Blow-ups of caloric measure in time varying domains and applications to two-phase problems

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in $\mathbb{R}^{n+1}$, $n \geq 2$, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems: 1) Let $\Omega_1$ and $\Omega_2$ be disjoint domains in $\mathbb{R}^{n+1}$, $n \geq 2$, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothesis on the set $E$ of mutual absolute continuity of the associated caloric measures $\omega_i$ with poles at $\bar{p}_i=(p_i,t_i)\in\Omega_i$, $i=1,2$. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of $\omega_1|_E$ is $n+1$ and the tangent measures of $\omega_1$ at $\omega_1$-a.e. point of $E$ are equal to a constant multiple of the parabolic $(n+1)$-Hausdorff measure restricted to hyperplanes containing a line parallel to the time-axis. 2) If, additionally, $\omega_1$ and $\omega_2$ are doubling, $\log \frac{d\omega_2|_E}{d\omega_1|_E} \in VMO(\omega_1|_E)$, and $E$ is relatively open in the support of $\omega_1$, then their tangent measures at {\it every} point of $E$ are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that in complementary $\delta$-Reifenberg flat domains, if $\delta$ is small enough and $\log \frac{d\omega_2}{d\omega_1} \in VMO(\omega_1)$, then $\Omega_1 \cap \{t<t_2\}$ is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian. 3) We establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure.

ABSTRACT. We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in R n+1 , n ≥ 2, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems: (1) Let Ω1 and Ω2 be disjoint domains in R n+1 , n ≥ 2, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothesis on the set E ⊂ ∂Ω1 ∩ ∂Ω2 of mutual absolute continuity of the associated caloric measures ωi with poles atpi = (pi, ti) ∈ Ωi, i = 1, 2. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of ω1|E is n + 1 and the tangent measures of ω1 at ω1-a.e. point of E are equal to a constant multiple of the parabolic (n + 1)-Hausdorff measure restricted to hyperplanes containing a line parallel to the time-axis.
(2) If, additionally, ω1 and ω2 are doubling, log dω 2 | E dω 1 | E ∈ VMO(ω1|E), and E is relatively open in the support of ω1, then their tangent measures at every point of E are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that if Ω1 is a δ-Reifenberg flat domain for δ small enough and Ω2 = R n+1 \ Ω1, and log dω 2 dω 1 ∈ VMO(ω1), then Ω1 ∩ {t < t2} is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian.
(3) Finally, we establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure. Assuming that Ωi, 1 ≤ i ≤ 3, are quasi-regular domains for both the heat and the adjoint heat equations, the set of points on ∩ 3 i=1 ∂Ωi, where the three caloric measures are mutually absolutely continuous has null caloric measure. In the course of proving our main theorems, we obtain new results on heat potential theory, parabolic geometric measure theory, and nodal sets of caloric functions, that may be of independent interest. CONTENTS 1. Introduction  In this paper, we study two-phase problems for harmonic measure associated with the heat equation, which is traditionally called caloric measure. We consider two disjoint open sets Ω + and Ω − in R n+1 , and ω + and ω − the associated caloric measures, so that their common boundary ∂Ω + ∩ ∂Ω − has positive ω + measure. Our main goal is to study how mutual absolute continuity of the respective caloric measures provides information about the infinitesimal structure of ∂Ω + ∩ ∂Ω − .
In the elliptic case, Bishop, Carleson, Garnett, and Jones studied in [BCGJ89] a twophase problem for simply connected planar disjoint domains: they showed that the two harmonic measures are mutually singular if and only if the intersection of the respective sets of inner tangent points has zero one-dimensional Hausdorff measure. More recently, in [KPT09], Kenig, Preiss, and Toro studied the higher-dimensional case, assuming that Ω + and Ω − = R n+1 \ Ω + are both NTA-domains (as defined by Jerison and Kenig in [JK82]). Combining the blow-up analysis at points of mutual absolute continuity in [KT06] with the theory of tangent measures along with the monotonicity formula of Alt, Caffarelli and Friedman (see [ACF84]), they showed that mutual absolute continuity of the interior and exterior harmonic measure ω + and ω − implies that the harmonic measure has Hausdorff dimension n and that as we zoom in at ω + -a.e. point of the common boundary, ∂Ω ± looks flatter and flatter. This blow-up technique was further improved by Azzam, the first named author, and Tolsa [AMT17] and the same authors along with Volberg in [AMTV19], who eventually showed that, without any further assumption on the domains, the harmonic measure restricted to the set of mutual absolute continuity is n-rectifiable. The connection between the Riesz transform and n-rectifiability was an element of major importance in the proofs of [AMT17] and [AMTV19] that allowed to improve on the results of [KPT09] from this perspective as well (since n-rectifiable measures have Hausdorff dimension n).
All the aforementioned results describe a.e. phenomena, while it is interesting to investigate conditions that ensure some nice limiting behavior of our blow-ups at every point. In [KT06], Kenig and Toro considered Ω + and Ω − to be complementary 2-sided NTA domains and log dω − dω + ∈ V M O(dω + ), and they showed that, in a sequence of arbitrarily small scales, the boundary around any point starts resembling the zero set of a harmonic polynomial (instead of a hyperplane). If the domains are δ-Reifenberg flat, for δ small enough, then the conclusion improves to a hyperplane implying that the domains are Reifenberg flat with vanishing constant. They also proved that the same conclusion holds without the δ-Reifenberg flatness assumption if we assume that Ω + is two-sided NTA, ∂Ω + is Ahlfors-David regular and the logarithms of the interior and exterior Poisson kernels are in VMO with respect to the surface measure on ∂Ω + . Badger went a step further in [Bad11], showing that those harmonic polynomials are always homogeneous, while later, in [Bad13], he explored the topological properties of sets where the boundary is approximated by zero sets of harmonic polynomials in this way. In [AM19], Azzam and the first named author, among others, extended the results in [KPT09], [KT06], and [Bad11] to general domains and elliptic measures associated with uniformly elliptic operators in divergence form with merely bounded coefficients that are also in the Sarason's space of vanishing mean oscillation with respect to ω + . For relevant results for elliptic operators with W 1,1 -coefficients see [TZ17].
