Proof. Choose a smooth cutoff \(\zeta_0\) equal to one on \(B(0,\rho)\), supported in \(B(0,2\rho)\), and identically \(1/2\) on a neighborhood \(V\) of a point \(\xi_1\) with \(\rho<|\xi_1|<2\rho\). It can be chosen by rescaling a fixed cutoff with an annular plateau. The phase \[q=Q_s+\zeta_0(q_s-Q_s)\] has Hessian \(I+O(\rho^2)\): the remainder \(q_s-Q_s\) and its derivatives of orders zero, one, and two are respectively \(O(\rho^4)\), \(O(\rho^3)\), and \(O(\rho^2)\) on the support of the cutoff. Fix \(\rho\) small enough that \(q''\) is positive definite everywhere.

After translating the target base point, let its height function be \(h_0\) near zero and put \(H=h_0''(0)>0\) and \(B=q''(\xi_1)>0\). Choose an invertible real matrix \(A\) with \(A^{\mathsf T}HA=B\); for example, \(A=H^{-1/2}B^{1/2}\). Define, near \(\xi_1\), \[\begin{align*} \psi(\xi) &=h_0\bigl(A(\xi-\xi_1)\bigr)+q(\xi_1)-h_0(0)\\ &\quad+ \bigl(\nabla q(\xi_1)-A^{\mathsf T}\nabla h_0(0)\bigr) \cdot(\xi-\xi_1). \end{align*}\] Thus \(\psi\) and \(q\) have the same value, gradient, and Hessian at \(\xi_1\). This change of graph is induced by an invertible affine map of ambient frequency space: its base part is \(x\mapsto \xi_1+A^{-1}x\), and its height part is addition of an affine function of the new base coordinates.

Let \(\vartheta\in C_c^\infty(B(0,2))\) equal one on \(B(0,1)\) and take \(0\le\vartheta\le1\). For a small fixed \(\epsilon>0\) put \[R_\epsilon(\xi)= \vartheta\bigl((\xi-\xi_1)/\epsilon\bigr) \bigl(\psi(\xi)-q(\xi)\bigr).\] Choose \(\epsilon\) so that its support is contained in \(V\) and the target graph is defined there. Taylor’s theorem and the agreement of the two-jets give \[\|D^jR_\epsilon\|_\infty\le C\epsilon^{3-j}, \qquad j=0,1,2.\] Consequently \(\phi=q+R_\epsilon\) remains uniformly convex for sufficiently small \(\epsilon\) and agrees with \(\psi\) on \(B(\xi_1,\epsilon)\). It equals \(Q_s\) outside a compact set, so all of its derivatives of order at least two are bounded.

To verify the required continuation form, observe that \(D=q_s-Q_s\) is strictly positive for \(0<|\xi|<1\). Shrink \(V\), if necessary, so that \(D\ge d_0>0\) there. The function \(R_\epsilon/D\), extended by zero outside \(V\), is smooth. Set \[\widetilde\zeta=\zeta_0+R_\epsilon/D.\] On the perturbation support, \(\zeta_0=1/2\) and \(|R_\epsilon/D|\le C\epsilon^3/d_0\). Choosing \(\epsilon\) smaller makes this last quantity at most \(1/4\). Hence \(0\le\widetilde\zeta\le1\), with the same inner and outer support properties as \(\zeta_0\), and \(\phi=Q_s+\widetilde\zeta(q_s-Q_s)\). Every choice was made using only the fixed target graph. ◻

Proof. Since \(N^2w\theta=1\), the vectors \(N^2z\) run through \((D_{\mathrm f}\theta)^{-1}\mathbb Z^2\). For each fixed \(\nu\), Equation (11) is the Fourier coefficient array of \(f\chi_\theta^\nu e(N^2t_I\phi)\) on a square of side \(D_{\mathrm f}\theta\). Parseval and Fourier inversion give \[\sum_z|(\mathsf P_{m,I}f)_{\nu,z}|^2 =(D_{\mathrm f}\theta)^2\|f\chi_\theta^\nu\|_2^2\] and, after synthesis, \(f(\chi_\theta^\nu)^2\). Sum over \(\nu\) and use \(\sum_\nu(\chi_\theta^\nu)^2=1\). The norm of synthesis follows from \(\mathsf S_{m,I}=(D_{\mathrm f}\theta)^{-2}\mathsf P_{m,I}^*\). These identities first hold for finite partial Fourier sums in \(L^2\) and then for their limits. ◻

