Oscillatory Integrals, Stationary Phase, and the Carleson Theorem¶

Job: develop the oscillatory integrals / stationary phase / non-stationary (IBP), the Carleson theorem ($L^2$ a.e. convergence / maximal $S^*$ / Lacey–Thiele time-frequency), the FIO / canonical relation (Hörmander), and the tiles / trees — with the Carleson via time-frequency / the stationary phase for the wave as the worked example.

Core¶

Oscillatory integrals and the stationary phase¶

Oscillatory / stationary / non-stationary	Description
Oscillatory integral	$I(\lambda)=\int e^{i\lambda\phi(x)}a(x)\,dx$ (the phase $\phi$, the amplitude $a$, the $\lambda\to\infty$ the asymptotic)
Stationary phase	$\nabla\phi(x_0)=0$ the critical, $I(\lambda)\sim e^{i\lambda\phi(x_0)}\lambda^{-n/2}a(x_0)(2\pi/i)^{n/2}(\det\mathrm{Hess}\,\phi)^{-1/2}$ (the Hessian)
Non-stationary / IBP	$\nabla\phi\ne0$ the $I(\lambda)=O(\lambda^{-N})$ (the rapid decay, the integration by parts)

The Carleson theorem and the pointwise convergence¶

Theorem (Carleson, 1966). The Fourier series of $f\in L^2(\mathbb{T})$ converges a.e. to $f$.

Carleson / maximal / pointwise / Kolmogorov	Description
Carleson	$S_Nf\to f$ a.e. for $f\in L^2(\mathbb{T})$ (the Carleson 1966, the Hunt $L^p$ $1<p<\infty$)
Maximal $S^*$	$S^*f=\sup_N
Pointwise	$S_Nf\to f$ a.e. (the Carleson, the Kolmogorov $L^1$ the counterexample)
Kolmogorov	the $L^1$ the counterexample (the $S_Nf\not\to f$ for some $f\in L^1$, the Kolmogorov)

The Carleson and the time-frequency¶

Time-frequency / tiles / BMO-$H^1$	Description
Time-frequency analysis	the Carleson the time-frequency (the Lacey–Thiele, the tiles / the trees)
Tiles / trees	the Carleson the tiles / the trees (the combinatorial, the Lacey–Thiele)
BMO / $H^1$	the Carleson the BMO / the $H^1$ the connection (the functional analysis / harmonic)

The oscillatory integrals and the microlocal / the FIO¶

FIO / stationary / canonical / microlocal	Description
FIO	$Tf=\int e^{i\phi(x,\xi)}a(x,\xi)\hat f(\xi)\,d\xi$ (the Fourier integral operator, the oscillatory, the microlocal)
Stationary phase / composition	the FIO the oscillatory, the stationary phase the composition (the Hörmander)
Canonical relation	$\Lambda\subset T^X\times T^Y$ the Lagrangian (the microlocal)

See it¶

Carleson theorem via the time-frequency / the tiles and the stationary phase for the wave. The Carleson: the $S_Nf\to f$ a.e. for $f\in L^2(\mathbb{T})$ (the Carleson 1966, the Hunt $L^p$ $1<p<\infty$). The time-frequency: the Lacey–Thiele (the $S^*f$ the weak (2,2) via the tiles / the trees, the combinatorial the time-frequency). The stationary phase for the wave: the $u(t,x)=\int e^{i(t|\xi|\pm x\xi)}\hat f(\xi)\,d\xi$ the wave solution (the oscillatory, the stationary phase the asymptotic, the $|t|^{-n/2}$ the dispersive decay, the Strichartz). The key point: the Carleson the time-frequency (the Lacey–Thiele the tiles), the stationary phase the wave (the dispersive decay the Strichartz) — the oscillatory the asymptotic / the pointwise the Carleson.

Check¶

Show that the stationary phase gives $I(\lambda)\sim e^{i\lambda\phi(x_0)}\lambda^{-n/2}a(x_0)(2\pi/i)^{n/2}(\det\mathrm{Hess}\,\phi)^{-1/2}$, and explain the Hessian.