It is interesting to understand if similar phenomena occur also in the context of parabolic PDEs and, in particular, the heat equation. The implementation of blow-up methods in the context of one-phase free boundary problems for the heat equation was initiated by Hofmann, Lewis, and Nyström in [HLN04]. They extended the work of Kenig and Toro showing that, in parabolic chord arc domains with vanishing constant, the logarithm of the density of caloric measure with respect to a certain projective Lebesgue measure (or else the Poisson kernel) is of vanishing mean oscillation, and also obtained a partial converse, which amounts to a one-phase problem in Reifenberg flat domains with parabolic uniformly rectifiable boundaries. The full converse was proved by Engelstein in [Eng17A], where a key fact of his proof was a delicate classification of "flat" blow-ups, which was an open problem in the parabolic setting. He also examined free boundary problems with conditions above the continuous threshold. This problem was already solved by Nyström (see [Nys06B] and [Nys12]) in graph domains under the assumption that that the Green function is comparable with the distance function from the boundary.
Results analogous to those in [KT06] for the heat equation were considered by Nyström in [Nys06A]. He proved that, if Ω and R n+1 \ Ω are parabolic NTA domains with parabolic Ahlfors regular boundary and the logarithms of the associated Poisson kernels are of vanishing mean oscillation, then a portion of ∂Ω suitably away from the poles of the caloric measures is Reifenberg flat with vanishing constant. Furthermore, it was shown in [Nys06C] that if one drops the Ahlfors regularity hypothesis and, instead of parabolic NTA, they ask for the domains to be δ-Reifenberg flat for δ > 0 small enough, then the same conclusion holds if one substitutes the assumption on the Poisson kernels with a vanishing mean oscillation hypothesis on the logarithm of the density dω − /dω + of the caloric measures associated with the two domains. We remark that the proof uses the construction of the Green function and caloric measure with pole at infinity. Finally, we refer to [Eng17B] for an interesting geometric result for planar NTA domains with an application to the previously discussed one-phase problem.
To the best of our knowledge, the present paper is the first one that studies free boundary problems in so general domains.
A proficient way to address the kind of questions we referred above is to analyze the fine structure of the boundary by "zooming-in" on boundary points. There are two ways to do that. The first one is by looking at the Hausdorff convergence of rescaled copies of the support of a measure as we zoom in, for which we follow the framework of Badger and Lewis [BL15].
Note that C r (x) is a euclidean right circular cylinder centered atx of height 2r 2 and radius r. Ifx =0, we simply write C r (x) = C r unless stated otherwise. We also need the following time-backwards and time-forwards versions of the parabolic ball: r (x, t) = {(y, s) ∈ R n+1 : y ∈ B r (x), t − r 2 < s < t} C + r (x, t) = {(y, s) ∈ R n+1 : y ∈ B r (x), t < s < t + r 2 }.
Let A ⊂ R n+1 be a set and let S be a collection of subsets of R n+1 . Givenx ∈ A and r > 0 we define where "dist" denotes the distance with respect to the parabolic norm · . We say that A is locally bilaterally well approximated by S , also abbreviated LBW A(S ), if for all ε > 0 and all K ⊂ A compact, there is r ε,K > 0 such that Θ S A (x, r) < ε fro allx ∈ K and 0 < r < r ε,K . In other words, forx ∈ A to be a S point means that, as we zoom in on A at the pointx, the set A resembles more and more an element of S . Remark that this element may change as we move from one scale to the next.
The second way is by investigating the weak convergence of rescaled copies of the measure itself. In [Pr87], Preiss developed the theory of tangent measures, which turned out to be vital in the study of two-phase problems for harmonic and elliptic measure; his results are at the core of [KPT09], [Bad11], [AMT17], [AMTV19], and [AM19]. The definition of a tangent measure can be readily adapted to the parabolic context, in which R n+1 is naturally endowed with a set of non-isotropic dilations.
If µ and ν are Radon measures in R n+1 , we define We say that ν is a tangent measure of µ at a pointx ∈ R n+1 and write ν ∈ Tan(µ,x) if ν is a non-zero Radon measure on R n+1 and there are sequences c i > 0 and r i ց 0 so that c i Tx ,r i [µ] converges weakly to ν as i → ∞.
Let us record here that a parabolic version of Besicovitch covering theorem is available for parabolic balls (see [It18]), which allows for the basic properties of tangent measures to hold in the parabolic setting by the same proofs as in the Euclidean case. For further details, we refer to Section 4.
The geometry of the blow-ups at a point of mutual absolute continuity of caloric measures is intimately related with that of nodal sets of caloric functions. We define the heat and adjoint heat equations by (1.1) Hu := ∆u − ∂ t u = 0, and H * u := ∆u + ∂ t u = 0, and a C 2,1 function satisfying Hu = 0 (resp. H * u = 0) is called (resp. adjoint) caloric function.