Proof. The disk of radius \(\theta\), with its two translations chosen independently, contains any prescribed index in both tests. Its weight is \(\theta^2\), so \(K_{\theta,r}(c)\ge\|c\|_\infty^2\) and \[G_{\theta,r}(c)^3 \ge w^2\|c\|_2^2\|c\|_\infty \ge w^2\sum_{\nu,z}|c_{\nu,z}|^3.\] For every atomic domination of \(c\), the triangle inequality in \(\ell^3\) now gives \[w^{2/3}\|c\|_3\le\sum_\ell G_{\theta,r}(a_\ell).\] Taking the infimum and summing its cube over bins with weight \(r\) proves Equation (16). A singleton has cost \(w^{2/3}|c_{\nu,z}|\), so the infimum is finite; the displayed lower bound makes it positive away from zero. Homogeneity, subadditivity, and absolute monotonicity follow directly from the definition, as does contraction under masks. ◻

Specialization of the cited theorem. Set \(m=N\). The scales in Equation (10) become \[\begin{array}{c|ccc} &\theta&r&w\\ \hline \text{input}&\delta&1&\delta\\ \text{output}&1&N^{-2}&N^{-2} \end{array} \qquad t_I=\frac{j}{N^2},\qquad rw^2=N^{-6}\quad\text{at output}.\] Thus the companion’s synthesis and analysis operators are exactly the displayed operators, with its window constant \(L_{\mathrm f}\) equal to our \(D_{\mathrm f}\). The initial gauge is exactly the last expression in Equation (21), and Equation (16) gives \[N^{-6}\sum_{\nu',z',j}|c_{\nu',z',j}|^3 \le\mathcal A_N(c)^3.\] All bounded label sets fit one fixed admissible complexity range for sufficiently large \(N\), since \(m=N\); the finitely many smaller dyadic scales may be included by enlarging the constant. The bound at any such fixed scale follows from finite-dimensional boundedness and \(\mathcal K_\kappa(d)\ge\|d\|_\infty^2\). Applying Equation (18) proves Equation (21). This argument uses arbitrary input arrays, so it also applies after any restriction of their support. ◻

Proof. Let \(\lambda\) be the finite complex measure on the graph of \(\phi\) whose density in base coordinates is \(g\). For \(v\in C_c^\infty(\mathbb R^3)\), Cauchy–Schwarz gives \[\begin{align*} \left|\int v\,d(\lambda*\lambda)\right| &\leq \|g\|_2^2 \left(\iint_{\mathbb R^2\times\mathbb R^2} |v(\xi+\xi',\phi(\xi)+\phi(\xi'))|^2\,d\xi\,d\xi'\right)^{1/2}. \end{align*}\] Put \(m=\xi+\xi'\) and \(w=(\xi-\xi')/2\); this linear change has absolute Jacobian one. Write \(w=r\omega\), where \(r>0\) and \(\omega\in S^1\), and set \[H_{m,\omega}(r)=\phi(m/2+r\omega)+\phi(m/2-r\omega).\] Uniform convexity implies \[H_{m,\omega}'(r) =\int_{-r}^{r}\omega^{\mathsf T}\phi''(m/2+s\omega) \omega\,ds \geq2cr.\] Thus \(H_{m,\omega}\) is strictly increasing for \(r>0\), and the substitution \(s=H_{m,\omega}(r)\) yields \[\begin{align*} &\iint |v(\xi+\xi',\phi(\xi)+\phi(\xi'))|^2\,d\xi\,d\xi' \\ &\quad=\int_{\mathbb R^2}\int_{S^1}\int_0^\infty |v(m,H_{m,\omega}(r))|^2r\,dr\,d\omega\,dm \leq\frac{\pi}{c}\|v\|_2^2. \end{align*}\] The omitted set \(r=0\) has measure zero. Consequently \(\lambda*\lambda\) has an \(L^2\) density with norm at most \((\pi/c)^{1/2}\|g\|_2^2\), by the \(L^2\) representation theorem. Its Fourier transform is \(G(-\,\cdot)^2\) in the sense of tempered distributions. Plancherel therefore gives Equation (46); this reasoning applies to the stated rough data and does not assume the convolution has a density in advance.

