# Curved boundaries and directional Fourier carriers Original exposition and proofs: GPT-6.1 Sol (OpenAI), Ultra. CC0 1.0. A high real frequency singles out the points where its direction is normal to a surface. Positive curvature makes these stationary points nondegenerate. Each point contributes a linear logarithmic Fourier profile, and the two points together determine a segment. We prove this description with uniform estimates, then show how the same localization gives a directional wavefront bound for arbitrary compact distributions. Basic references are Guillemin and Sternberg's *Semi-classical Analysis* and Hörmander's *The Analysis of Linear Partial Differential Operators I* and *II*. The proof uses the available full expansion, Theorem 5.1 of Stationary phase and critical manifolds, and its Proposition 2.1. The profile prerequisites are Joint logarithmic-frequency limits, Local compactness and Hartogs bounds, and the exact joint maximum law, Theorem 6.1 of Isolated atoms and separated singularities. Fourier inversion, compact-distribution growth and multiplication are those of Compact Fourier division and multiplicity-sensitive annihilators. We assume multivariable calculus, smooth partitions of unity and the inverse function theorem, as in the stationary-phase prerequisite. ## 1. What a direction can retain For a compact distribution \(u\) on \(\mathbb R^n\), use \[ F_u(\zeta)=\langle u,e^{-ix\cdot\zeta}\rangle,\qquad L_u(z,c)=\frac{\log|F_u(c+z\log|c|)|}{\log|c|}, \quad c\in\mathbb R^n,\quad |c|>2. \tag{1.1} \] Along an escaping sequence the PSH compactness alternative gives a proper local \(L^1\) limit \(U\), or locally uniform convergence to \(-\infty\). A proper limit has horizontal envelope and recession indicator \[ M_U(\eta)=\sup_{x\in\mathbb R^n}U(x+i\eta),\qquad h_U(\eta)=\lim_{t\to\infty}\frac{M_U(t\eta)}t. \tag{1.2} \] The linked horizontal-envelope proof in Joint logarithmic-frequency limits makes \(h_U\) the support function of a nonempty compact convex set. For a collapsed profile set \(h_U=-\infty\), with empty carrier. Write \(\mathcal J(u)\) for the resulting indicator family, and \(\mathcal J_\theta(u)\) when the witnessing centers additionally satisfy \(c/|c|\to\theta\). For a compact set \(E\), write \[ H_E(\eta)=\sup_{x\in E}x\cdot\eta,\qquad H_\varnothing=-\infty. \tag{1.3} \] Convexification does not change this function. A proper profile of a distribution supported in \(E\) has \(h_U\leq H_E\), by the compact Fourier growth estimate and the full profile support theorem. A proper constant profile has indicator zero and carrier \(\{0\}\); it is different from a collapsed profile. Finite joint profiles always use the same escaping centers. We shall use the following exact preceding result: if two compact distributions have disjoint singular supports, then every joint indicator triple for their sum and its two summands satisfies \(h=\max(h_1,h_2)\). Every prescribed coordinate extends to a finite joint extraction. This is Theorem 6.1 of the linked lesson, including its proper and collapsed cases. It is a statement about indicators, rather than a pointwise formula for the logarithm of a sum. ## 2. Rapid real decay survives a logarithmic complex shift An open cone here is invariant under multiplication by positive real scalars. Saying that \(F_v\) is rapidly decreasing in a cone \(\Gamma\) means that for every integer \(N\geq0\), \[ |F_v(\xi)|\leq C_N(1+|\xi|)^{-N},\qquad \xi\in\Gamma. \tag{2.1} \] **Lemma 2.1 (conic localization and complex windows).** Let \(v\) be a compact distribution satisfying (2.1). Multiplication by any compact smooth function preserves (2.1) on every cone whose unit directions have closure inside \(\Gamma\). Furthermore, for every compact set of such directions, every \(M,N\), and all sufficiently large \(q\), \[ \sup_{\substack{|z|\leq M\\ \theta\text{ in that set}}} |F_v(q\theta+z\log q)|\leq C_{M,N}q^{-N}. \tag{2.2} \] *Proof.