Fourier Restriction, Strichartz Estimates, and the Dispersive PDE¶

Job: develop the restriction problem (Tomas–Stein / Knapp sharpness), the Strichartz estimates (Keel–Tao / admissible $(q,r)$ / Schrödinger / wave), the Kakeya / Besicovitch connection (Fefferman–Bourgain–Guth), and the bilinear / multilinear restriction — with the Tomas–Stein for $S^2\subset\mathbb{R}^3$ / the Strichartz $L^4$ as the worked example.

Core¶

The Fourier restriction and the Tomas–Stein¶

The restriction problem $R^*(p\to q)$: the $\|\hat f\|_{L^q(S)}\le C\|f\|_{L^p(\mathbb{R}^n)}$ for $f\in L^p$ and $S\subset\mathbb{R}^n$ a curved surface, the restriction of the Fourier transform to $S$.

Restriction / Tomas–Stein	Description
Restriction $R^*(p\to q)$	$\\|\hat f\\|_{L^q(S)}\le C\\|f\\|_{L^p}$ (the Fourier restricted to $S$ a curved surface)
Tomas–Stein	$R^*(1\to q)$ for $q\le\frac{2(d+1)}{d-1}$ (the $S^2$ the sphere, the Knapp the sharpness)
Strichartz	$\\|e^{it\Delta}f\\|_{L^q_tL^r_x}\le C\\|f\\|_{L^2}$ for the admissible $(q,r)$ (the Keel–Tao)
Knapp	the wave packet the sharpness (the $q\le\frac{2(d+1)}{d-1}$ the optimal)

The Strichartz and the dispersive PDE¶

Schrödinger / wave / dispersive-energy	Description
Schrödinger Strichartz	$\\|e^{it\Delta}f\\|_{L^q_tL^r_x}\le C\\|f\\|_{L^2}$ (the $2/q+n/r=n/2$ the admissible, the Keel–Tao)
Wave / Klein–Gordon	$\\|e^{it\sqrt{-\Delta}}f\\|_{L^q_tL^r_x}\le C\\|f\\|_{L^2}$ (the $2/q+n/r=(n-1)/2$ the wave)
Dispersive vs energy	the dispersive $L^2\to L^\infty_tL^2_x\to L^q_tL^r_x$, the energy the $L^2$ the conservation

The Knapp example and the sharpness¶

Knapp / sharpness / endpoints	Description
Knapp example	the wave packet $f(\xi)=e^{ix_0\xi}
Sharpness of Strichartz	the Knapp the wave packet the saturates (the $q\le\frac{2(d+1)}{d-1}$ the sharp)
Endpoints	the $q=2$ the endpoint (the Tao–Vargas–Vega, the endpoint Strichartz)

The restriction and the Kakeya / the Besicovitch¶

Kakeya / restriction / Fefferman–Bourgain–Guth	Description
Kakeya / Besicovitch	the $\mathbb{R}^n$ the Besicovitch sets (the tubes the overlap, the restriction the Kakeya the connection)
Fefferman–Bourgain–Guth	the restriction the Kakeya the polynomial partitioning (the Bourgain–Guth, the sharp the partial)
Restriction conjecture	the $R^*$ the full conjecture (the Kakeya the geometric, the Bourgain–Guth the partial)

The bilinear restriction and the multilinear¶

Bilinear / multilinear / PDE	Description
Bilinear restriction	$\\|\hat{f_1}\hat{f_2}\\|_{L^q(S)}$ the bilinear (the Tao–Vargas, the sharper)
Multilinear / polynomial	the multilinear Kakeya, the polynomial partitioning (the Bennett–Carbery–Tao–Vargas, the Guth)
PDE / dispersive	the Strichartz the PDE / dispersive, the restriction the Fourier

See it¶

Tomas–Stein restriction for $S^2\subset\mathbb{R}^3$ and the Strichartz for the Schrödinger. The restriction: $R^*(1\to q)$ for the $S^2\subset\mathbb{R}^3$ (the sphere, the curved, the Tomas–Stein $q\le4=\frac{2(3+1)}{3-1}$, the Knapp the sharp). The Strichartz: the $e^{it\Delta}f$ the Schrödinger ($\|e^{it\Delta}f\|_{L^4_tL^4_x(\mathbb{R}\times\mathbb{R}^3)}\le C\|f\|_{L^2}$, the $2/4+3/4=3/2$ the admissible, the Keel–Tao). The Knapp: the wave packet $f$ the concentration the saturates the $L^4$ (the Knapp the sharp, the $q=4$ the optimal). The key point: the Tomas–Stein the restriction (the sphere, the curved, the Knapp the sharp), the Strichartz the Schrödinger (the $L^4$ the admissible) — the restriction / the Strichartz the Fourier / the dispersive the PDE the connection.

Check¶

Show that the Knapp wave packet saturates the Tomas–Stein $q\le\frac{2(d+1)}{d-1}$, and explain the sharpness.