For \(g=h\chi_1^{\nu_*}\), compact support and Hölder’s inequality bound \(\|g\|_2\) by a fixed multiple of \(\|h\|_3\). Furthermore \[\|F\|_\infty\leq\|g\|_1,\qquad \|\nabla F\|_\infty \leq2\pi\int |(\xi,\phi(\xi))|\,|g(\xi)|\,d\xi.\] Differentiation under the integral is valid because the integrands are supported in a fixed compact set. Another application of Hölder’s inequality proves Equation (47). ◻

Proof. First take any finite collection \(\mathcal Q\) of occupied cubes and choose \(x_Q\in Q\cap A_\mu\) for each of them. By Equation (47), on \(B(x_Q,r_\mu)\), where \(r_\mu=\mu/(2C_{\mathrm b})\leq1/2\), one has \(|F|>\mu/2\). These balls have overlap at most \(125\): if a point belongs to one of them, its center lies within distance one, and at most \(5^3\) unit lattice cubes can contain such centers. Separation of the selected points is not required. Therefore \[\#\mathcal Q\,\frac{\pi}{96C_{\mathrm b}^3}\mu^7 \leq\sum_{Q\in\mathcal Q}\int_{B(x_Q,r_\mu)}|F|^4 \leq125\|F\|_4^4\leq125C_{\mathrm b}^4.\] For example, one may take \(C_X=\max\{1,12000C_{\mathrm b}^7/\pi\}\). This bound holds for every finite collection of occupied cubes. An infinite collection would have finite subsets of arbitrarily large cardinality, a contradiction. Thus the full collection is finite with the same bound. Finally \(C_X\mu^{-7}\leq\mu^{-8}\) when \(\mu\leq C_X^{-1}\). ◻

Proof. If \(X\) is empty there is nothing to cover. Otherwise, using open balls throughout, set \[n_j(x)=\#\bigl(X\cap B(x,R_j)\bigr),\qquad x\in X.\] For each \(x\) there is an index \(0\leq j<J\) with \[ n_{j+1}(x)\leq\mu^{-\beta}n_j(x). \tag{52}\] Indeed, failure at every index would imply \(n_J(x)>\mu^{-\beta J}n_0(x)\geq\mu^{-\beta J}>\mu^{-8}\), since \(\beta J>8\), contrary to \(\#X\leq\mu^{-8}\). Assign each \(x\) one such index \(j\) and the unique integer \(k\geq0\) for which \[2^k\leq n_j(x)<2^{k+1}.\] There are at most \(J(1+8\log_2(1/\mu))\) nonempty groups with a common pair \((j,k)\).

In one group, choose a maximal pairwise disjoint collection of the open balls \(B(x,R_j)\), and denote its centers by \(Y\). The collection is finite. Each unselected ball intersects a selected one, so every center in the group is within distance \(2R_j\) of a point of \(Y\). A point of the unit cube containing that center is within a further distance \(\sqrt3\). Since \(R_j\geq\mu^{-100}\), the balls \(\{B(y,4R_j):y\in Y\}\) cover all cubes assigned to the group.

Make a finite graph on \(Y\), joining distinct \(x,y\) when \(|x-y|<R_{j+1}/2\). For a fixed vertex \(x\), the original balls \(B(y,R_j)\) of all its neighbors are disjoint and lie in \(B(x,R_{j+1})\): Equation (48) gives \(R_j<R_{j+1}/2\). Each neighbor ball contains at least \(2^k\) points of \(X\), whereas Equation (52) gives \[n_{j+1}(x)<\mu^{-\beta}2^{k+1}.\] The degree of \(x\) is therefore less than \(2\mu^{-\beta}\). A greedy coloring uses at most \(3\mu^{-\beta}\) colors. Centers of the same color are separated by at least \(R_{j+1}/2\), and \[\frac{R_{j+1}}2 =\frac{\mu^{-100}}2R_j \geq\mu^{-40}(4R_j)\] by Equation (48). Each color thus supplies a family satisfying the separation condition, with at most \(\#X\leq\mu^{-8}\) balls and \(R=4R_j\geq\mu^{-100}\).