Let Θ denote the set of caloric functions vanishing at0, P (d) the set of the caloric polynomials vanishing at0, and F (d) the set of homogeneous caloric polynomials of degree d. In Lemma 6.1, we show that, for any caloric function h on R n+1 , there exists a unique Radon measure ω h such that The measure ω h is called caloric measure ω h associated to h. In fact, this is the unique adjoint caloric measure with pole at infinity in Ω ± = {h ± = 0} and h ± is the Green function with pole at infinity. If h is an adjoint caloric function in R n+1 , we define the adjoint caloric measure analogously.
Thus, we define Given a caloric function h, we use the notation Σ h := {h = 0}. Furthermore, we indicate The families Θ * , P * (d), F * (d),T * , P * (d), F * (d), T * Σ , P * Σ (d), and F * Σ (d) are defined analogously for adjoint caloric functions, polynomials, and homogeneous polynomials. Set, moreover, F Σ := {V ⊂ R n+1 : V is an n-plane through0 containing a line parallel to the t-axis} Potential theory is also very important in problems relating the metric properties of caloric measure to the geometry of the boundary but, in order to take full advantage of it, we need to ensure that the boundary is regular enough so that the continuous Dirichlet problem is well-posed in that domain. One can find several geometric conditions in the literature that imply the desired regularity of the boundary; e.g., the parabolic Reifenberg flatness (see the definition before Corollary 1.3), the non-tangential accessibility (see e.g. [Nys06A,), and the time-backwards Ahlfors-David regularity (see [GH20]). A well-known condition which implies regularity of the boundary in the Euclidean setting and has been used in the harmonic measure framework is the so-called Capacity-Density Condition (see e.g. [AMT17]). The CDC ensures that the capacity of the complement of Ω in each ball centered at the boundary is large in a scale invariant way. We will now introduce two similar notions of thickness that, in the parabolic context, serve the same purpose as the CDC.
Given ρ > 0 andx = (x, t) ∈ R n+1 we define the heat ball centered atx with radius ρ to be the set where Γ(·, ·) stands for the fundamental solution of the heat equation (see Section 3). We We say thatx ∈ ∂Ω belongs to the parabolic boundary of Ω (resp. adjoint parabolic boundary) and writex ∈ PΩ (resp.
We also refer to Section 3 for the definition of Cap(·), the thermal capacity of a set. Cap E(ξ j ; r 2 j ) ∩ Ω c ≥ c r n j .
If F = {ξ}, then we say that Ω satisfies the TBCPC atξ. We remark that c,ξ j , and r j may depend on F .
Let Ω + and Ω − be two disjoint open sets in R n+1 with S ′ Ω + ∩ S ′ Ω − = ∅ and assume that F ⊂ S ′ Ω + ∩ S ′ Ω − is compact. Then we say that Ω + and Ω − satisfy the joint TBCPC at F if there exists a sequence (ξ j , r j ) j≥1 ⊂ F × (0, 1) so that r j → 0 and (1.3) holds for Ω ± , then there exists a subsequence of (ξ j , r j ) j≥1 so that (1.3) holds for Ω ∓ (with possibly different constant c). We may define the time-forwards CPC (TFCPC) by replacing the heat balls by adjoint heat balls in the definitions above.
Analogously, we say that Ω satisfies the Time-Forwards Capacity Density Condition (TFCDC) if, for sime c > 0, The joint TBCPC guarantees regularity of the parabolic boundary for the heat equation and is a much weaker condition than TBCDC, which, in turn, is a natural setting for our problems (see e.g. [AMT17]). The joint TBCPC assumption is a sufficient condition for our blow-up arguments to work and, at the moment, it is not clear how to remove it from our hypotheses. In the case of harmonic and elliptic measures, it has been proved in [AMTV19, Lemma 5.3] and [AM19, Lemma 4.13] that mutual absolute continuity of the interior and the exterior measures implies an even stronger version of (1.3) in terms of the (n + 1)dimensional Lebesgue measure of euclidean balls. This method, however, is based on a dichotomy argument and cannot be generalized directly to the caloric measure because it is not enough to obtain that the exterior of the domain is "large" in the whole parabolic ball; we need to know that this is true in the time-backwards cylinder, which complicates things significantly.
Criteria for parabolic Wiener regularity have been extensively studied in the literature (see Section 3 for the definition and more references). This property is particularly important to our purposes because we need to extend the Green function by zero to the complement of the domain in a continuous way. In order to work in more generals domains, the intrinsic difficulties of heat potential theory constitute a challenging obstacle to overcome. For instance, we remark that Harnack inequality is time-directed (see e.g. [Wa12, Corollary 1.33]) and that it is possible to construct domains so that the capacity of the set of the irregular points for the heat equation is positive (see [TW85,p.336]). The most general class of domains in which (some of) our arguments work seem to be the so-called quasi-regular domains for the heat operator (resp. adjoint heat operator), which amounts to domains for which the set of irregular points of the essential boundary for the heat equation (resp. adjoint heat equation) is polar (see Section 3). This is in accordance to the elliptic theory where, indeed, the set of irregular points is always polar (which is not the case in the parabolic context).
Before we state our first main theorem, which is a generalization of [KPT09] and [AM19, Theorem I]. we need the notion of parabolic Hausdorff dimension of a Borel measure ω. This is defined as where "dim" stands for the parabolic Hausdorff dimension (see Section 4).
