Unique Continuation Problem on RCD Spaces. I

In this note we establish the weak unique continuation theorem for caloric functions on compact $RCD(K,2)$ spaces and show that there exists an $RCD(K,4)$ space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point. We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.

• The heat equation satisfies the strong unique continuation property on an RCD(K, N) space (X, d, m) if, for any solution u ∈ W 1,2 (X × [0, T ]) of (0.1), if u vanishes up to infinite order at any (x 0 , t 0 ) ∈ X × (0, T ], then u ≡ 0. Note that we restrict t 0 to be away from 0 in the definition of strong unique continuation as it is always possible to solve the heat equation starting from some initial data u(•, 0), which vanishes up to infinite order (spatially) at some x 0 ∈ X.This would easily imply that the heat flow also vanishes up to infinite order at (x 0 , 0) in the smooth case.We mention that unique continuation type results are related to giving an upper bound for the measure of the nodal set of non-trivial solutions and that, recently, there has also been work to establish lower bounds in the nonsmooth setting, see [CF21,DPF21].
Our first result in Section 1 gives the validity of the weak unique continuation property for compact RCD(K, 2) spaces: Theorem 0.4.Let (X, d, m) be a compact RCD(K, 2) space.The heat equation on X satisfies the weak unique continuation property.
As in [DZ21], where the same theorem was shown for harmonic functions, the main idea to handle the spatial direction is to leverage the C 0 -Riemannian structure of non-collapsed RCD(K, 2) spaces from [LS18,OS94].To handle the time direction, we also show the time analyticity of solutions of (0.1) following the recent [Z21].
In [DZ21], an RCD(K, 4) space was constructed on which there exists non-trivial harmonic functions which vanish up to infinite order at some point.The heat flow of any such harmonic function would immediately give a counterexample to strong unique continuation for the heat equation as well.As such, we will be primarily interested in the strong unique continuation property for non-stationary solutions of the heat equation in this paper.We extend our result from [DZ21] as follows in Sections 3 and 5 respectively: Theorem 0.5.There exists an RCD(K, 4) space and a non-trivial eigenfunction on it with eigenvalue µ = 0 which vanishes up to infinite order at one point.
Theorem 0.6.There exists an RCD(K, 4) space and a non-stationary solution of the heat equation on it which vanishes up to infinite order at one point.
Besides these results, we also give frequency estimate on metric horns in Sections 2 and 4 and establish a strong unique continuation type result for eigenfunctions and caloric functions on metric horns, where the classical strong unique continuation property fails by the results of [DZ21].
We have to use the geometry setting of this equation to deal with the difficulty of coefficients which are non-Lipschitz.For discussions on smooth manifolds with Ricci curvarure lower bound and some related questions, see for example Therefore, it suffices to consider 2-dimensional Alexandrov spaces of curvature at least K and collapsed RCD(K, 2) spaces.Alexandrov spaces are known to have a generalized Riemannian structure by [OS94].We refer to [OS94] for relevant definitions.
Theorem 1.2.([OS94]) Let (X, d) be an n-dimensional Alexandrov space and denote S X as the set of singular points.Then there exists a C 0 -Riemannian structure on X \ S X ⊂ X satisfying the following: (1) There exists an X 0 ⊂ X \ S X such that X \ X 0 is of n-dimensional Hausdorff measure zero and that the Riemannian structure is (2) The metric structure on X \ S X induced from the Riemannian structure coincides with the original metric of X.
In a coordinate neighborhood U given by the C 0 -Riemannian structure of X, with corresponding metric g ij , a solution u of (0.1) satisfies ).Note that since g ij may not be Lipschitz, the results of [L90] do not apply.We will instead use techniques which are special in the 2-dimensional case.For more details, see for example [CH62].
We proceed by taking local isothermal coordinates.To be precise, consider functions σ, ρ : U → R 2 satisfying where derivatives are taken with respect to the coordinate chart for U.The existence of such functions is given by [M38], see also [BN54,CH62].This is equivalent to solving the complex equation We have (1.10) After rearranging the terms, ˆU×[0,T] and so in the new coordinates, we have dx dy dt = 0. (1.12) Thus in the coordinate (ρ, σ) the equation becomes ∆u − a∂ t u = 0, (1.13) where a is measurable and Hölder on a full measure subset.Note that as now we have a possibly discontinuous coefficient in front of ∂ t , the result in [L90] cannot be used.We will instead use a geometrical argument to deal with this difficulty.Now we are ready to state the main result of this section.
If u vanishes on a non-empty open set Γ ⊂ X × (0, T ], then u ≡ 0 on proj Proof.By a similar discussion as in [DZ21], we can only consider the non-collapsed case.We assume that u vanishes on a non-empty open set V × (t 1 , t 2 ).Since u satisfies the heat equation, we have the following a priori estimates (cf.[G20,Remark 5.2.11]): We first prove that u vanishes on V × [0, T ] by showing that u(x, •) is analytic with respect to t.For the corresponding arguments on Riemannian manifolds see for example [Z21].For the reader's convenience we sketch the argument here.
As in [LZ19], for any (x 0 , t 0 ) with 0 < t 0 < T , k > 0 and j ≤ k, we consider So that u is defined on these sets, we extend u to be a solution of (0.1) on X × [0, ∞).
By [MN19], see also [CN12], we may choose a cutoff function φ 1 (x, t) supported in H 2 j such that φ 1 = 1 on H 1 j , satisfying Indeed, φ 1 may be chosen to be a product of two cutoff functions on the spatial and time coordinates respectively, so that the spatial and time derivatives are well-defined without further measure theoretic arguments.It follows that ˆH2 where the last inequality follows from integration by parts, Young's inequality and (1.20).This gives that ˆH1 Arguing as before, we have ˆH1 By using (1.21),(1.24)inductively on j, we obtain where C at the end also depends on the L 2 -norm of u 0 .
From [HK00, BB11, R12] we know that the (p, p)-Poincaré inequality holds on RCD(K, N) space for any p ≥ 1 (see, for example, [K15, Section 4] for a careful discussion of Poincaré inequalties in the RCD(K, N) setting).Thus Moser iteration ([HK00, Lemma 3.10]) shows that for R < 1 there exist a mean value inequality Thus by taking R = √ t 0 √ 2k and using (1.25), we obtain (1.27) Using volume comparison, this implies that u(x 0 , •) is analytic with respect to t.In particular, since u vanishes on V × (t 1 , t 2 ) by assumption, u also vanishes on V × [0, T ].We now prove the second assertion of the theorem by contradiction.Assume X is compact and u ≡ 0.
Let φ k be eigenfunctions of −∆ corresponding to eigenvalues λ k with φ k L 2 = 1 and 0 = λ 0 < λ 1 ≤ λ 2 ≤ ... → ∞, see [H18] for a discussion on this.It follows from the estimates obtained in the appendix of [AHTP18] that u admits the representation (1.28) Let j be the first index for which a j = 0.As it does not affect the argument, we assume for simplicity that the dimension of the eigenspace corresponding to λ j is 1.This gives that for any x 0 ∈ V , 0 = lim t→∞ e λ j t u(x 0 , t) = a j φ j (x 0 ), (1.29) where we have used [AHTP18, Proposition 7.1] to bound the values of φ k (x 0 ) for k > j to obtain the second equality.As we assumed that a j = 0, we conclude that φ j (x 0 ) = 0.This shows that φ j (x 0 ) = 0 for all k and x 0 ∈ V .From [DZ21], this implies that φ j ≡ 0, which is a contradiction.It follows that u ≡ 0. Note that similar arguments also work for Dirichlet problem on non-compact space.
Remark 1.30.The previous argument actually shows the time analyticity of caloric functions with respect to time on any RCD(K, N) space, since that part of the argument does not require any assumptions on dimension.
Remark 1.31.In [L90], Fourier transform was used to reduce the problem to a solution of an elliptic equation on X × R. In this case, one does not have weak unique continuation for elliptic equations on RCD(K, 3) spaces, so a different argument had to be used.
Finally, we recall a well-known counterexample given by Miller [M74] which indicates that in general we cannot expect weak unique continuation for parabolic operators with timedependent coefficients even if the time-slices of the corresponding metric have a uniform Ricci curvature bound.