Multiplying the group and color counts proves Equation (50). A dyadic \(N\) with \(4R\leq N^2<16R\) exists, and satisfies \[N<4\sqrt R=8\mu^{-50(j+1)}\leq8\mu^{-50J}.\] The argument uses neither the absolute locations of the points nor the diameter of \(X\). In particular, different clusters may be arbitrarily far apart. The use of open balls also resolves boundary cases: unselected balls meet selected balls by maximality, and equality in the graph’s distance threshold leaves the desired weak separation inequality intact. ◻

Proof. By homogeneity it suffices to take \(\|h\|_3\leq1\). In Proposition 10 choose \(\kappa=\eta=s=\theta\). These choices lie in the permitted parameter ranges, and \[\gamma_0=10\kappa+\eta+\tfrac32s=\tfrac{25}{2}\theta.\] For \(J\) in Equation (49), we have \[ J\leq\frac{34}{\tau},\qquad 50J\gamma_0\leq\frac{17}{800}\tau<\frac{\tau}{4}. \tag{55}\] Consequently the scale in every family furnished by Lemma 13 satisfies \[N^{\gamma_0}\leq8^{\gamma_0}\mu^{-\tau/4} \leq2\mu^{-\tau/4},\qquad \log(2+N)\leq53J\bigl(1+\log(1/\mu)\bigr).\] Applying Proposition 10 to each family and summing over the cover yields, for \(0<\mu\leq\mu_0\), \[ |A_\mu|\leq C P(\theta,\theta)M(\theta)^{3/2} J^3\bigl(1+\log(1/\mu)\bigr)^3\mu^{-3-\tau/2}. \tag{56}\] Here the colors cost \(\mu^{-\beta}=\mu^{-\tau/4}\), while the scale factor costs at most \(\mu^{-\tau/4}\). The group count contributes one factor of \(J(1+\log(1/\mu))\), and the logarithm squared in Proposition 10 contributes the other two. All constants and \(\mu_0\) are independent of \(\tau\).

The layer-cake identity, valid by Tonelli even before finiteness of the norm is known, is \[\|F\|_p^p=p\int_0^\infty\mu^{p-1}|A_\mu|\,d\mu.\] For its portion below \(\mu_0\), Equation (56) and \(p=3+\tau\) reduce the needed integral to \[\begin{align*} \int_0^1\mu^{-1+\tau/2}\bigl(1+\log(1/\mu)\bigr)^3\,d\mu &=\frac2\tau+\frac{12}{\tau^2} +\frac{48}{\tau^3}+\frac{96}{\tau^4} \leq158\tau^{-4}. \tag{57}\end{align*}\] For example, the substitution \(u=\log(1/\mu)\) turns the integral into \(\int_0^\infty e^{-\tau u/2}(1+u)^3\,du\), which gives the displayed identity by integrating each monomial. Since \(p\leq4\) and \(J^3\leq34^3\tau^{-3}\), the small-level contribution is at most \(C P(\theta,\theta)M(\theta)^{3/2}\tau^{-7}\).

For the remaining levels, again by Tonelli, \[\begin{align*} p\int_{\mu_0}^\infty\mu^{p-1}|A_\mu|\,d\mu &\leq\int_{\{|F|>\mu_0\}}|F|^p \leq\mu_0^{p-4}\|F\|_4^4 \leq\mu_0^{-1}C_{\mathrm b}^4. \end{align*}\] This is uniform for \(3<p\leq4\) and is absorbed into the previous bound because \(P,M\geq1\) and \(\tau\leq1\). Taking the \(p\)th root and restoring homogeneity proves Equation (54). If \(p>4\), directly use \[\|F\|_p^p\leq\|F\|_\infty^{p-4}\|F\|_4^4 \leq C_{\mathrm b}^p\|h\|_3^p.\] ◻