Moreover, if Ω ± also satisfy the TFCDC, The theorem above (and similarly all the theorems we prove) can be formulated for adjoint caloric measures if we assume the joint TFCPC and the TBCDC in place of the joint TBCPC and TFCDC and let E be in PΩ + ∩ PΩ − . We point out that caloric measure is not necessarily doubling in domains such as the ones considered in Theorem I and also that it is natural to study mutual absolute continuity of ω + and ω − on the lateral part of the boundary because it cannot occur elsewhere. For more details on this matter, we refer to Section 9.
Secondly, we prove the caloric equivalent of Tsirelson's theorem [Ts97] about triplepoints for harmonic measure. Tsirelson's method is based on the fine analysis of filtrations for Brownian and Walsh-Brownian motions, although, more recently, Tolsa and Volberg [TV18] obtained the same result using analytical tools and, in particular, blow-up arguments, which is exactly the method we follow as well.
be three disjoint open sets which are quasi-regular for both H and H * . Let also ω i be the caloric measure in Ω i with pole atp i ∈ Ω i and assume Then ω i (E) = 0 for i = 1, 2, 3.
Note that Theorem II does not need neither the joint TBCPC assumption on the domains nor regularity for H * . Quasi-regularity is all we need to assume since, by an interior approximation argument that we prove in Section 3, we show that it suffices to study the problem in regular domains.
Under a pointwise VMO-type condition on dω − /dω + on a particular subset of the boundary E, we prove a local analogue of [AM19, Theorem II].

Theorem III.
Let Ω + and Ω − be two disjoint domains in R n+1 which are quasi-regular for H and regular for H * and let ω ± be the caloric measures associated with Ω ± with poles p ± ∈ Ω ± . Let also E ⊂ P * Ω + ∩ P * Ω − ∩ supp ω + be a relatively open set in supp ω + such that ω + (E) > 0 and ω − | E ≪ ω + | E . Assume that Ω + and Ω − satisfy the joint TBCPC at all points of E and, if we set f = dω − | E dω + | E to be the Radon-Nikodym derivative of ω − | E with respect to ω + | E , we assume that forξ ∈ E, and Tan(ω + ,ξ) = ∅, then there is k ≥ 1 such that Tan(ω + ,ξ) ⊂ F (k) and Furthermore, if Ω + and Ω − also satisfy the TFCDC, then Before we go any further, let us introduce the space of Vanishing Mean Oscillation. Given a Radon measure ω in R n+1 , we denote and we say that f ∈ VMO(ω) if f ∈ L 1 loc (ω), and The condition (1.7) implies a vanishing mean oscillation assumption on dω − /dω + on E. However, in general, these assumptions are not equivalent. For more details, we refer to [AM19, Section 7].
The next theorem is a global version of Theorem III under the additional assumption that ω ± is doubling. This is the analogue of [AM19, Theorem III], which, in turn generalized, the main results of [KT06] and [Bad11]. We prove that at sufficiently small scales, the support of ω + resembles the zero set of an adjoint caloric polynomial uniformly on compact subsets of the set of mutual absolute continuity. This assertion can be formulated both in terms of Θ P Σ (d) ∂Ω ± and the functional d 1 (·, P(d)), which is essentially a distance between measures and the set P(d) (see (4.4) for the exact definition).

Theorem IV.
Let Ω + , Ω − , ω + , ω − , f and E be as in the statement of Theorem III. If If Ω + and Ω − also satisfy the TFCDC, then Theorem IV also applies directly to the study of (parabolic) Reifenberg flat domains, and gives an alternative proof of a result that Nyström proved with different techniques in [Nys06C]. We recall that Ω ⊂ R n+1 is δ-Reifenberg flat, δ > 0, if for R > 0 andξ ∈ ∂Ω there exists a n-plane Lξ ,R throughξ containing a line parallel to the time-axis and such that wheren is the normal vector to Lξ ,R pointing into Ω atξ.
The analogous result for harmonic measure can be found in [KT06,Corollary 4.1], while for the original theorem for caloric measure we refer to [Nys06C, Theorem 1].
Discussion of the proofs. In the current paper, we follow the same strategy as in [AM19], although there are many significant challenges along the way that we overcome to adapt this method successfully to the parabolic setting. In Section 3, we exhibit various properties of thermal capacity, the most important of which is the backwards in time self-improvement of a pointwise time-backwards capacity density condition on a particular scale. This is the building block of the proofs of several PDE estimates around the boundary that are absolutely necessary for our blow-up arguments in Section 8. Those results can find applications in future works and are new and interesting on their own. In Section 4, we develop the required parabolic GMT framework and confirm that, due to a parabolic Besicovitch covering theorem, the theory of tangent measures translates almost unchanged to the parabolic setting. Moreover, we show that the blow-ups of the "parabolic surface measure" on euclidean rectifiable sets are parabolic flat, meaning that there is a plane that contains a line parallel to the time axis such that the blow-up measure is the parabolic Hausdorff measure restricted to that plane. Han and Lin [HL94] had proven that if h is caloric and Σ h = {h = 0}, then in any ball centered on Σ h , it holds that Σ h can be written as the union of its regular set, which is a smooth n-submanifold and it singular set, which is an (n − 1)-rectifiable set. Unlike the harmonic case, this is not enough for our purposes and so, in Section 5, we investigate the finer structure of the regular set separating space and time-singularities. We