# Boundary regularity and conic decomposition *Written by GPT-6 Astra (OpenAI), October 2026. Public domain (CC0).* A singular covector can be read at one point on the boundary of a complex tube. This lesson proves the exact correspondence: Fourier regularity in the direction \(\theta\) is equivalent to derivative bounds near the boundary point \(x-i\theta\). The correspondence works for smooth, analytic and all the derivative-growth classes considered here. It then gives a decomposition subordinate to any finite closed conic cover, together with the precise ambiguity on cone intersections. The main inputs are [Derivative growth and finite-order localization](derivative-growth-and-finite-order-localization.md), Lemmas 1 and 5 and Proposition 3; [Cones and derivative-growth wavefronts](cones-and-derivative-growth-wavefronts.md), the definition and Theorems 3, 5–7; [Holomorphic decompositions of tempered distributions](holomorphic-decompositions-of-tempered-distributions.md), Lemma 1, Theorems 3–5 and Proposition 6; and [Continuing the spherical kernel](continuing-the-spherical-kernel.md), Corollary 8 and its uniform analytic remainder. The [Schwartz and Fourier foundation](../prerequisites/U011-free-foundations/schwartz-fourier-foundations-U017.md), F1–F5, supplies Fourier inversion and pairings. Integral Taylor remainders are proved in [When a kernel is smooth](when-a-kernel-is-smooth.md#taylor-estimates-on-compact-neighborhoods), and polydisk Cauchy bounds in [Holomorphic boundaries in convex cones](holomorphic-boundaries-in-convex-cones.md). The integration foundation, [Sections 15–16](../prerequisites/U011-free-foundations/banach-foundation-bridges.md), supplies the finite measures and limiting operations below. ## The tube and its two notions of boundary regularity Use \(\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx\), with inverse factor \((2\pi)^{-n}\), and complex-linear distribution pairings. Let \(n\ge1\), let \(S^{n-1}\) have its ordinary surface measure \(\sigma\) (two unit masses for \(n=1\)), and put \[ T=\{x+iy:|y|<1\},\qquad I(\xi)=\int_{S^{n-1}}e^{-\omega\cdot\xi}\,d\sigma(\omega), \qquad a=\frac{n-1}{2}. \tag{1} \] For \(u\in\mathcal S'(\mathbb R^n)\) its spherical representation is \[ U(z)=(2\pi)^{-n}\widehat u\left(\frac{e^{iz\cdot\xi}}{I(\xi)}\right), \qquad z\in T. \tag{2} \] The earlier representation theorem proves holomorphy, polynomial growth in \(x\) and \((1-|y|)^{-1}\), continuous distributional boundary values \(B_\omega=U(\,\cdot+i\omega)\), and \[ u=\int_{S^{n-1}}B_\omega\,d\sigma(\omega). \tag{3} \] These integrals are pairings against each Schwartz test; the boundary family has a common finite Schwartz-seminorm bound. Fix an admissible scale \(L\): \(L_0=1\), \(k\le L_k\), \(L_k\) is nondecreasing and \(L_{k+1}\le QL_k\). Say that \(U\) is **\(L\)-regular at \(p\in\partial T\)** if a complex ball \(D(p,r)\) and constants \(A,B\ge1\) satisfy \[ |\partial_x^\alpha U(z)|\le A(BL_k)^k, \qquad k=|\alpha|,\quad z\in D(p,r)\cap T. \tag{4} \] Degree zero means a bound by \(A\). Say that it is **smooth at \(p\)** if one ball works for bounds \(|\partial_x^\alpha U|\le C_\alpha\) for every \(\alpha\). The constants may depend arbitrarily on \(\alpha\), but the ball must not shrink with \(\alpha\). Holomorphy makes these also bounds for all real and imaginary derivatives, since \(\partial_{y_j}U=i\partial_{x_j}U\). ## Fourier bounds become boundary bounds **Theorem 1.** For \(u\in\mathcal S'\), \(x_0\in\mathbb R^n\), \(\theta\in S^{n-1}\), and every admissible \(L\), \[ (x_0,\theta)\notin\operatorname{WF}_L(u) \quad\Longrightarrow\quad U\text{ is }L\text{-regular at }x_0-i\theta. \tag{5} \] The analogous implication holds for the ordinary smooth wavefront and smooth boundary regularity. **Proof for \(L\).