Local smoothing for the Euclidean wave equation

1. Interpolation at the two ends

With $h=N^{-1}$, Plancherel gives a uniform $L^2_X\to L^2_{T,X}$ bound, while $L^\infty_X\to L^\infty_{T,X}$ costs $O(h^{-(n-1)/2})$. For the latter, split into caps of width $h^{1/2}$ and remove the linear phase; each remaining kernel has uniformly bounded $L^1$ norm. Riesz–Thorin yields

Section 7.2, (7.3) · pp. 24–25

2. Spending the positive loss on summation

For $g=J^{s_+(p)+\varepsilon}f$, the coefficient on the $j$th annulus is $2^{-j\varepsilon}(1+j)^{\beta(p)}$, an absolutely summable sequence. Mihlin handles low frequencies; Plancherel directly gives the lossless $p=2$ case. Corollary 7.1 refines this sum to logarithmic Sobolev bounds with $B>\beta(p)+1/2-1/p$ (or equality with an additional iterated-logarithm condition), but does not generally give $B=C=0$.

Section 7.2, Corollary 7.1 · pp. 25–26

1. Preserving interference and producing the critical power

Two auxiliary flows of a symmetric first-order system produce $A_0=\chi\chi^*$. The vector $\chi$ is formed by summing the Gaussian-weighted waves before taking the square. Every real column is divergence free, and $(\partial\Phi)A_0=0$. For frequency $N=h^{-1}$ in a fixed small cap, the time component recovers the wave when the auxiliary radius is comparable to $h$.

The fiber defining function satisfies $H_{\rm fib}(r)>0$. The null line of the full $H(r)$ is orthogonal both to the boundary normal and to the image of $H(|\Phi|^2)$. Smooth and cut off the current, then add a small cofactor current to obtain $\mu=A_{TT}>0$. For $B_0=\lambda|\Phi|^2$, $\lambda=h^{-30n}$, solve

The positive defining Hessian and density gap satisfy Theorem A.1. The spatial background has rank one, so $\det(B_{\rm fib}+\alpha I)=(\beta+\alpha)\alpha^{n-1}$ with $\beta\asymp\lambda$. A density lower bound $m$ on a small ball forces $-u\gtrsim h^2m^{1/(n-1)}$; consequently $(-u)\mu$ detects $m^{n/(n-1)}$, hence the wave's $2n/(n-1)$ power.

Proposition 2.2, (2.15), Lemma 4.1 · pp. 9–10, 14 Section 4.2, Theorem A.1 · pp. 15, 38–41

2. Cancelling the indefinite determinant, retaining only boundary flux

The Laplacian of $X_0=\int_{\Omega_z}(-u)\mu$ contains a full determinant of uncontrolled sign. Correct it by

Integration by parts with the two conservation laws, for the current and the Hessian cofactor, cancels that determinant. The null line makes the cofactor's normal–normal coefficient vanish, including on the moving boundary. Proposition 4.2 identifies $\Delta F+\int A:\Sigma$ with the wave's boundary flux, with $A,\Sigma\ge0$. Lemma 4.3 gives

so the correction retains the required quantity.

Proposition 4.2, Lemma 4.3 · pp. 16–18

3. Comparing decay at the wall with recovery in the interior

Write $z=(T,\tau)\in\mathbb R\times\mathbb R^2$ and $a=Kh$. The flux-carrying wall stays a fixed angular distance from the cap, giving Gaussian decay $e^{-c_0|\tau|/h}$. Inverting the interior multiplier costs $e^{C\gamma^2|\tau|/h}$. Choose the aperture $\gamma$ small first, then the fixed $K$ large, so the wall decay dominates. The comparison function and small-ball determinant comparison yield Proposition 5.1:

The comparison uses a density lower bound throughout the small ball and the ceiling $\mu^{1/(n-1)}\le\lambda$.

Proposition 5.1, (5.11)–(5.12) · pp. 18–21

4. Accounting for both logarithms and removing the period

Let $H(T,s)$ be the circular mean of $F$ at $|\tau|=s$. Lemma 6.1 with $b=2a$, $r=2b$ gives

The steps are an outward annular comparison, a mean inequality at fixed radius, and an inward comparison. A radial planar harmonic function is affine in $\log s$, so each passage costs one inverse logarithm. Exponential decay uniformly bounds the weighted integral of the positive potential. Cancel the factors $a^2$ and take $p_c$th roots to obtain Proposition 6.3. Sum a fixed finite family of caps. Outside $|X|\ge3$, nonstationary phase bounds the wave kernel by $O(h^M(1+|X|)^{-M})$. Local periodization on unit cubes and bounded overlap then transfer the estimate to $\mathbb R^n$.

