Proof. In a flat chart containing the convolution disks, the operation is ordinary spatial convolution at fixed angle. Its kernel, and its first spatial derivative multiplied by \(\varepsilon\), have uniformly bounded absolute integrals in either spatial variable. Parallel transport preserves angular measure. The usual kernel bound on \(L^2\) therefore proves the asserted norms. Local commutation, positivity, and the \(L^\infty\) estimate follow from the same chart description.

For a smooth compactly supported input, the expression in (4) tends to zero directly: for small scales its convolution remains away from the tips and its first derivatives are uniformly bounded in \(L^2\). Such inputs are dense in \(L^2(SM)\); the uniform scaled-derivative bound proves the claim for any fixed input. The ordinary approximate-identity argument gives local convergence.

For the third assertion, a contributing kernel pair has base points at distance at most \(\varepsilon\). If one point is in the support of \(g\), both lie strictly inside the safe region. Reversal of the short segment reverses its parallel transport and leaves the radial kernel unchanged. This gives symmetry. In a common flat chart the kernel satisfies \(Y_x K_\varepsilon(x-y)=-Y_y K_\varepsilon(x-y)\), with the angles identified by parallel transport. Kernel differentiation and Fubini give (5). No derivative of \(g\) is required.

The support estimate follows from the triangle inequality for \(r\). In particular, the convolution vanishes in a neighborhood of the boundary of the safe region, so its extension by zero creates no boundary derivative. Distributional local commutation proves (6); \(Xg\) is supported in the same closed spatial support as \(g\), and the \(L^2\) kernel bound gives the final inequality. ◻

Proof. For each coefficient, integration by parts gives \[ \left\lVert\partial w_j\right\rVert_2^2=\left\lVert\bar\partial w_j\right\rVert_2^2 =\tfrac14\left\lVert\nabla w_j\right\rVert_2^2. \tag{8}\] Indeed, the difference of the first two squared norms is proportional to the integral of \(dw_j\wedge d\overline{w_j}\). The primitive \(w_j\,d\overline{w_j}\) is global because its constant unitary transition factors cancel, and its support is compact in the regular surface. Stokes gives zero. Spatially smooth fields suffice initially; ordinary local Sobolev approximation gives the stated generality.

Put \(D_j=\left\lVert\partial w_j\right\rVert_2^2\). By (2) and (8), \[D_j-D_{j+2} =\operatorname{Re}\left\langle(Xw)_{j+1},\partial w_j-\bar\partial w_{j+2}\right\rangle.\] The pairing is over \(E\), since its first factor vanishes elsewhere. Moreover \(D_k\to0\) as \(k\to+\infty\), because the horizontal derivatives of \(w\) are square summable in their Fourier modes. Telescoping in steps of two and applying Cauchy–Schwarz gives \[D_j\le\left\lVert Xw\right\rVert_2 \left(\sum_{k\ge0} \left\lVert\mathbf 1_E(\partial w_{j+2k}-\bar\partial w_{j+2k+2})\right\rVert_2^2\right)^{1/2}.\] Pointwise in the base variable, the sum inside is bounded by \[2\sum_{\ell\in\mathbb Z}(|\partial w_\ell|^2+|\bar\partial w_\ell|^2) =\sum_{\ell\in\mathbb Z}|\nabla w_\ell|^2.\] Parseval and (8) prove (7). ◻

Proof. Take a smooth spatial cutoff \(\chi_\varepsilon\) equal to zero for \(r\le(K+1)\varepsilon\), equal to one for \(r\ge(K+2)\varepsilon\), with \(|\nabla\chi_\varepsilon|\le C/\varepsilon\). For small \(\varepsilon\), its transition regions are disjoint cone annuli. Set \(w=\chi_\varepsilon S_\varepsilon h\), extended by zero near the vertices. This field has compact spatial support and square-integrable horizontal derivatives. Local commutation gives \[Xw=(X\chi_\varepsilon)S_\varepsilon h.\] Thus \(Xw\) is supported in the transition annuli \(E_\varepsilon\), whose total area is \(O(\varepsilon^2)\). On these annuli the kernel derivative estimate and boundedness of \(h\) give \[|Xw|+|\nabla w|\le CH/\varepsilon.\] Both norms on the right of (7) are at most \(CH\). Hence \(\left\lVert\nabla w_j\right\rVert_2\le CH\), independently of \(\varepsilon,j\).