* Fix \(\chi_0\in C_c^\infty\) equal to one near \(\operatorname{supp}v\), supported in \(\{|x|\leq R\}\). For \(z=\alpha+i\beta\), put \(\ell=\log q\), \(b=q\theta+\ell\alpha\), and \[ g_{q,\beta}(x)=\chi_0(x)e^{\ell\beta\cdot x}. \tag{2.3} \] The Fourier multiplication identity gives \[ F_v(b+i\ell\beta) =F_{g_{q,\beta}v}(b) =(2\pi)^{-n}\int_{\mathbb R^n} F_{g_{q,\beta}}(t)F_v(b-t)\,dt. \tag{2.4} \] One way to justify this identity is to insert the absolutely convergent Fourier inversion of \(g_{q,\beta}\) into the distribution pairing. The derivatives of that inversion converge uniformly through the finite order of \(v\), since its Fourier transform is Schwartz. The polynomial real growth of \(F_v\) makes the displayed integral absolutely convergent. For each integer \(k\), differentiation of (2.3) and integration by parts yield, uniformly for \(|\beta|\leq M\), \[ |F_{g_{q,\beta}}(t)| \leq C_{k,M}q^{RM}(1+\ell)^k(1+|t|)^{-k}. \tag{2.5} \] In detail, powers of each coordinate \(t_j\) are Fourier transforms of derivatives of \(g_{q,\beta}\). The product rule bounds those derivatives in \(L^1\) by \(Cq^{RM}(1+\ell)^k\). Summing the coordinate bounds and the undifferentiated bound gives (2.5). Choose \(\varepsilon>0\) so small that for the given compact direction set, \(|\alpha|\leq M\), and large \(q\), all \(b-t\) with \(|t|\leq\varepsilon q\) lie in \(\Gamma\) and satisfy \(|b-t|\geq q/2\). This follows from positive distance on the unit sphere to the complement of \(\Gamma\), and \(\ell M/q\to0\). On this part of the integral, (2.1) with exponent \(L\) and (2.5) with one fixed \(k>n\) give \[ \int_{|t|\leq\varepsilon q} |F_{g_{q,\beta}}(t)F_v(b-t)|\,dt \leq C_{L,M}q^{RM-L}(1+\ell)^k. \tag{2.6} \] On the complementary part, compact-distribution growth gives \(|F_v(b-t)|\leq C(1+q+|t|)^m\leq C_\varepsilon(1+|t|)^m\). Take \(k>m+n\) in (2.5). Integration in polar coordinates gives the bound \[ C_{k,M}q^{RM+m+n-k}(1+\ell)^k. \tag{2.7} \] For any desired \(N\), first take \(L>RM+N+1\) and \(k>RM+m+n+N+1\). The corresponding logarithmic factors are \(o(q)\). Equations (2.6) and (2.7) prove (2.2). For multiplication by a fixed \(\chi\), use \(F_{\chi v}=(2\pi)^{-n}F_\chi*F_v\). The same two parts of the integral work with the Schwartz estimates for \(F_\chi\), with no factor \(q^{RM}\). This proves preservation of real conic rapid decay. \(\square\) **Lemma 2.2 (a rapid remainder preserves the profile).** Suppose \(a,b\) are compact distributions and, along given centers \(c_j\), \(q_j=|c_j|\to\infty\), their Fourier difference satisfies \[ \sup_{|z|\leq M}|F_a(c_j+z\log q_j)-F_b(c_j+z\log q_j)| \leq C_{M,N}q_j^{-N} \tag{2.8} \] for all \(M,N\). Their proper limiting profiles and their collapsed limits on this sequence are the same, with canonical PSH representatives. *Proof.* If one profile collapses uniformly locally, its Fourier transform is bounded by every prescribed \(q_j^{-N}\) on a fixed parameter compact set on a tail. The triangle inequality and (2.8) give the same assertion for the other transform. Thus collapse is equivalent. Suppose a subsequence for \(a\) has proper local \(L^1\) limit \(U\). The locally uniform upper bounds from compact-distribution growth allow a further extraction for \(b\). It cannot collapse, by the preceding argument and the finite integral of \(U\) on a ball. Let its proper limit be \(V\). A further subsequence converges almost everywhere in both coordinates, on an exhaustion by parameter balls. At almost every point \(z\), \(U(z)\) is finite and \[ |F_a(c_j+z\log q_j)|=q_j^{\,U(z)+o(1)}. \tag{2.9} \] Choose an integer \(N>-U(z)+2\). Equation (2.8) makes the Fourier difference divided by \(F_a\) tend to zero. Hence \(F_b/F_a\to1\) at this point, and the difference of the two normalized logarithms tends to zero. Therefore \(V=U\) almost everywhere. Equality of their canonical PSH representatives follows from their common local integral averages. Every extraction for \(b\) has the same further limit; the compactness alternative