** The definition gives compact distributions \(u_N\), all equal to \(u\) on a ball \(B(x_0,r)\), with one compact support and one finite-order bound. There are an open cone \(\Gamma\) about \(\theta\), constants \(C,B\ge1\), and an integer \(M\ge0\), such that \[ |\widehat u_N(\xi)|\le C(1+|\xi|)^M\quad(\xi\in\mathbb R^n), \qquad |\widehat u_N(\xi)|\le C(BL_N/|\xi|)^N\quad(\xi\in\Gamma). \tag{6} \] Set \(v_N=u-u_N\). The finite-order bound and compact support imply a common finite Schwartz-seminorm bound for \(u_N\); adding the fixed bound for \(u\) gives one for \(v_N\). Every \(v_N\) vanishes on \(B(x_0,r)\). The uniform off-support theorem, Corollary 8 of the preceding lesson, therefore extends the representations \(V_N\) of \(v_N\) holomorphically to one full neighborhood of \(p=x_0-i\theta\), with common bounds on smaller compact sets. Polydisk Cauchy on a fixed smaller ball gives \[ |\partial_x^\alpha V_N(z)|\le C_0 H^k k! \le C_0H^k L_k^k \quad(k=|\alpha|), \tag{7} \] uniformly in \(N\); for \(k=0\) read both rightmost factors as one. The last inequality uses \(k!\le k^k\le L_k^k\). The representation of \(u_N\) is the ordinary absolutely convergent integral \[ U_N(x+iy)=(2\pi)^{-n}\int e^{ix\cdot\xi-y\cdot\xi}\frac{\widehat u_N(\xi)}{I(\xi)}\,d\xi. \tag{8} \] Indeed the Fourier transform of a compact distribution is its smooth polynomially bounded transform, by F5. On compact subsets of \(T\), the reciprocal estimate \(I(\xi)^{-1}\le C(1+|\xi|)^a e^{-|\xi|}\) makes every differentiated integrand integrable. Thus differentiation and (2) give \(U=U_N+V_N\) and \[ |\partial_x^\alpha U_N(x+iy)| \le C\int e^{-|\xi|-y\cdot\xi} |\xi|^k(1+|\xi|)^a|\widehat u_N(\xi)|\,d\xi. \tag{9} \] Because \(\theta\) is an interior direction of \(\Gamma\), compactness of the complementary unit directions gives \(\delta>0\) such that \(\theta\cdot\xi\le(1-2\delta)|\xi|\) outside \(\Gamma\). For \(|y+\theta|<\delta\), \[ -|\xi|-y\cdot\xi\le-\delta|\xi| \quad(\xi\notin\Gamma). \tag{10} \] If the complementary directions are empty there is no such integral. Otherwise choose an integer \(b\ge M+a\). Polar integration bounds the complementary part of (9) by \(C\int_0^\infty e^{-\delta t}t^{k+n-1}(1+t)^b\,dt\). Expanding the integer power and integrating by parts gives a sum of \((k+n+j-1)!\delta^{-k-n-j}\), \(0\le j\le b\). Each is at most \(C_1D^k L_k^k\): for a fixed integer \(h\ge n+b\), use \((k+h)!\le(k+h)^{k+h}\le L_{k+h}^{k+h}\), then the fixed-shift Lemma 1 of the growth lesson. All constants are independent of \(N,k,x,y\). Inside \(\Gamma\), \(|y|<1\) gives \(e^{-|\xi|-y\cdot\xi}\le1\). Choose once and for all an integer \[ d>M+a+n+1,\qquad N=k+d. \tag{11} \] On \(|\xi|\le L_N\), the first bound in (6), polar integration and \(L_N\ge1\) give \[ C\int_0^{L_N}t^{k+n-1}(1+t)^{M+a}\,dt \le C_2 L_N^{k+M+a+n}\le C_2 L_N^N. \tag{12} \] On \(|\xi|>L_N\), the second bound in (6) gives \[ \begin{aligned} C_3 B^N L_N^N\int_{L_N}^\infty t^{k+a+n-1-N}\,dt &=\frac{C_3 B^N}{d-a-n}L_N^{k+a+n}\\ &\le C_4 B^N L_N^N. \end{aligned} \tag{13} \] The denominator is positive and fixed. The fixed-shift lemma gives \(B^{k+d}L_{k+d}^{k+d}\le C_5D_1^kL_k^k\), including \(k=0\). Combining (7), (9)–(13) proves (4) on one sufficiently small ball about \(p\). Although the auxiliary \(U_N\) depends on \(k\), their sum with \(V_N\) is always the same function \(U\). This is why choosing \(N=k+d\) is legitimate. \(\square\) **Proof for smooth regularity.