For each fixed \(j\), local \(L^2\) convergence of \(w_j\) to \(h_j\) and weak compactness of its globally bounded gradients identify the weak gradient limit with \(\nabla h_j\), by testing in regular charts. Lower semicontinuity gives (9). This argument neither assigns traces at the cone tips nor asserts square summability over \(j\). ◻

Proof. Write \(H=\left\lVert f\right\rVert_\infty\). The case \(H=0\) is immediate. We know \(Xf=0\), so 4 applies. Fix \(m\in\mathbb Z\) throughout the following argument.

The endpoint equation. Define \[ U=\sum_{k\ge0}P_{m+2k}f\in L^2(SM),\qquad B=e^{i(m-1)\theta}\bar\partial f_m\in L^2(SM). \tag{10}\] The latter expression is a global field of mode \(m-1\). The recurrence (2) implies \[ XU=B \tag{11}\] distributionally. One precise verification uses finite tails \(U_N\): \[XU_N=B+e^{i(m+2N+1)\theta}\partial f_{m+2N}.\] The terminal fields \(R_N\) are uniformly bounded in \(L^2\) by 4, and their angular modes tend to infinity. For any fixed \(\varphi\in L^2(SM)\), \[|\left\langle R_N,\varphi\right\rangle| \le CH\left\lVert P_{m+2N+1}\varphi\right\rVert_2\longrightarrow0.\] Thus they converge weakly to zero; their norms need not tend to zero. Since \(U_N\to U\) in \(L^2\), (11) follows. On safe disks, smoothing gives \[ X Y S_\varepsilon U=Y S_\varepsilon B. \tag{12}\] Our aim is to prove \(\left\langle B,P_{m-1}Yf\right\rangle=0\). The recurrence (2) gives \(P_{m-1}Yf=-2iB\), so that identity forces \(B=0\); the construction below obtains it without assuming \(Yf\in L^2\).

Long segments with clearance. Fix \(s>0\), put \(T=s/\varepsilon\), and fix \(L>K+4\). This time scale keeps the discarded mass of order \(s\), while the endpoint error will vanish as \(\varepsilon\to0\) at fixed \(s\). Let \(G_\varepsilon\) be the set of regular states whose trajectories satisfy \(r(\Psi_t z)>L\varepsilon\) throughout \(0\le t\le T\). The distance function is \(1\)-Lipschitz along a unit-speed path. A time mesh of spacing at most \(\varepsilon\), with at most \(T/\varepsilon+2\) points, detects every failure of this condition inside \(\{r\le(L+1)\varepsilon\}\). Measure preservation and (3) therefore give \[ \nu(SM\setminus G_\varepsilon)\le C(T\varepsilon+\varepsilon^2) =C(s+\varepsilon^2). \tag{13}\] Null sets of singular states may be ignored.

Set \(a_\varepsilon=f\mathbf 1_{G_\varepsilon}\) and form the negative-time average \[ g_\varepsilon=\frac1T\int_0^T a_\varepsilon\circ\Psi_{-t}\,dt. \tag{14}\] It has absolute value at most \(H\). If an integrand is nonzero at \(z\), then \(z\) lies at time \(t\) on a good segment starting at \(\Psi_{-t}z\). Consequently \(g_\varepsilon=0\) on \(\{r<L\varepsilon\}\). Invariance of \(f\), Minkowski’s inequality, and (13) show that \[ \left\lVert g_\varepsilon-f\right\rVert_2\le CH\sqrt{s+\varepsilon^2},\qquad \left\lVert Xg_\varepsilon\right\rVert_2\le\frac{2H}{T}. \tag{15}\] For the second assertion, finite-time averaging gives the weak identity \[Xg_\varepsilon=\frac{a_\varepsilon-a_\varepsilon\circ\Psi_{-T}}{T}.\] It follows by testing (14) against a compactly supported smooth regular field, changing variables by the flow, and integrating its derivative in \(t\). It does not differentiate the indicator of \(G_\varepsilon\).