Lemma 6.1, Proposition 6.3 · pp. 21–23 Section 7.1 · p. 24

1. Recovering local extension near a focus

The time average $M_Nf=\int\psi(T)e^{iNT}U(T)\chi(|D|/N)f\,dT$ selects a frequency shell of width one. Its leading kernel is $cN^{(n-1)/2}|Z|^{-(n-1)/2}e^{iN|Z|}\psi(|Z|)$. Insert $f_N(r\omega)=e^{-iNr}\psi(r)g(\omega)$ and linearize the distance phase at $X/N$ near the focus. Taking $N$ to be a sufficiently large fixed power of $R$ bounds the error, while $n/p_c=(n-1)/2$ cancels the scaling factor. Only $\log N\asymp\log R$ remains.

Lemma 9.2 · p. 28

2. Randomizing caps and dualizing the maximal function

Modulated disjoint $\delta$-caps have extension bounded below on tubes of length $\delta^{-2}$ and width $\delta^{-1}$. Independent signs turn the expected square into the sum of the squared cap extensions, which bounds a weighted sum of tube indicators from below. Combining this with local extension gives Lemma 9.4's overlap estimate at $q=p_c/2=n/(n-1)$. Grouping tube centers into cubes keeps the constant uniform for arbitrary locations. A $\delta$-net of directions, $L^n$–$L^q$ duality and the supremum over centers give the Kakeya maximal estimate.

Lemma 9.4, Section 9.3 · pp. 29–30

3. Varying-radius covers and the projective Nikodym argument

A single scale only gives $|F_\delta|\gtrsim[\log(1/\delta)]^{-2(n-1)}$. Hausdorff dimension requires grouping a cover by balls of different radii. Section 9.3 obtains infinite Hausdorff measure for the gauge $r^n[\log(1/r)]^{2(n-1)}[\log\log(1/r)]^\gamma$, $\gamma>n-1$, hence full dimension.

In Section 9.4, $N_\delta f(x)$ is the supremum of tube averages over unit segments through $x$, and

A projective map sends portions a fixed distance from the base point into tubes whose direction records that base point; its Jacobian is locally bounded above and below. Pieces near the base point are summed by dyadic distance $s$ with weight $O(s)$. The same covering argument gives dimension $n$ for a Borel set meeting a suitable unit segment through every point of a positive-measure set in uniformly positive length. No converse implication from dimension alone to a maximal estimate is asserted.

Sections 9.3–9.4, (9.6)–(9.7) · pp. 30–32

1. Removing the loss and interpolating exponents

The inequality $[\log(2+R)]^A\lesssim_\varepsilon R^\varepsilon$ and duality give local $L^{p_c'}\to L^{p_c'}$ restriction. Since $1<p_c'<2$, [Tao99] applies. Duality gives $E:L^s\to L^s$ for every $s>p_c$. When $a\ge b$, use the finite measure of the sphere. When $a<b$, set $s=1+b/a'$ and interpolate with $L^1\to L^\infty$; the hypothesis $b>(n+1)a'/(n-1)$ guarantees $s>p_c$.

Section 9.1, proof of Corollary 9.3 · p. 29 [Tao99, Theorem 1.2, p. 2]

2. Integrating the cone slices

For $G_\rho(\omega)=G(\rho\omega,\rho)$, $E_\Sigma G=\sqrt2\int_1^2e^{it\rho}\rho^{n-1}E(G_\rho)(\rho x)\,d\rho$. Apply Hausdorff–Young in time, Minkowski using $b\ge b'$, and spatial dilation. The final $L^{b'}_\rho$ norm is controlled by $L^a_\rho$ because $a>b'$, which follows from the stated hypotheses. For $b=\infty$, Hölder directly bounds the defining integrals on both compact surfaces.

Section 9.1, proof of Corollary 9.3 · p. 29

1. The time average costs an inverse power of shell thickness

For $\sigma>s_+(p)$, $2\le p<\infty$ and $g$ in a fixed frequency annulus, $\|U(t)g\|_{L^p(\mathbb R^n\times\{R\le|t|\le2R\})}\lesssim R^{1/p+\sigma}\|g\|_p$. Fourier inversion writes $A_{\delta,r}=\psi((1-|D|/r)/\delta)$ as $\delta r\int e^{itr}\widehat\psi(\delta rt)U(t)g\,dt/(2\pi)$. Time Hölder and the long-time estimate bound the supremum on $r\in[1,2]$ by $O(\delta^{-\sigma})$. Dyadic radius octaves and Littlewood–Paley for $p\ge2$ give the supremum over all radii.