** Take one compact smooth cutoff \(\chi=1\) near \(x_0\) such that \(\widehat{\chi u}\) decreases faster than every inverse power on a fixed \(\Gamma\). Write \(u=\chi u+(u-\chi u)\). The second representation is holomorphic on a full neighborhood of \(p\), by the same off-support theorem. In (9) use the polynomial bound of the one compact distribution outside \(\Gamma\), where (10) applies. For a derivative of order \(k\), choose a decay power \(q>k+a+n\) inside \(\Gamma\); the integral of \(t^{k+n-1}(1+t)^{a-q}\) is finite. The real and angular neighborhoods used here are independent of \(k\). This gives all the fixed-order bounds required for smooth boundary regularity. No estimate uniform in \(k\) is asserted in this case. \(\square\) ## Reading a singularity from one boundary point For a finite complex Borel measure \(\mu\) on the sphere, let \[ u_\mu=\int B_\omega\,d\mu(\omega),\qquad E=\operatorname{supp}\mu. \tag{14} \] This is a tempered distribution by the common boundary seminorm bound. The earlier angular theorem and Proposition 6 already prove \[ \operatorname{WF}_L(u_\mu)\subset \{(x,\xi): -\xi/|\xi|\in E, \ U\text{ is not }L\text{-regular at }x-i\xi/|\xi|\}. \tag{15} \] **Lemma 2 (smooth angular bound).** Replacing \(L\)-regular by smooth in (15) gives the same inclusion for \(\operatorname{WF}(u_\mu)\). **Proof.** If \(-\xi/|\xi|\notin E\), the earlier analytic angular bound excludes \((x,\xi)\) from \(\operatorname{WF}_A(u_\mu)\), hence from \(\operatorname{WF}(u_\mu)\) by the proved class comparison. Suppose instead that \(U\) is smooth at \(x_0+i\omega_0\), where \(\omega_0=-\xi/|\xi|\). Take a smaller ball inside the ball of all-order bounds. Its intersection with \(T\) is convex. The bound on derivatives of order \(|\alpha|+1\), together with the Cauchy–Riemann equations, makes \(\partial_x^\alpha U\) Lipschitz there. Its values therefore extend uniquely and continuously to the closure: interior approximations are Cauchy by that bound, and any two approximations have the same limit. Each derivative retains its original bound. For real \(x\) near \(x_0\) and \(\omega\) in a small closed spherical cap about \(\omega_0\), denote the extended value of \(U(x+i\omega)\) by \(b_\omega(x)\). At height \(r\omega\), \(r<1\), apply the fundamental theorem along a small real coordinate segment. Passing to \(r\uparrow1\) uniformly on that segment gives the same identity with the extended first derivative. Repeating for every \(\alpha\) proves that \(b_\omega\) is smooth with these derivatives and their common bounds. Uniform convergence on compact real sets identifies this function with the distributional boundary \(B_\omega\) there. Choose an open cap \(O\) about \(\omega_0\) whose closure is such a cap. Split \(\mu=1_O\mu+1_{S^{n-1}\setminus O}\mu\). Dominated convergence against the finite total variation measure makes the first contribution smooth near \(x_0\). The second measure's support avoids \(\omega_0\), so its contribution is analytically regular in the direction \(\xi\). For completeness, the smooth sum rule needed in this last step follows directly from the Fourier cutoff argument already used in the conic lesson. If \(v,w\) are both smooth-wavefront regular at \((x_0,\xi)\), select their two compact cutoff witnesses and a new cutoff supported where both original cutoffs are one. Multiplication preserves rapid decay on a smaller cone by the convolution estimate: for compact \(f\), \[ \widehat{bf}(\zeta)=(2\pi)^{-n} \int\widehat b(\zeta-\eta)\widehat f(\eta)\,d\eta. \tag{16} \] On the cone of rapid decay, distribute any output weight using \(1+|\zeta|\le(1+|\zeta-\eta|)(1+|\eta|)\) and take both decay exponents greater than the output exponent plus \(n+1\). Off that cone, separated unit directions give \(|\zeta-\eta|\ge c(|\zeta|+|\eta|)\); a cutoff decay exponent greater than the output exponent plus the polynomial order of \(\widehat f\) plus \(n+1\) is integrable. Formula (16) follows by Fourier inversion of the test in the compact distribution's finite-order norm. This proves the claimed preservation and makes the sum of the two new transforms rapidly decreasing. A smooth function has no smooth wavefront because its compact cutoff has rapidly decreasing transform. The analytic-to-smooth inclusion proved in the conic lesson applies to the other summand. Thus the two contributions above have no smooth wavefront at \((x_0,\xi)\), proving the lemma. \(\square\) **Theorem 3 (exact boundary characterization).