Integrating the smoothed equation. We justify use of the fundamental theorem in (12) along good segments, despite the absence of angular smoothing. For each fixed \(\varepsilon\), choose a countable cover by flat charts enlarged to contain the required convolution disks. For almost every angular slice in each chart, the smoothed fields have smooth spatial representatives and their differential identity holds everywhere in position. Overlap identities have the same property; include their exceptional angular sets in this countable collection.

During a visit to a flat chart a regular orbit has constant chart angle. If it met one of the exceptional slices, it would remain in that slice for an open time interval, which contains a rational time. For each chart and rational time the set of such initial states is null, by measure preservation. Removing their countable union makes the calculus valid along every chart visit of almost every good segment. Smooth spatial representatives agree on overlaps, so their identities can be integrated along the whole segment. This argument also handles its endpoints by regular continuation there.

Changing variables in the pairing with (14), and using the fundamental theorem, now gives \[ \left\langle Y S_\varepsilon B,g_\varepsilon\right\rangle =\frac1T\int_{G_\varepsilon}\overline{f(z)} \bigl[(Y S_\varepsilon U)(\Psi_T z)-(Y S_\varepsilon U)(z)\bigr]\,d\nu(z). \tag{16}\] Both endpoints are safe. By Cauchy–Schwarz and measure preservation, the absolute value of the right side is at most \[\frac{2H}{s}\,\varepsilon\left\lVert Y S_\varepsilon U\right\rVert_{L^2(r>K\varepsilon)}=o(1) \qquad(\varepsilon\to0,\ s\text{ fixed}),\] by (4). Here \(U\) is fixed.

Put \(w_\varepsilon=S_\varepsilon g_\varepsilon\), with the supported extension of 2. Its adjoint identity converts (16) into \[ \left\langle B,Yw_\varepsilon\right\rangle=o(1). \tag{17}\] The same lemma and (15) give \[\left\lVert\nabla w_\varepsilon\right\rVert_2\le CH/\varepsilon,\qquad \left\lVert Xw_\varepsilon\right\rVert_2\le CH/T.\] It has compact spatial support in \(M\). Applying 3 with \(E=M\) yields \[ \left\lVert\nabla(w_\varepsilon)_j\right\rVert_2^2\le CH^2/s\qquad(j\in\mathbb Z). \tag{18}\]

The first limit: fixed \(s\). Take a sequence \(\varepsilon\to0\) and a subsequence along which \(w_\varepsilon\) converges weakly in \(L^2\) to \(g^{(s)}\). The closed convex set \(\{|u|\le H\}\subset L^2\) is weakly closed, so \(\left\lVert g^{(s)}\right\rVert_\infty\le H\). For a fixed smooth compactly supported test field \(\varphi\), symmetry of convolution gives, for sufficiently small \(\varepsilon\), \[\left\langle w_\varepsilon-g_\varepsilon,\varphi\right\rangle =\left\langle g_\varepsilon,S_\varepsilon\varphi-\varphi\right\rangle\longrightarrow0.\] The uniform \(L^2\) bound and density of these tests show that \(g_\varepsilon\) has the same weak \(L^2\) limit. Therefore \[ Xg^{(s)}=0,\qquad \left\lVert g^{(s)}-f\right\rVert_2\le CH\sqrt{s}. \tag{19}\] The first equality follows either from (15) or the bound for \(Xw_\varepsilon\); the second follows by weak lower semicontinuity.

Only coefficient derivatives at modes \(m\) and \(m-2\) contribute to (17), because \(B\) has mode \(m-1\). For fixed \(s\), (18) allows a further subsequence on which both these gradients converge weakly in their global \(L^2\) spaces. Testing in regular charts identifies the limits as the gradients of the corresponding coefficients of \(g^{(s)}\). Since \(B\) is fixed in \(L^2\), passing to the limit gives \[ \left\langle B,P_{m-1}Yg^{(s)}\right\rangle=0. \tag{20}\] Here this projected derivative means the combination of the two coefficient derivatives in (2); no full \(L^2\) bound on \(Yg^{(s)}\) is asserted.