The oscillatory $I(\lambda)=\int e^{i\lambda\phi(x)}a(x)\,dx$ with $\nabla\phi(x_0)=0$ (the critical point $x_0$). The Taylor: $\phi(x)=\phi(x_0)+\frac12(x-x_0)^T H (x-x_0)+\dots$ ($H=\mathrm{Hess}\,\phi(x_0)$ the Hessian). The $I(\lambda)\approx e^{i\lambda\phi(x_0)}a(x_0)\int e^{i\lambda\frac12 y^T H y}\,dy$ (the Gaussian, the $a(x)\approx a(x_0)$). The Gaussian: $\int e^{i\lambda\frac12 y^T H y}\,dy=(2\pi/i)^{n/2}\lambda^{-n/2}(\det H)^{-1/2}$ (the Fresnel, the $\det H\ne0$ the non-degenerate). So $I(\lambda)\sim e^{i\lambda\phi(x_0)}\lambda^{-n/2}a(x_0)(2\pi/i)^{n/2}(\det\mathrm{Hess}\,\phi)^{-1/2}$ (the stationary phase, the $\lambda^{-n/2}$ the decay, the $\det H$ the Hessian). The Hessian: the $\det H\ne0$ the non-degenerate (the stationary phase the standard), the $\det H=0$ the degenerate (the different, the higher-order / the caustics).

Explain the Carleson theorem and the Lacey–Thiele proof via the time-frequency / the tiles.

The Carleson: $S_Nf(\theta)=\sum_{|n|\le N}\hat f(n)e^{in\theta}\to f(\theta)$ a.e. for $f\in L^2(\mathbb{T})$ (the pointwise the convergence). The maximal $S^*f(\theta)=\sup_N|S_Nf(\theta)|$ (the maximal the partial sums). The Carleson: the $S^*f$ the weak (2,2) ($\|S^*f\|_{L^{2,\infty}}\le C\|f\|_2$, the weak (2,2)) $\Rightarrow$ the $S_Nf\to f$ a.e. (the density the $L^2$ + the maximal the weak). The Lacey–Thiele: the $S^*$ the weak (2,2) via the time-frequency / the tiles. The tiles: the time-frequency tiles (the $(I,\omega)$ the interval $I$ the time, the $\omega$ the frequency, the dyadic), the trees (the organized tiles, the combinatorial the structure). The Lacey–Thiele: the $S^*f$ the sum over the tiles, the tiles / the trees the combinatorial the bound, the weak (2,2) the estimate. The BMO / $H^1$: the Carleson the BMO / the $H^1$ the connection (the time-frequency the BMO the dual, the functional analysis / harmonic). So the Carleson the $L^2$ a.e. (the Lacey–Thiele the time-frequency / the tiles).

Pitfalls¶

The Carleson the $L^2$ a.e. (the $L^1$ the Kolmogorov the counterexample, the Lacey–Thiele the tiles) — the $L^2$ the Carleson, the $L^1$ the Kolmogorov (the counterexample), the Lacey–Thiele the tiles.
The stationary phase the Hessian (the non-degenerate the stationary, the degenerate the different) — the $\det H\ne0$ the standard, the $\det H=0$ the degenerate (the caustics).
The FIO the canonical relation (the Hörmander, the microlocal) — the FIO the oscillatory, the canonical relation the Lagrangian, the Hörmander.
The time-frequency / the tiles / the trees (the Lacey–Thiele the combinatorial) — the tiles the time-frequency, the trees the organized, the Lacey–Thiele the bound.

Refs¶

Carleson, "On convergence and growth of partial sums of Fourier series" (Acta 1966); Hunt, "On the convergence of Fourier series" (1968)
Lacey–Thiele, "A proof of boundedness of the Carleson operator" (Math. Res. Lett. 2000)
Hörmander, "Fourier integral operators I" (Acta 1971); the FIO / the canonical relation
Stein, Harmonic Analysis (the oscillatory / the stationary phase); the microlocal

Next¶

FA index · Foundations · Restriction & Strichartz · Hardy & BMO · Abstract Harmonic & Pontryagin · Microlocal analysis · PDE / dispersive