Proof. Choose a nonempty interior chart; a nonempty smooth surface with boundary also has such a chart. Write it in the affine graph form \[\Psi(\xi)=v_0+L(\xi,\phi(\xi)),\qquad \xi\in D,\] where \(L\) is invertible, and write \(d\sigma=J_\Sigma(\xi)\,d\xi\) on this chart. Take a fixed nonzero \(g\in C_c^\infty(D)\) and define \(f(\Psi(\xi))=g(\xi)/J_\Sigma(\xi)\) on the chart, with \(f=0\) elsewhere. This is a smooth surface datum with \[B_f:=\sup_{3\leq p\leq4}\|f\|_{L^p(\Sigma)}<\infty.\] Let \[G(y',t)=\int g(\xi)e\big(y'\cdot\xi+t\phi(\xi)\big)\,d\xi, \qquad A=\|g\|_2^2>0.\] Horizontal Plancherel and its derivative identity give, for \(t\geq0\), \[\begin{align*} \int_{\mathbb R^2}|G(y',t)|^2\,dy'&=A,\\ \int_{\mathbb R^2}|y'|^2|G(y',t)|^2\,dy' &=\frac1{(2\pi)^2} \left\|\nabla\big(g e(t\phi)\big)\right\|_2^2 \leq B_g(1+t)^2, \end{align*}\] where one may take \(B_g=2(2\pi)^{-2}\|\nabla g\|_2^2+2\|g\nabla\phi\|_2^2>0\). Set \(K=\max\{1,(2B_g/A)^{1/2}\}\). The integral outside the disk \(|y'|<K(1+t)\) is at most \(B_g/K^2\leq A/2\), so at least \(A/2\) of the squared norm lies inside that disk. Hölder’s inequality on the disk implies \[ \int_{\mathbb R^2}|G(y',t)|^p\,dy' \geq (A/2)^{p/2}(\pi K^2)^{1-p/2}(1+t)^{2-p} \geq c_g(1+t)^{2-p}, \tag{59}\] uniformly for \(3\leq p\leq4\), with the explicit positive choice \[c_g=\min\left\{ (A/2)^{3/2}(\pi K^2)^{-1/2}, (A/2)^2(\pi K^2)^{-1} \right\}.\] Integrating Equation (59) over \(t\geq0\) yields \(\|G\|_p^p\geq c_g/(p-3)\).

The affine coordinates introduce exactly the factors \[E_\Sigma f(x)=e(x\cdot v_0)G(L^{\mathsf T}x),\qquad \|E_\Sigma f\|_p^p=|\det L|^{-1}\|G\|_p^p.\] Hence, on setting \(d_g=|\det L|^{-1}c_g>0\), we obtain \[C_{\Sigma,p}^{\mathrm{best}} \geq B_f^{-1}d_g^{1/p}(p-3)^{-1/p} \geq B_f^{-1}\min\{d_g^{1/3},d_g^{1/4}\}(p-3)^{-1/p}.\] This proves Equation (58), with every choice of chart and datum fixed independently of \(p\). ◻

Proof of Corollary 2. On the compact paraboloid \(\Sigma=\{(y,|y|^2):|y|\le1\}\), write \(\,d\sigma=w(y)\,dy\), where \(w(y)=\sqrt{1+4|y|^2}\). Apply Theorem 1 to the datum \(g(y)/w(y)\) on \(\Sigma\). A fixed change of output variables then gives \[\left\|\int_{|y|\le1}g(y)e^{i(\tau|y|^2-\xi\cdot y)}\,dy\right\|_{L^p(\mathbb R^3_{\xi,\tau})} \le C_p\|g\|_{L^p(\{|y|\le1\})}.\] This is the compact-frequency extension estimate in Lee–Rogers–Seeger (Lee et al. 2013, Equation (1.3)) with both exponents equal to \(p\). Their Theorem 1.1, with spatial dimension two and Besov summation exponent two, yields \[\|e^{it\Delta}f\|_{L^p(\mathbb R^2\times[0,1])} \le C_p\|f\|_{B^p_{\alpha,2}(\mathbb R^2)}, \qquad \alpha=2(1-2/p)-2/p=2-6/p.\] For an inhomogeneous Littlewood–Paley partition \((P_k)_{k\ge0}\), the uniform annular multiplier bound \(\|P_kf\|_p\le C_s2^{-ks}\|(I-\Delta)^{s/2}f\|_p\) gives \[\|f\|_{B^p_{\alpha,2}} \le C_s\left(\sum_{k\ge0}2^{-2k(s-\alpha)}\right)^{1/2} \|f\|_{W^{s,p}}.\] The series converges for \(s>\alpha\). This proves Equation (2) on Schwartz functions, and density gives the stated bounded extension. ◻