The second limit: \(s\to0\). We do not use the deteriorating bound (18) for this limit. Instead, \(g^{(s)}\) is bounded by \(H\) and solves the exact local equation in (19), so 4 gives \[\left\lVert\nabla(g^{(s)})_j\right\rVert_2\le CH,\] uniformly in \(s\). Choose \(s_n\downarrow0\) and the limits just constructed for each \(s_n\). By (19), they converge strongly to \(f\) in \(L^2\). Their relevant gradients are bounded in \(L^2\), and every weak subsequential limit is identified locally with the corresponding gradient of \(f\). Thus (20) implies \[\left\langle B,P_{m-1}Yf\right\rangle=0.\] The equation \(Xf=0\) in mode \(m-1\) says \(\partial f_{m-2}=-\bar\partial f_m\). By (2), \[P_{m-1}Yf=-2iB,\qquad 0=\left\langle B,-2iB\right\rangle=2i\left\lVert B\right\rVert_2^2.\] Hence \(\bar\partial f_m=0\). Since \(m\) was arbitrary, the recurrence also gives \(\partial f_j=0\) for every \(j\), proving the proposition. ◻

Proof of 1. Relabel the vertices so that \(\alpha/\pi\) is irrational. Let \(A\) be a measurable invariant set as in the theorem, and lift its indicator to a bounded invariant function \(f\) on \(SM\). By 5, each coefficient has zero flat gradient. In every regular chart it therefore has a constant representative, and these constants match under the unitary transitions. Each coefficient is a parallel section.

Continuation around the vertex of angle \(\alpha\) has holonomy rotation by \(-2\alpha\) modulo \(2\pi\), as follows by developing its punctured cone of angle \(2\alpha\). A parallel coefficient of mode \(j\) must consequently be fixed by the phase \(e^{2ij\alpha}\). If \(j\ne0\), irrationality of \(\alpha/\pi\) makes that phase different from one, so the coefficient vanishes. The zeroth coefficient is constant on the connected surface \(M\). Fourier completeness shows that \(f\) is constant almost everywhere. As an indicator with the same mass as \(A\), it has integral zero or one. Thus \(\mu(A)\in\{0,1\}\). ◻

Proof. Place \(\overline Q\) in a sufficiently large flat torus. Its boundary is Lipschitz, with three smooth open sides and a finite codimension-two singular set; incident sides meet transversely. Thus the triangle meets the piecewise-smooth hypotheses in (Burq 2003, sec. 2), (Hassell and Zelditch 2004, Definition 3.1), and (Hezari 2016, Definition 1.16), including the last reference’s metric-extension condition. Theorem 1 supplies the required billiard-flow ergodicity. The first conclusion is the interior quantum-ergodicity theorem of Zelditch and Zworski (Zelditch and Zworski 1996), in the Dirichlet and Neumann formulation recalled in (Burq 2003, Theorem 2 and Remark 2.1).

For the boundary conclusion, discard the null sets of vertex hits and tangential collisions. The collision section is \(B^*\Gamma\), with invariant measure \(ds\,d\eta\). Boundary-flight coordinates identify the flow, up to null sets, with the suspension of its collision map under the positive free-flight time \(\tau\), and Liouville measure is proportional to \(ds\,d\eta\,dt\) for \(0\le t<\tau\). The roof \(\tau\) is positive almost everywhere and bounded by the diameter of \(Q\). Hence an invariant collision set of positive measure with positive-measure complement would suspend to a nontrivial invariant flow set, contradicting Theorem 1. The collision map is therefore ergodic. Hassell–Zelditch’s boundary theorem (Hassell and Zelditch 2004, Theorem 1.1, equations (1.3)–(1.5), and Section 7) now gives the second conclusion: its prefactor is \(4/(\operatorname{vol}(S^1)\operatorname{Area}(Q)) =2/(\pi\operatorname{Area}(Q))\). Their inward normal instead of our outward normal changes the Dirichlet data by a sign and leaves the quadratic pairing unchanged.

Finally, Hezari’s nodal-domain theorem for two-dimensional ergodic billiards with piecewise-smooth boundary (Hezari 2016, Theorem 1.6) gives the third conclusion for real eigenfunctions. The finitely many zero-frequency terms have no effect on density. The three density-one sets may be replaced by their finite intersection when a simultaneous subsequence is desired. ◻