therefore gives local \(L^1\) convergence to \(U\) on the original proper subsequence. Interchanging \(a,b\) proves the converse. \(\square\) ## 3. Two normal points on a positively curved boundary Let \(X\subset\mathbb R^n\), \(n\geq2\), be nonempty, bounded, open and convex, and let \(\Sigma=\partial X\) be smooth. Use the outward unit normal \(\nu\), with the second fundamental form defined by \(d\nu\) on tangent vectors, so it is positive on a round sphere. Assume this form is positive definite at every point. Write \(d=n-1\) and \(\kappa(x)=\det d\nu(x)>0\). **Lemma 3.1 (the normal points and their Hessians).** For each \(\theta\in S^{n-1}\) there are exactly two stationary points of \(x\mapsto-\theta\cdot x\) on \(\Sigma\): \[ p_+(\theta)=\nu^{-1}(\theta),\qquad p_-(\theta)=\nu^{-1}(-\theta). \tag{3.1} \] The maps \(p_\pm\) are smooth. They are distinct, and the Hessian signatures of the displayed phase are \(+d\) at \(p_+\) and \(-d\) at \(p_-\). In orthonormal tangent coordinates its absolute determinant is \(\kappa(p_\pm)\). *Proof.* A linear functional \(\theta\cdot x\) has a maximum and a minimum on the compact convex body \(\overline X\). Both lie on its boundary. A tangent derivative at either extremum vanishes, and the outward orientation gives normals \(\theta\) at a maximum and \(-\theta\) at a minimum. At every smooth point of a convex body's boundary the outward tangent hyperplane supports the body. To verify this, its local interior lies on the negative side of the normal coordinate, up to the smooth boundary graph. If some point of the body lay strictly on the positive side of the tangent plane, the segment from the boundary point toward an interior point sufficiently close to that point would enter the local interior with a strictly positive first normal derivative. The local graph has zero first derivative there, which is a contradiction. Thus every point with normal \(\theta\) maximizes the functional. Two distinct maximizers would give a boundary line segment in their supporting face. At an interior point of that segment the tangent vector in the segment direction is nonzero, and differentiating the constant normal component of its straight parametrization gives zero second fundamental form in that direction. Positive definiteness forbids this. The minimizer is unique by the same argument. The normal map is therefore bijective. Its differential is invertible by the curvature assumption, and the inverse function theorem makes its inverse smooth locally and hence globally. Distinct maximum and minimum points follow because \(X\) has nonempty interior. At a point \(p\), choose orthonormal tangent coordinates and write the surface as \(p+t+g(t)\nu(p)\), with \(g(0)=0\) and \(Dg(0)=0\). Differentiating the unit normal at zero gives \(d\nu=-g''(0)\). At \(p_+\), the phase Hessian is \(-g''(0)=d\nu\), while at \(p_-\) it is \(g''(0)=-d\nu\). The signatures and determinants follow. The area density at \(t=0\) is one. Compact parameter neighborhoods give uniform inverse-Hessian and derivative bounds. \(\square\) Set \[ D_\theta=[p_-(\theta),p_+(\theta)],\qquad H_{D_\theta}(\eta)= \max\{p_-(\theta)\cdot\eta,p_+(\theta)\cdot\eta\}. \tag{3.2} \] The segment joins points with opposite normals. Its direction need not be parallel to \(\theta\). Also \(D_{-\theta}=D_\theta\). ## 4. A single cap in a complex logarithmic window Let \(a\in C^\infty(\Sigma)\) be strictly positive and \(u=a\,dS\). Fix \(\theta_0\). Choose two disjoint smooth cap cutoffs \(\chi_+,\chi_-\), equal to one near \(p_+(\theta_0),p_-(\theta_0)\), and supported in disjoint small surface charts. Shrink a neighborhood \(V\) of \(\theta_0\) so that these cutoffs remain one near \(p_\pm(\theta)\) for every \(\theta\in\overline V\). Put \(u_\pm=\chi_\pm a\,dS\). **Theorem 4.1 (uniform cap asymptotic).** For each fixed \(M\), uniformly for \(\theta\in\overline V\), \(|z|\leq M\), as \(q\to\infty\), \[ F_{u_\pm}(q\theta+z\log q)= q^{-d/2}C_\pm(\theta) e^{-iq\theta\cdot p_\pm(\theta)} e^{-i(\log q)z\cdot p_\pm(\theta)} \left(1+O_M\!