Proof. Shrink the product neighborhood so that its input domain is a ball. Since \(M\) is a Hessian in \(y\), commuting its derivatives in \(\partial_tM=cM\) gives \[(\partial_{y_k}c)M_{ij}=(\partial_{y_i}c)M_{kj}.\] Multiplication by \((M^{-1})_{j\ell}\), summed over \(j\), yields \((\partial_{y_k}c)\delta_{i\ell} =(\partial_{y_i}c)\delta_{k\ell}\). For each \(k\), choose \(\ell=i\ne k\); hence \(\nabla_yc=0\). The input ball is connected, so \(c=c(t)\). Set \[\rho(t)=\exp\!\left(\int_0^t c(v)\,dv\right),\qquad h(y)=\partial_t\psi(0;y),\qquad S(t)=\int_0^t\rho(v)\,dv.\] The equation for \(M\) now implies \(D_y^2(\partial_t\psi-\rho h)=0\). Thus \(\partial_t\psi=\rho(t)h(y)+b(t)\cdot y+d(t)\) for smooth functions \(b,d\). Integrating and using \(\psi(0;y)=0\) gives \[\psi(t;y)=S(t)h(y)+B(t)\cdot y+D(t),\qquad B(t)=\int_0^t b(v)\,dv,\quad D(t)=\int_0^t d(v)\,dv.\] Since \(S'=\rho>0\), it has a smooth local inverse \(t=t(s)\). The output change \(\kappa(u,s)=(u-B(t(s)),t(s))\) therefore satisfies \[\phi(\kappa(u,s);y)=u\cdot y+s h(y)+r(s),\qquad r(s)=D(t(s)).\] Here \(D_y^2h=M(0,y)\) is definite. No input coordinate change is needed.

Put \(z=(u,s)\) and \(A(z,y)=a(\kappa(z);y)\). Extend this fixed amplitude smoothly by zero, and periodically in \(z\) across a fixed cube \(Q\) of side \(L\) whose interior contains its output support. On \(Q\), \[A(z,y)=\sum_{k\in\mathbb Z^3}e^{2\pi i k\cdot z/L}A_k(y), \qquad \sum_k\|A_k\|_\infty<\infty.\] Indeed, integration by parts in \(z\) gives \(\|A_k\|_\infty\le C_m(1+|k|)^{-m}\) for any fixed \(m>3\). All coefficients have a common compact frequency support, and the constants are independent of \(N\). Choose a compact graph patch \(\Sigma\), parametrized by \(\omega(y)=(y,h(y))\), with smooth boundary and parameter domain containing this support, and write \(\,d\sigma=w(y)\,dy\), where \(w(y)=\sqrt{1+|\nabla h(y)|^2}\). For \[g_k(\omega(y))=A_k(y)f(y)/w(y)\] we have \[\|g_k\|_{L^p(\Sigma)}^p =\int |A_kf|^p w^{1-p}\,dy \le\|A_k\|_\infty^p\|f\|_p^p.\] The factor \(e^{iNr(s)}\) is unimodular, and each remaining Fourier term is \(e^{2\pi i k\cdot z/L}E_\Sigma g_k(Nz/(2\pi))\). Minkowski’s inequality and Theorem 1, applied globally to each term, bound their sum on \(Q\) by \[(2\pi/N)^{3/p}C_{\Sigma,p} \sum_k\|A_k\|_\infty\|f\|_p.\] Changing back by \(\kappa\) contributes only the bounded factor \(\sup|\det D\kappa|^{1/p}\) on the fixed output support. Absolute coefficient summability and the local \(L^1\) integrability of \(f\) justify the series and its interchange with the integral. This proves Equation (60), with no power loss in \(N\). ◻