demonstrate that anyx ∈ R x = Σ h ∩ {|∇h| = 0} has a neighborhood so that Σ h is given by an admissible n-dimensional smooth graph, while for anyx ∈ R t = Σ h ∩ {|∇h| = 0} ∩ {|∂ t h| = 0}, we can find a neighborhood around it in which R t is contained in a smooth (n − 1)-graph. Those results are fundamental to several measure-theoretic arguments and are used repeatedly in the rest of the paper. To the best of our knowledge, this description of the timeregular set is novel in the literature. In Section 6, we prove the existence and uniqueness of the caloric measure associated with a caloric function of the form dω h = −∂ νt hdσ h , where σ h is the "surface measure" on Σ h . In the elliptic setting, this is a straightforward application of the Gauss-Green formula on sets of locally finite perimeter, which is not the case here. The aim of Section 7 is to prove that, if the parabolic tangent measures Tan(ω,ξ) to a given Radon measure ω are caloric measures associated with caloric polynomials and there is a measure corresponding to a homogeneous caloric polynomial of degree k, then all the elements of Tan(ω,ξ) are caloric measures associated with a homogeneous caloric polynomial of the same degree. The connectivity arguments for tangent measures translate unchanged from [AM19], although, this is not the case for the analogues of the main lemmas from [Bad11]. In fact, it is not clear to us how to adapt Badger's proofs directly to our case, although, his ideas inspired us to come up with new ones, which we find simpler even for harmonic functions. Section 8 groups the main one and two-phase blow-up lemmas, for which we follow the approach in [AMT17], [AMTV19], and [AM19]. However, there are substantial obstacles such as the compact embedding of a "parabolic" Sobolev space W 1,2 in L 2 , the lack of analyticity in time, the unique continuation arguments, and the locality of Lemma 8. 4. Finally, in Section 9 we provide the proofs of the four main theorems using the results from the previous sections. Acknowledgements. We would like to thank Jonas Azzam, Max Engelstein, Luis Escauriaza, Steve Hofmann, Tuomo Kuusi, Andrea Merlo, and Xavier Tolsa for several fruitful discussions on topics related to the current paper. The first named author would like to dedicate this paper to the memory of his teacher, mentor, and dear friend, Professor Michel Marias.
Given an open set Ω, and a pointξ ∈ Ω, we denote by Λ(ξ, Ω) (resp. Λ * (ξ, Ω)) the set of pointsx ∈ Ω that are lower (resp. higher) thanξ relative to ∂Ω, in the sense that there is a polygonal path γ ⊂ Ω joiningξ tox, along which the time variable is strictly decreasing (resp. increasing). By a polygonal path, we mean a path which is a union of finitely many line segments.
Following [Wa12, Definition 8.1], we define the normal boundary if Ω is unbounded and the abnormal boundary ∂ a Ω = {x ∈ ∂Ω : ∃ ε > 0 such that C − ε (x) ⊂ Ω} . The abnormal boundary is further decomposed into ∂ a Ω = ∂ s Ω ∪ ∂ ss Ω, where ∂ s Ω stands for the singular boundary and ∂ ss Ω for the semi-singular boundary, which are defined respectively by 8.40], ∂ a Ω is contained in a sequence of hyperplanes of the form R n × {t}. The essential boundary is defined as ∂ e Ω = ∂ n Ω ∪ ∂ ss Ω = ∂Ω \ ∂ s Ω, replacing ∂Ω by ∂Ω ∪ {∞} if Ω is unbounded. Finally, following [GH20], we define the quasi-lateral boundary to be where (BΩ) T min is the time-slice of BΩ with t = T min and (∂ s Ω) Tmax is the time-slice of ∂ s Ω with t = T max . Observe that for a cylindrical domain U × (T min , T max ), where U ⊂ R n is a domain in the spatial variables, the quasi-lateral boundary coincides with the lateral boundary. By [GH20, Lemma 1.14], both ∂ e Ω and SΩ are closed sets.
Given f : R n+1 → R we write Df = (∇f, ∂ t f ) for the (full) gradient of the function f and, D α,ℓ f = ∂ α 1 x 1 · · · ∂ αn xn ∂ ℓ t f for higher order derivatives, where α ∈ Z n + and ℓ ∈ Z + . If E ⊂ R n+1 and f is a continuous function with compact support in E ⊂ R n+1 , then we write f ∈ C c (E). If Ω is an open set, we denote by C m, m 2 (Ω) the class of functions such that f (·, t) ∈ C m (Ω t ) for any fixed t ∈ (T min , T max ) and f (x, ·) ∈ C m 2 (T min , T max ) for any fixedx ∈ R n+1 such thatx ∈ Ω. If f is C m in both space and time variables, we will simply write that f ∈ C m (Ω). Finally, we say that We write a b if there is C > 0 so that a ≤ Cb and a t b if the constant C depends on the parameter t. We write a ≈ b to mean a b a and define a ≈ t b similarly. Sometimes we also use the notation − F dµ for the average µ(F ) −1 F dµ over a set F ⊂ R n+1 with respect to the measure µ.
If E ⊂ R n+1 is a Borel set and H d stands for the Euclidean d-Hausdorff measure in R n+1 for d ≤ n − 1, we define the d-"surface measure" on E as the measure dσ = dσ t dt, If s ∈ [2, n + 2] and 0 < δ ≤ ∞, we set for E ⊂ R n+1 , and, as in the Euclidean case, define the parabolic s-Hausdorff measure by which is a Borel measure by the Carathéodory criterion.

HEAT POTENTIAL THEORY AND PDE
Let Ω ⊂ R n+1 be an open set. Define the parabolic operator (3.1) H a u := a∆ − ∂ t , and H * a u := a∆ + ∂ t , for a > 0. When a = 1, we simply write H 1 = H and H * 1 = H * for the heat and the adjoint heat operator respectively. Forx = (x, t) ∈ R n+1 , we denote by the Gaussian kernel. Note that Γ a is the fundamental solution associated with H a , i.e., it satisfies (1) H a Γ a (x, t) = δ0(x, t), in the distributional sense, for {t > 0}, where δ0 stands for the Dirac mass at0; (2) Γ a (x, t) = δ 0 (x), for t = 0; for some constant C h > 0 depending on n and a.

