 13, No. 2, 403–433 (2020; Zbl 1436.35340)]. The main ingredient here is a decoupling inequality and is based on T. Wolff’s approach to the local smoothing conjecture [T. Wolff, Geom. Funct. Anal. 10, No. 5, 1237–1288 (2000; Zbl 0972.42005)]. The key ingredient (to which Section 5 is devoted) is an extension of the celebrated Bourgain-Demeter Decoupling Inequality [J. Bourgain and C. Demeter, Ann. Math. (2) 182, No. 1, 351–389 (2015; Zbl 1322.42014)] that was extended by the authors to the variable coefficient setting.
This survey covers many topics related to several conjectures central to modern harmonic analysis and presents some of the most advanced techniques in the field. It can be considered a must-read for people willing to keep up to date with those topics.
For the entire collection see [Zbl 1470.42001].
Reviewer: Philippe Jaming (Bordeaux)
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B37 | Harmonic analysis and PDEs |
Citations:
Zbl 0212.46601; Zbl 0232.47055; Zbl 0754.58037; Zbl 0754.35004; Zbl 0626.42012; Zbl 1436.35340; Zbl 0972.42005; Zbl 1322.42014References:
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