\left(\frac{(1+\log q)^2}{q}\right)\right), \tag{4.1} \] where \[ C_\pm(\theta)= (2\pi)^{d/2}e^{\pm i\pi d/4} \frac{a(p_\pm(\theta))}{\sqrt{\kappa(p_\pm(\theta))}}. \tag{4.2} \] In particular \(C_\pm\) are bounded away from zero on \(\overline V\). If \(\theta_j\to\theta_0\), then \[ L_{u_\pm}(z,q_j\theta_j)\longrightarrow -\frac d2+p_\pm(\theta_0)\cdot\operatorname{Im}z \tag{4.3} \] uniformly on each parameter compact set. *Proof.* In each fixed chart write the Fourier integral with the real phase \(\phi_\theta(x)=-\theta\cdot X(x)\) and amplitude \[ A_{q,z}(x)=\chi_\pm(X(x))a(X(x))J(x) e^{-i(\log q)z\cdot X(x)}, \tag{4.4} \] where \(J\) is the area density. All chart supports are fixed and compact. Lemma 3.1 supplies one smooth critical point, a uniformly invertible Hessian, and a uniform gradient lower bound away from a smaller critical neighborhood. Thus the full uniform real stationary-phase theorem applies for every chosen order \(N\), with \(q\) its large real parameter. Choose \(R\) bounding \(|X|\) on the chart. The product and chain rules give, for every fixed \(k\), \[ \|A_{q,z}\|_{C^k} \leq C_{k,M}q^{RM}(1+\log q)^k,\qquad |z|\leq M. \tag{4.5} \] The expansion's coefficient of order \(j\) uses only derivatives of the amplitude through order \(2j\) at the critical point. Each such derivative contains the same exponential \(e^{-i(\log q)z\cdot p_\pm(\theta)}\), multiplied by a polynomial in \((\log q)z\) of degree at most \(2j\). The coefficients of these polynomials are uniformly bounded. Since the leading density is positive and bounded away from zero, the ratio of the \(j\)-th term to the leading term is bounded by \[ C_{j,M}q^{-j}(1+\log q)^{2j}. \tag{4.6} \] For each fixed finite \(N\), summing \(1\leq j2RM+2\). The finite logarithmic power is eventually less than any fixed positive power of \(q\), so (4.7) is \(O(q^{-1})\), and is absorbed by (4.6). This choice may depend on \(M\). A one-term expansion without this higher-order argument would not suffice for uniform relative control. The leading coefficient is intrinsic. At each critical point the orthonormal graph calculation in Lemma 3.1 gives area density one, absolute Hessian determinant \(\kappa\), and signature \(\pm d\). The usual change-of-coordinate Jacobians in the available stationary-phase formula give exactly (4.2) in any chart. This proves (4.1). The bracket in (4.1) is uniformly nonzero on each fixed parameter compact set for sufficiently large \(q\). Take its modulus and logarithm. The oscillatory real factor has modulus one, while the complex factor contributes \((\log q)p_\pm(\theta)\cdot\operatorname{Im}z\). The term \(\log|C_\pm(\theta)|/\log q\) tends uniformly to zero, as does the logarithm of the bracket divided by \(\log q\). Smoothness of \(p_\pm\) proves (4.3). \(\square\) ## 5. The whole surface has a segment for each direction **Theorem 5.1 (all directional carriers of a curved boundary).** Under the hypotheses of Sections 3 and 4, \[ \mathcal J_\theta(a\,dS)=\{H_{D_\theta}\}, \qquad \mathcal J(a\,dS)= \{H_{D_\theta}:\theta\in S^{n-1}\}. \tag{5.1} \] Every profile is proper. The statement also holds in dimension one for an interval and positive weights at its two boundary points. *Proof.* Work first near one direction \(\theta_0\). With the two cutoffs already chosen, write \[ u=u_++u_-+r,\qquad r=(1-\chi_+-\chi_-)a\,dS. \tag{5.2} \] The support of \(r\) stays away from both normal points for directions in \(\overline V\). Cover that compact support by finitely many surface charts. In every chart the real phase \(-\theta\cdot X(x)\) has gradient bounded below uniformly for \(\theta\in\overline V\). For \(|z|\leq M\) the amplitudes obey bounds of the form (4.5). Apply Proposition 2.1 of the stationary-phase prerequisite with as many integrations by parts as needed. For every \(M,N\) this gives \[ \sup_{\substack{|z|\leq M\\\theta\in\overline