Proof
The lower class 8.26], any f ∈ C(∂ e Ω) is resolutive. Therefore, for anyx ∈ Ω, the map f → u f (x) is linear and, by the Riesz representation theorem, there exists a unique probability measure ωx on ∂ e Ω, which is called caloric measure, such that More generally, if f is an extended real-valued function andx ∈ Ω and ∂eΩ f dωx exists, then and is finite, then f is ωx-integrable and (3.4) holds (see [Wa12,Theorem 8.32]). In the last statement, if we also assume that f is Borel, then we obtain that f is resolutive.
If f ∈ C(∂ e Ω), we say that u is a solution of the classical Dirichlet problem with data f if u is caloric and it holds that If there exists a solution to the classical Dirichlet problem in Ω with data f , then f is resolutive and u f is the PWB solution for f in Ω (see [Wa12,Theorem 8.26]).
Let G Ω (·, ·) be the non-negative real-valued function defined in Ω × Ω as where h Ω (·,p) is the greatest thermic minorant of Γ(· −p), which is non-negative (see [Wa12,Definition 3.65] and the paragraph before this definition). We call G(·,p) the Green function in Ω with pole atp ∈ Ω and, by [Wa12,Theorem 8.53], if ψp = Γ(· −p)| ∂eΩ , we also have the representation where the last equality follows from the fact that Γ(·,p) is continuous on ∂ e Ω and thus resolutive. It is straightforward to see that In fact, more is true: if C r := B r × (0, r) is a cylinder of radius r > 0, then there exists c 1 > 0 and c 2 > 0 (independent of r) such that Note that G Ω (·, ·) is lower semi-continuous on the diagonal {(p,p) : p ∈ Ω} and continuous everywhere else in Ω. In fact, it is a supertemperature in Ω and a temperature in Ω \ {p}. Recall that Λ * (p, Ω) is the set of pointsx ∈ Ω for which there is a polygonal path γ ⊂ Ω joiningp tox, along which the time variable is strictly increasing. By [Wa12, Theorem 6.7], it holds that G Ω (·,p) > 0 in Λ * (p, Ω) and G Ω (·,p) = 0 in Ω \ Λ * (p, Ω), for anȳ p ∈ Ω. Ifξ ∈ ∂ n Ω (resp. ∂ ss Ω) is a regular point for H, we have that limx →ξ G Ω (x,p) = 0 (resp. limx →(ξ,t + ) G Ω (x,p) = 0). When the domain Ω where the Green function is defined is clear from the context, we will drop the subscript and just write G(·, ·).
Let us remark that the Riesz measure associated with the Green function G Ω (x,ȳ) in Ω is the caloric measure ωx Ω in Ω (see [Do84]), i.e., Given a Borel measure µ on R n+1 and an open set Ω ⊂ R n+1 , defines a non-negative supertemperature in Ω provided the integral is finite in a dense subset of Ω and is called the heat potential of µ in Ω.
Given a compact set K ⊂ R n+1 , we define its thermal capacity as Equivalently, if u = R 1 (K, Ω) is the smoothed reduction 1 over K in Ω (see [Wa12, Definition 7.23]) and µ u is the associated Riesz measure, then Cap(K, Ω) = µ u (K) (see [ This definition can be extended to arbitrary sets S ⊂ R n+1 : the inner thermal capacity of S is given by If the inner and the outer thermal capacities of S coincide, we say that S is capacitable and we denote Cap(S, Ω) := Cap − (S, Ω) = Cap + (S, Ω). In the case Ω = R n+1 , we simply write Cap(S). If E is a Borel set, then, by [Wa12, Theorems 7.15 and 7.49], it is capacitable.
Observe that if Z 1 ⊂ Z and Z is polar, then Z 1 is polar itself, and a set S ⊂ R n+1 is polar if and only if Cap(S) = 0 (see [Wa12,Theorem 7.46]). Moreover, if a set is contained in a horizontal hyperplane its capacity is equal to its n-dimensional Lebesgue measure (see [Wa12,Theorem 7.55]). For more properties of polar sets we refer the reader to [Wa12,Chapter 7].
An open set Ω is quasi-regular (resp. regular) if the set of irregular points of ∂ e Ω is polar (resp. empty). We remark here that there are domains so that the capacity of the set of the irregular points of ∂ e Ω for the heat equation is positive (see [TW85,p. 336]). This comes in contrast to the corresponding result in elliptic potential theory, where the set of irregular boundary points has zero capacity and thus, is polar.
Lemma 3. 4. Let C r be a cylinder of radius r centered atξ 0 ∈ R n+1 and let K ⊂ C r be a compact set such thatξ 0 ∈ K. If s ∈ (0, 2], then there exists c 2 > 0 (independent of K) such that Proof. If we set ρ = min(diam(K), r), it holds that K ⊂ C ρ ∩ C r . By the parabolic version of Frostman's lemma, whose proof is analogous to the Euclidean one and is omitted (see e.g. [Mat95,Lemma 8.8]), we can find a Borel measure µ ′ supported on K, such that • µ ′ (C r (ȳ)) r n+s , for any r > 0 andȳ ∈ R n+1 , and Let G 2r be the Green function in C 2r and define µ = µ ′ ρ −s . We claim that the corresponding heat potential G 2r µ(x) 1 for anyx ∈ C 2r . Indeed, notice that Using (3.8) along with that µ ′ is supported on K and has (n + s)-growth, we have that , arguing as above, we infer that concluding the proof of the claim.