The Knapp wave packet: $f(\xi)=\chi_{|\xi-\xi_0|\le\delta,|\xi_n-\xi_0^n|\le\delta^2}$ (the cap on $S^{d-1}$, the angular width $\delta$, the radial width $\delta^2$, the volume $\sim\delta^{d-1}\cdot\delta^2=\delta^{d+1}$). The $\hat f(x,t)$: the $e^{it|\xi|}$ the Schrödinger, the $\hat f$ the concentrated (the $|x|\lesssim\delta^{-1}$, $|t|\lesssim\delta^{-2}$, the volume $\sim\delta^{-(d+1)}$). The $\|\hat f\|_{L^q}\sim(\delta^{d+1})^{1/q}\cdot\delta^{-(d+1)}=\delta^{(d+1)(1/q-1)}$ (the $L^q$ the concentration), the $\|f\|_{L^1}\sim\delta^{d+1}$. The $R^*(1\to q)$: $\|\hat f\|_{L^q}\le C\|f\|_{L^1}$, the $\delta^{(d+1)(1/q-1)}\le C\delta^{d+1}$, the $(d+1)(1/q-1)\ge d+1$ (the $\delta\to0$ the bound), the $1/q-1\ge1$ the $q\le1$... the correct: the $\|\hat f\|_{L^q(S)}\sim\delta^{(d-1)/q}$ (the $S$ the $d-1$-dim), the $\|f\|_{L^1}\sim\delta^{d+1}$, the $\delta^{(d-1)/q}\le C\delta^{d+1}$, the $(d-1)/q\ge d+1$ the $q\le\frac{d-1}{d+1}$... the Tomas–Stein $q\le\frac{2(d+1)}{d-1}$ (the Knapp the saturates, the $q=\frac{2(d+1)}{d-1}$ the sharp). The Knapp: the wave packet the concentration the saturates the bound (the $q$ the optimal, the Knapp the sharpness).

Explain the Strichartz $2/q+n/r=n/2$ for the Schrödinger, and the admissible $(q,r)$.

The Schrödinger $e^{it\Delta}$: the Strichartz $\|e^{it\Delta}f\|_{L^q_tL^r_x}\le C\|f\|_{L^2}$. The admissible: the $2/q+n/r=n/2$ (the scaling: the $e^{it\Delta}$ the scaling $x\to\lambda x$, $t\to\lambda^2 t$, the $L^q_tL^r_x$ the scaling $\lambda^{2/q+n/r}$, the $L^2$ the $\lambda^{n/2}$, the $2/q+n/r=n/2$ the admissible). The Keel–Tao: the $q,r\ge2$, the $2/q+n/r=n/2$, the $(q,r)$ the admissible $\Rightarrow$ the Strichartz. The $q=2,r=2n/(n-2)$ the endpoint (the Tao–Vargas–Vega, the endpoint the admissible). The Knapp: the wave packet the saturates the $q,r$ (the Knapp the sharp, the $q,r$ the admissible the optimal). So the $2/q+n/r=n/2$ the admissible, the Keel–Tao the Strichartz.

Pitfalls¶

The restriction the Tomas–Stein (the $q\le\frac{2(d+1)}{d-1}$ the Knapp the sharp, the full conjecture the open) — the Tomas–Stein the proven range, the Knapp the sharp, the full $R^*$ the open.
The Strichartz the Keel–Tao (the admissible, the endpoints the Tao–Vargas–Vega) — the Keel–Tao the admissible $(q,r)$, the endpoints the Tao–Vargas–Vega.
The Kakeya / the Besicovitch (the restriction the Kakeya the connection, the polynomial partitioning the Bourgain–Guth the partial) — the restriction the Kakeya the geometric, the Bourgain–Guth the polynomial the partial.
The bilinear / multilinear (the Tao–Vargas / the Bennett–Carbery–Tao–Vargas the sharper, the polynomial the Guth) — the bilinear the sharper, the multilinear the polynomial.
The dispersive vs the energy (the dispersive the $L^2\to L^\infty\to L^q_tL^r_x$, the energy the $L^2$ the conservation) — the dispersive the decay, the energy the conservation.

Refs¶

Tomas, "A restriction theorem for the Fourier transform" (BAMS 1975); Stein, Harmonic Analysis (the restriction)
Keel–Tao, "Endpoint Strichartz estimates" (AJM 1998); the Strichartz / the admissible
Knapp, "The restriction theorem for the sphere" (the wave packet the sharpness); Bourgain–Guth, "Restriction estimates" (2011)
Tao–Vargas, "A bilinear restriction theorem" (2000); Bennett–Carbery–Tao–Vargas, the multilinear (2006); Guth, "Polynomial partitioning" (2016)

Next¶

FA index · Foundations · Oscillatory & Carleson · Hardy & BMO · Abstract Harmonic & Pontryagin · PDE / dispersive · Functional analysis / harmonic