Elliptic Frequency estimate on Metric horn
In this section we will give a frequency estimate on the metric horn, which allows us to prove a form of unique continuation.Recall from [DZ21] that the standard formulation of strong unique continuation does not hold at the horn tip.The form of unique continuation we will prove in this section will therefore assume a higher order of decay at the horn tip, see Remark 2.30 at the end of the section for more details.
On a weighted warped product (X, dr 2 + f 2 (r)g S n−1 , e −ψ(r) dvol), given function ϕ, we have that away from r = 0.In the case of the standard metric horn, From the equation of Laplacian on metric horn, in particular, for ϕ = r α , we have that Given an eigenfunction u with ∆u = λu, define scale-invariant quantities I(r), E(r) and the frequency function U(r) with respect to the level sets of the function d p , where p is the horn tip.For r > 0, we denote and define where m = e ψ (r)dvol is the weighted volume measure of the metric horn and m r = e ψ (r)dvol r is the corresponding weighted area measure on ∂B r .
We first compute the derivative of I. We remark that all computations are done away from the cone tip p, so no regularity issues will arise from p itself.Let φ be any smooth function compactly supported on (0, ∞), we have This shows that , a.e.t ∈ (0, 1), and so by using (2.5), For E, from the definition (2.11) and the coarea formula, we have that Now consider the frequency function U(r).We have that From Cauchy-Schwarz inequality we know that