If we normalize µ so that G 2r µ(x) ≤ 1, we deduce that µ is an admissible measure for the definition of thermal capacity on compact sets. Therefore, , and the proof of the lemma is now complete.
As a and b are arbitrary, we may approximate R − a,b (ξ 0 ; r) by a sequence of compact subsets of the form R − a k ,b k (ξ 0 ; r) and obtain our result.
Forx = (x, t) ∈ R n+1 , we define the heat ball centered atx with radius ρ > 0 to be the set It is not hard to see that E(x, t; ρ) is a convex body, axially symmetric about the line {x}×R, and

Let us set
to be the closed heat annulus of radius ρ. It is clear that ifȳ ∈ A(ξ 0 ; ρ/2, ρ) satisfies 0 < t − s < ρ/2, then it holds that Remark 3.6. Note that if ρ = r 2 , α ∈ (0, 1), and t − s = αr 2 , then the region given by (3.19) is an annulus in the spatial variable which can be covered by at most c ′ n many (spatial) cubes of sidelength √ αr.
There are parabolic analogues of the Wiener criterion that determine whether a point ξ ∈ ∂ n Ω is regular in terms of the capacity of the complement of Ω that lies either in heat balls centered atξ 0 or in time-backwards parabolic cylinders R − ar (ξ 0 ). To be precise,

The necessity was proved by Evans and Gariepy [EG82, Theorem 1] and the sufficiency by
This follows from (3.20) and [Br90, Theorem 1.13]. Alternatively,ξ 0 is regular if and only if In fact ifξ 0 is regular, then (3.22) holds for all λ > 1.
Remark 3. 7. In [EG82] and [Lanc73], the authors do not specify that this criterion only works for points on ∂ n Ω, although it is clear this is the case. Observe that, by definition, if ξ 0 ∈ ∂ a Ω, there exists ρ > 0 small enough so that E(ξ 0 ; ρ) ∩ Ω c = ∅ and thus, the integral in (3.21) clearly converges. Nevertheless, Watson [Wa14, Theorem 4.1] proved that a point ξ 0 ∈ ∂ ss Ω is regular if and only if there exists r 0 > 0 such that then there exists a ∈ (0, 1/2) depending on n such that Proof. If we set 2 A different necessary and sufficent condition is given by [Land69].
then, for anyx ∈ E k , it holds Thus, we can cover E k by at most c(n)k n/2 number of cylinders C j (k) of radius 2 −k r, which, in view of the subadditivity of capacity and Lemma 3.3, infers that Since k≥1 k n/2 2 −kn converges, we can find M > 0 depending on n so that Cap(E M ) ≤ cr n , which implies (3.23) for a = 2 −M .
Similarly we can prove the following lemma.
As backwards cylinders R − a (ξ 0 ; r) appear more naturally in applications, it is interesting to know a version of the Wiener's criterion with R − a (ξ 0 ; r) instead of heat balls. This can be obtained using the following lemma (compare it with [GZ82, Theorem 3.1]).
Lemma 3.11. Let Ω be an open set andξ 0 ∈ ∂ n Ω. If there exists a constant a > 0 such that wherec is given in the proof of Lemma 3. 10. The converse direction follows from Lemma 3.9.
We will now introduce a class of regular domains that has played an important role in (free) boundary value problems for harmonic and elliptic measure.
Letξ 0 ∈ SΩ. If there exists a ∈ (0, 1] and c > 0 such that then, we say that Ω has the time backwards cylindrical capacity density condition at the pointξ 0 . Although the TBCDC looks more general, because of Lemmas 3.10 and 3.8, we can see that the two conditions are in fact equivalent. Note that, by Lemma 3.11 and Remark 3.13, if Ω has the TBCDC at the pointξ 0 ∈ SΩ, thenξ 0 is a regular point and belongs to ∂ n Ω.
, we obtain the range of r in (3.28).
A crucial property of our definitions of TBCDC is their invariance under parabolic scaling.
Proof. Let us observe that forζ ∈ S Ω and every r > 0 we have that Therefore, the measure ρ n µ is an admissible measure for Cap(K) and thus, The proof of the converse inequality is similar and we omit it. This proves (3.31), which, in turn, is valid if we replace the backwards in time cylinders C − r/ρ by R − a (ζ; r/ρ), the truncated cylinders defined in (3.25). Thus, the TBCDC for Ω readily follows from the TBCDC for Ω.
The next lemma was proved in [GH20, Lemma 2.2] in the particular case of domains with Ahlfors-David regular boundaries that, in addition, satisfy the time-backwards Ahlfors-David regular condition. In light of (3.15), those domains satisfy the TBCDC.
Remark 3. 16. Lemma 3.15 holds for the adjoint caloric measure if we replace the cylinders R − a (ξ 0 , r) and R + a (ξ 0 , r) with their reflections across the hyperplane passing through their centers orthogonal to the time axis.
If Ω is regular there is the above lemma has a much easier proof based on the reduction function. Also, the dilation factor M 0 need not be taken large enough and just M 0 = 2 does the job.

Proof.
As in the proof of the previous lemma, we assume thatξ 0 =0 and use the notation R − a,r and R + a,r . Moreover, we assume that Cap(K) > 0 since otherwise (3.35) is trivial. We also denote where in the last step we used the weak Harnack inequality since u is a non-negative supertemperature and thus, a non-negative supercaloric function (see [Lieb96, Corollary 6.24, p. 128]).