Lemma 10.2, (10.3) · pp. 32–33

2. Strict order and three extensions

Choose $s_+(p)<\sigma<\alpha$. Splitting the multiplier near its boundary into shells of thickness $\delta=2^{-j}$ costs $\sum_j\delta^{\alpha-\sigma}$; the smooth remainder is controlled by the Hardy–Littlewood maximal function. This proves the maximal bound and the fixed-radius bound for $p\ge2$. Self-adjointness gives the latter for $1<p<2$; the integrable Bessel kernel gives $p=1,\infty$. Convergence for Schwartz functions with compact Fourier support and density give $L^p$ convergence. In the maximal range $2\le p<\infty$, almost-everywhere approximation uniform in $r$ gives pointwise convergence.

Proof of Theorem 10.1 · p. 33

1. Three powers: differentiation, the square norm and smoothing

On a shell of thickness $\delta$, radial differentiation costs $\delta^{\alpha-1}$, the square integral restores $\delta^{1/2}$, and the wave costs $\delta^{-\sigma}$. Lemma 11.2 combines Plancherel in the radius variable with the long-time wave estimate. For $H=L^2([1,2],ds)$, the shell operator has $L^p\to L^p(H)$ norm $O(\delta^{1/2-\sigma})$ and $L^2\to L^2(H)$ norm $O(\delta^{1/2})$.

Lemma 11.2 · p. 34

2. Combining different scales inside the square

Lemma 11.3 combines dilates under decay bounds for a Hilbert-valued kernel and its first derivative. On a test cube, only $O(1+\log(1/\delta))$ intermediate scales use the $L^p$ bound; Hölder costs $[1+\log(1/\delta)]^{1/2-1/p}$. Distant scales use a kernel derivative or $L^2$. Choosing $\sigma<\alpha-1/2$ leaves $\delta^{\alpha-1/2-\sigma}$, absorbing this logarithm. The multiplier away from the boundary vanishes quadratically at zero, giving another summable series. For $p=2$, directly $\|G^\alpha f\|_2^2=\alpha(2\alpha-1)^{-1}\|f\|_2^2$. Fatou and monotone convergence pass from finite sums to the full operator, and density extends the linear map into $L^p_x(L^2(dr/r))$.

Lemma 11.3, proof of Theorem 11.1 · pp. 35–36

1. The interior range of paraboloid extension

Parametrize the cap by $(\delta v,\sqrt{1-\delta^2|v|^2})$ and rescale physical variables to $(x/\delta,-2t/\delta^2)$. The new phase is $q_\delta(v)=2(1-\sqrt{1-\delta^2|v|^2})/\delta^2=|v|^2+O(\delta^2)$. At $p_c=2(d+1)/d$, the Jacobian and input powers cancel. Taking $\delta$ to be a fixed negative power of $R$ preserves $[\log(2+R)]^{2/p_c}$ in the local estimate (12.4). For [Kim17]'s compact-surface epsilon removal, stationary phase gives Fourier decay of a smooth density on a slightly larger patch. Duality and interpolation then give the interior range in (12.1).

Proof of Corollary 12.1, (12.4) · pp. 36–37 [Kim17, Theorem 5.3, p. 10]

2. The critical Schrödinger estimate retains a logarithm on a frequency ball

Localize frequency-ball data to spatial cubes of side $N$. Treat distant input by nonstationary phase for $S(t)P_N$. For the nearby input $h=f1_{30Q}$, the Fresnel kernel expresses the wave through $\mathcal E[g](-\lambda x/(2t),\lambda^2/(4t))$, where $g(v)=h(\lambda v)$ and $\lambda\asymp N$. Keep the multiplier outside, as $P_NS(t)h$, because the spatial truncation creates high frequencies. On $t\in[1/2,1]$, the Jacobian and input norm leave $\lambda^{d-(2d+2)/p_c}=1$, so only the logarithm in (12.4) remains. Bounded overlap of cubes, dyadic time intervals and conjugation give any compact $I$.

Proof of (12.2) · pp. 37–38

3. Exact mixed-norm order strictly above the critical exponent

Choose $p_c<a<p$. Equation (12.1) gives $\mathcal E:L^a\to L^a$. Apply [LRS13, Corollary 1.2] with $p_0=q_0=a$ and both spatial exponents equal to $p$. This gives $B^s_{p,p}\to L^p_xL^r_t$ at exactly $s=d(1-2/p)-2/r$ for $p\le r\le\infty$. Littlewood–Paley and $\ell^p\le\ell^2$ for $p>2$ give the Sobolev input bound. The room to choose $a<p$ distinguishes this conclusion from the critical logarithmic estimate.

Proof of (12.3) · p. 38 [LRS13, Corollary 1.2, p. 3]