Failure of strong unique continuation of eigenfuctions on metric horn
In this section we will prove the following theorem: Theorem 3.1.There exists an RCD(K, 4) space and a non-trivial eigenfunction on it with eigenvalue µ = 0, which vanishes up to infinite order at one point.
Proof.Consider any modified metric horn X constructed in [DZ21, Section 6], which uses techniques in [WZ21].Let us denote the eigenfunctions on Assume ϕ is an L 2 function which is smooth away from the tip, then ϕ may be decomposed as Therefore, for any eigenfunction on metric horn with eigenvalue −µ we have, for the decomposition (3.4) and each i, for r sufficiently close to 0. We note that we do not have this formula for arbitrary r since the modified metric horn was constructed using a gluing procedure.
For the radial part, that is for µ 0 = 0, where J ν , Y ν are Bessel functions.For a discussion of the properties of Bessel functions, see [M10, Appendix B].We will use the fact that as r → 0. Since ϕ is assumed to be in L 2 , we have that f 0 is in a weighted L 2 space where the weight for small r is of the form r c .Taking this into consideration with the asymptotics of J and Y , we conclude that and so If we consider the solution with data k i (r), we have e is also a solution of (3.13).So we get 1 2 From ODE theory we have that k i form a basis of the solutions of (3.13).This tells us that As before, we have that f i is in a weighted L 2 space from assumption, this tells us that From (3.16) we have the decay rate f i (r) = O(e −Cr −ǫ ).This tells us that there exists eigenfunctions on RCD(K, 4) spaces which are not zero but vanish up to infinite order at one point.
Remark 3.20.In fact this argument tells us that any eigenfunction on this metric horn with zero integral on each link {r = r 0 } vanishes up to infinite order.
the normalized coefficient c µ,i for the eigenfunction f i (r)ϕ i (θ) with L 2 -norm 1 corresponding to eigenvalue µ satisfies c µ,i ≤ c(n, ǫ, µ i )e (3.25) Thus if we denote the pasting radius as r, by the discussion we have As we know that when transform back As outside r the space is a cone, the function is of the form f From the asymptotics of Bessel functions we can see that if we consider the eigenfunction on B R , the normalized coefficient satisfies

parabolic frequency estimate on metric horn
In this section we give a parabolic frequency estimate and the corresponding unique continuation type result on the metric horn.For simplicity, we consider the metric horn which is not modified.For previous discussions on parabolic frequency see for example [P96,CM22].

Failure of Strong unique continuation property of caloric on metric horn
In this section we will prove the following theorem: Theorem 5.1.There exists an RCD(K, 4) space and a non-stationary solution of heat equation on it which vanishes up to infinite order at one point.
Similar as in the elliptic case, the heat equation becomes like (5.2) Denote the eigenfunctions on S n−1 as {ϕ i } ∞ i=1 such that ∆ S n−1 ϕ i = −µ i ϕ i , µ i > 0 (5.3) ˆSn−1 ϕ 2 i dS = 1.(5.4) We can decompose it as ϕ(r, θ, t) = f 0 (r, t) + ∞ i=1 f i (r, t)ϕ i (θ).So we have We first consider the solution of heat equation on the compact modified metric horn constructed in [DZ21, Section 6], where p is the tip of the metric horn.We denote the eigenfunctions with eigenvalue ν j with spherical components ϕ i as q ν j (r, θ) = g j (r)ϕ i (θ).
Then we have that f i (r, t) = j c j e −ν j t g j (r).As the estimate is independent of k, we can see that f i vanishes up to infinite order at tip.Remark 5.9.The argument above actually shows that all caloric functions on the modified metric horn which does not contain the radial part f 0 (r, t) vanishes up to infinite order at the tip.