** For all \(u\in\mathcal S'(\mathbb R^n)\) and admissible \(L\), \[ \begin{aligned} \operatorname{WF}_L(u)\cap(\mathbb R^n\times S^{n-1}) &=\{(x,\theta):U\text{ is not }L\text{-regular at }x-i\theta\},\\ \operatorname{WF}(u)\cap(\mathbb R^n\times S^{n-1}) &=\{(x,\theta):U\text{ is not smooth at }x-i\theta\}. \end{aligned} \tag{17} \] For \(\operatorname{WF}_A\), absence of the covector is equivalent to holomorphic continuation of \(U\) to a full complex neighborhood of \(x-i\theta\). **Proof.** Theorem 1 proves the forward regularity implications. In (15) take \(\mu=\sigma\) and use the exact average (3); this proves the reverse implication for every \(L\). Lemma 2 with the same average proves the reverse implication for smooth regularity. It remains to justify the analytic continuation statement. For \(L_k=k+1\), the factorial comparison in Proposition 3 of the growth lesson converts (4) to \[ |\partial_x^\alpha U(z)|\le A H^k k!,\qquad k=|\alpha|, \tag{18} \] on one ball intersected with \(T\). The preceding Lipschitz argument extends every derivative continuously to the boundary point \(p\). Form the series \[ F(p+h)=\sum_{\alpha\ge0} \frac{\partial_x^\alpha U(p)}{\alpha!}h^\alpha. \tag{19} \] The sum of absolute values at degree \(k\) is at most \(A(H|h|_1)^k\), by the multinomial identity. Consequently (19) converges normally when \(H|h|_1<1\), defines a holomorphic function there, and may be differentiated termwise on smaller polydisks. To see the last assertion without a convergence shortcut, a derivative of fixed multi-order introduces at most a fixed power of the degree and a fixed power of the inverse smaller radius; a geometric series times that degree power converges. Choose interior points \(p_j\to p\). For any fixed \(z\in T\) sufficiently close to \(p\), the segment from \(p_j\) to \(z\) lies in the original ball intersected with \(T\) for all large \(j\), by convexity. Apply the integral Taylor formula along this complex segment, using holomorphy to identify its derivatives. The remainder after degrees below \(m\) is bounded by \(A(H|z-p_j|_1)^m\), exactly as in the real Taylor estimate. Shrink the neighborhood of \(p\) so that this ratio is less than a fixed number below one for all large \(j\). First let \(j\to\infty\) in the finite Taylor polynomial and its remainder bound, and then let \(m\to\infty\). This proves \(F(z)=U(z)\) throughout the intersection with a smaller neighborhood. Thus (19) is the required extension. Conversely a holomorphic extension is bounded on a compact polydisk about \(p\); the proved Cauchy derivative estimate on a smaller polydisk gives \(C H^k\alpha!\le C H^k k!\le C H^k(k+1)^k\). This is (4) and completes both directions. \(\square\) ## Splitting directions with closed cones Write \(\mathcal W\) for any one of \(\operatorname{WF}\) or \(\operatorname{WF}_L\), including \(\operatorname{WF}_A\). The same operators below work simultaneously for all these choices. **Theorem 4 (finite conic decomposition).** Suppose \(\Gamma_1,\ldots,\Gamma_m\) are closed cones in \(\mathbb R^n\setminus\{0\}\) whose union is the whole punctured space. There are linear maps \(P_j:\mathcal S'\to\mathcal S'\), depending only on the ordered cover, such that \[ \sum_{j=1}^m P_j u=u,\qquad \mathcal W(P_j u)\subset \mathcal W(u)\cap(\mathbb R^n\times\Gamma_j). \tag{20} \] **Proof.** Put \(F_j=\Gamma_j\cap S^{n-1}\). Define the disjoint Borel pieces \[ E_1=F_1,\qquad E_j=F_j\setminus\bigcup_{\ell