V}} |F_r(q\theta+z\log q)|\leq C_{M,N}q^{-N}. \tag{5.3} \] For example, an integration order \(k>RM+N+1\) absorbs the polynomial exponential weight and its finite logarithmic derivative factor. A finite partition of unity adds only finitely many such estimates. Lemma 2.2 therefore identifies the profiles of \(u\) with those of \(s=u_++u_-\) on every sequence whose directions approach \(\theta_0\). The two cap distributions have disjoint ordinary supports and hence disjoint singular supports. Theorem 4.1 gives proper affine limits with indicators \(p_\pm(\theta_0)\cdot\eta\). The preceding exact joint maximum theorem, after extending an extracted profile of \(s\) to a finite joint tuple, gives \[ h_s(\eta)= \max\{p_+(\theta_0)\cdot\eta,p_-(\theta_0)\cdot\eta\} =H_{D_{\theta_0}}(\eta). \tag{5.4} \] It also excludes collapse of \(s\): the maximum of the two proper indicators is proper. Thus \(u\) cannot collapse on such a sequence. For any profile of \(u\), compactness of the unit sphere permits a further subsequence of its witnessing centers with a limiting direction \(\theta_0\). The prescribed proper limit or collapsed status remains unchanged under further extraction. The argument above forces properness and gives its indicator \(H_{D_{\theta_0}}\). This proves one inclusion in the full family. Conversely, fix any \(\theta_0\) and take \(c_j=q_j\theta_0\) with \(q_j\to\infty\). Compactness gives an extracted profile of \(s\), and finite joint extraction retains both known cap limits. Equation (5.4) makes that profile proper with the prescribed segment indicator. Lemma 2.2 transfers it to \(u\). This proves the other inclusion and the asserted directional equality. For \(n=1\), write \(X=(b_-,b_+)\). Its surface measure is counting measure on the two endpoints, so \(u=a_-\delta_{b_-}+a_+\delta_{b_+}\), with \(a_\pm>0\). The finite-support theorem in Isolated atoms and separated singularities gives the single indicator \(H_{[b_-,b_+]}\), with no collapsed profile. Both directions select this same segment. \(\square\) Theorem 5.1 determines the indicator family. It does not identify every total limiting profile with \(-d/2+H_{D_\theta}(\operatorname{Im}z)\), or assert convergence at Fourier zeros. The individual caps have the uniform limit (4.3); the sum is treated through a joint indicator theorem. ## 6. A directional bound from the wavefront set For a distribution \(u\), define \((x_0,\xi_0)\notin\operatorname{WF}(u)\), with \(\xi_0\ne0\), if some \(\chi\in C_c^\infty\), equal to one near \(x_0\), has \(F_{\chi u}\) rapidly decreasing in an open cone containing \(\xi_0\). This definition makes the complement open: the same cutoff works at nearby points, and a smaller cone works at nearby directions. Thus \(\operatorname{WF}(u)\) is closed and invariant under positive scaling of its second coordinate. No point outside \(\operatorname{supp}u\) lies in it, since a cutoff there gives the zero distribution. For a compact distribution and a unit direction, set \[ E_\theta(u)=\{x:(x,\theta)\in\operatorname{WF}(u)\},\qquad K_\theta(u)=\operatorname{conv}E_\theta(u). \tag{6.1} \] The fiber \(E_\theta(u)\) is compact. Its convex hull is compact if nonempty, by the finite-dimensional compact-convex-hull theorem; use the empty hull otherwise. **Theorem 6.1 (directional wavefront bound).** A profile witnessed by centers with direction limit \(\theta\) has \[ h(\eta)\leq H_{K_\theta(u)}(\eta) \quad\text{for every }\eta\in\mathbb R^n. \tag{6.2} \] If the fiber is empty, every profile on such a sequence collapses. Consequently every \(h\in\mathcal J(u)\) is bounded above by the support function of a directional wavefront fiber hull for at least one nonzero direction. *Proof.* Suppose first that \(E_\theta(u)\) is nonempty. For \(\varepsilon>0\), choose a compact smooth cutoff \(\chi\) equal to one near this fiber and supported within its open \(\varepsilon\)-neighborhood. Put \(w=\chi u\) and \(v=(1-\chi)u\). Every point of the compact support of \(v\) has a wavefront-free neighborhood for the direction \(\theta\). Choose finitely many such neighborhoods, cutoffs \(\chi_j=1\) there and corresponding Fourier-decay cones \(\Gamma_j\). A smooth finite partition of unity near \(\operatorname{supp}v\), with each coefficient \(\rho_j\) compactly supported where \(\chi_j=1\), writes \[ v=\sum_j\rho_j(1-\chi)\chi_j u. \tag{6.3} \] For completeness, choose nonnegative bumps supported in the neighborhoods and positive on a smaller finite cover of that compact set. Their sum is positive near it. Divide each bump by that sum there, and multiply by one additional compact cutoff equal to one near the set and supported where the sum is positive. Extending by zero gives the required coefficients and identity of distributions. Each \(\chi_j u\) has real rapid decay in \(\Gamma_j\). Lemma 2.1 preserves that decay after the fixed smooth multiplier \(\rho_j(1-\chi)\). The finitely many smaller cones contain one common cone about \(\theta\). Their finite sum \(v\) is rapidly decreasing in that cone. The same lemma gives every inverse-power bound on its logarithmic complex windows whenever \(c/|c|\to\theta\). If a profile \(U\) of \(u=w+v\) is proper, Lemma 2.2 gives that same proper profile for \(w\). Its compact-support indicator bound yields \[ h_U(\eta)\leq H_{\operatorname{supp}w}(\eta) \leq H_{E_\theta(u)}(\eta)+\varepsilon|\eta|. \tag{6.4} \] Let \(\varepsilon\downarrow0\). Convexification leaves the support function unchanged, proving (6.2). A collapsed indicator already satisfies it. If \(E_\theta(u)=\varnothing\), apply the finite-cover construction directly to \(\operatorname{supp}u\). It gives real rapid decay for \(F_u\) in a cone about \(\theta\), and (2.2) makes every normalized logarithm tend uniformly to \(-\infty\) locally. The empty-fiber assertion follows. Finally, take any witnessing sequence for a member of \(\mathcal J(u)\) and extract a direction limit on the unit sphere. Proper local \(L^1\) convergence or collapsed convergence is retained. Equation (6.2) then gives the desired nonzero direction. \(\square\) **Corollary 6.2 (the curved surface has exactly its normal wavefront).** For the positive smooth density of Theorem 5.1, \[ \operatorname{WF}(a\,dS)= \{(x,t\nu(x)):x\in\Sigma,\ t\in\mathbb R\setminus\{0\}\}. \tag{6.5} \] Thus \(E_\theta(u)=\{p_+(\theta),p_-(\theta)\}\), and (6.2) is sharp. *Proof.* At a surface point and a direction not parallel to its normal, choose a sufficiently small surface cutoff. The phase gradient has a positive lower bound in a smaller cone. The nonstationary estimate gives real rapid decay, so that pair is absent from the wavefront set. Away from the surface the distribution is zero. Now take \(p\in\Sigma\) and \(\theta=\nu(p)\) or \(-\nu(p)\). If a cutoff \(\chi=1\) near \(p\) made \(F_{\chi u}\) rapidly decreasing in a cone about \(\theta\), multiply by a smaller smooth cap cutoff \(\rho\), equal to one near \(p\), with support where \(\chi=1\). Lemma 2.1 would make \(F_{\rho u}\) rapidly decreasing in a smaller cone. But Theorem 4.1 at \(z=0\) gives a nonzero leading coefficient times \(q^{-d/2}\) along \(q\theta\), contradicting rapid decrease. Both normal signs therefore occur. Positive scaling and the two signs give all the stated covectors. The fiber description follows from Lemma 3.1, and sharpness follows from Theorem 5.1. \(\square\) ## References - Victor Guillemin and Shlomo Sternberg, *Semi-classical Analysis*, lecture notes, 2010. [Online reading](https://math.mit.edu/~vwg/semiclassGuilleminSternberg.pdf). Background on oscillatory integrals and stationary phase. - Lars Hörmander, *The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis*, second edition, Springer, 1990. Background on distributions, wavefront sets and stationary phase. - Lars Hörmander, *The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients*, Springer, 1983. Background on logarithmic Fourier profiles and convolution geometry.