By the definition of u, we have that for any non-negative supertemperature v in C βr (ξ 0 ) such that v ≥ 1 on K, it holds that v ≥ u and thus, (3.35) is true for v as well. Now, if w is a subtemperature in C βr (ξ 0 ) so that w = 0 on K and w ≤ 1 in C 2r (ξ 0 ), then 1 − w is a nonnegative supertemperature in C βr (ξ 0 ) that is identically 1 on K and (3.36) readily follows. Finally, by the regularity of ∂Ω ∩ C βr (ξ 0 ), we may extend ωx(C βr (ξ 0 )) by 1 in C βr (ξ 0 ) \ Ω and use [Wa12,Theorem 7.20] to show that it is a non-negative supertemperature in C βr (ξ 0 ) that is 1 on K and non-neagtive in C βr (ξ 0 ) and use (3.35) to show (3.34) for M 0 = 2.
Remark 3. 18. One can show that (3.36) holds for upper semicontinuous weakly subcaloric functions as well. Indeed, if we follow the proof of Lemma 3.15 substituting Γ(·, ·) with the Green function G C βr (ξ 0 ) (·, ·) and using (3.8) along with the weak minimum principle 3 (instead of the resolutivity of characteristic functions of Borel sets) we can show that if v is a non-negative supercaloric function in Ω ∩ C βM r (ξ 0 ) such that lim infx →ξ v ≥ 1 for everȳ ξ ∈ ∂ n Ω ∩ C βM r (ξ 0 ), then v satisfies (3.35). The rest of the proof is the same as before and we skip the details.
for some constant c > 0 depending on n and a.
and note that a k−1 = βa k . By (3.36), it holds that In particular, the latter inequality holds in C a 1 r (ξ 0 ). We apply (3.36) once again in C a 2 r (ξ 0 ) and get that The latter inequality holds in C a 3 r (ξ 0 ) and we may apply (3.36) in C a 4 r (ξ 0 ). By iteration, if M is an integer such that a 2M +2 ≤ ρ ≤ a 2M , we get that Therefore, since Given T > 0, we define E(T ) := {(x, t) ∈ R n+1 : t < T }. Moreover, for an open set Ω,x ∈ ∂Ω and r > 0 we define Ω r := Ω r (x) := Ω ∩ C r (x) and Ω r (T ) := Ω r ∩ E(T ).
Lemma 3. 20. Let Ω ⊂ R n+1 be an open set satisfying the TBCDC, and letξ 0 ∈ SΩ and 0 < r < (t 0 − T min )/2. Then for any non-negative function u, which is either weakly subcaloric or subtemperature in Ω 2r (T 1 ) vanishing continuously on C 2r (ξ 0 )∩∂ e Ω∩E(T 1 ), it holds that Proof. For subtemperatures, this is a direct consequence of Lemma 3.19 (with a slightly larger cylinder on the right hand-side of (3.39)), while for weakly subcaloric functions one can follow the proof of [GH20, Lemma B.2] (which still works for TBCDC domains).
A pretty standard result in elliptic theory is that polar sets have zero harmonic measure. The same is true for caloric measure as well. As we were not able to find an appropriate reference, we will write the proof for completeness. Proof. Fixx ∈ Ω. Since E ∩ ∂Ω is polar as a subset of the polar set E, without loss of generality, we may assume that E ⊂ ∂Ω. As we have that Cap(E) = 0, by (3.12), there exists a sequence of open sets S j ⊃ E such that lim j→∞ Cap − (S j ) = 0. Set now E := j S j , and note that E is Borel, E ⊂ E, and Cap( E) = 0. Therefore, without loss of generality, we may assume that E is Borel.
Since v i is supertemperature on Ω i , it is lower semicontinuous, and remains so when we extend it by zero to Ω c i . Thus, for eachx ∈ F there is a closed cylinder let C i (x) be the cylinder of the same center and half the radius of C ′ i (x). Let {C j } be a Besicovitch subcovering (see Lemma 4.8) and if E j is the closed ellipsoid of revolution around the axis of the cylinder C ′ j that is inscribed in C ′ j , we define Note that Ω i is open. Indeed, to show that Ω c i is closed considerx k ∈ Ω c i , k ≥ 1 and x k →x. Then we need to showx ∈ Ω c i . If there is a subsequence contained in Ω c i , we are done. Otherwise, assume thatx k ∈ H\Ω c i = H ∩ Ω i . Ifx k ∈ E j for infinitely many k, thenx ∈ E j and we are done since E j is closed and E j ⊂ Ω c i . Otherwise, supposex k is not in any E j more than finitely many times. By the bounded overlap property, if j(x k ) is such thatx k ∈ E j(x k ) , then diam(E j(x k ) ) ↓ 0 as k → ∞, and since the ellipsoids are centered on F ⊂ Ω c i ,x ∈ Ω c i ⊂ Ω c i , and we are done. Thus, Ω i is open. Set V j = R n+1 \ E j and note that ifz 1 j = (z 1 , t 1 ) andz 2 j = (z 2 , t 2 ), are the points on ∂E j ∩ ℓ j , where ℓ j is the line containing the axis of C j , so that t 1 < t 2 , it holds that z 1 ∈ ∂ ss V j and everyx ∈ ∂E j \ {z 1 j } ⊂ PV j is in ∂ n V j . Moreover, it is clear by (3.21) and Remark 3.7 that everyx ∈ ∂E j is regular for V j . Therefore, by [Wa12,Corollary 8.47], it holds that ∂E j ∩ Ω i consists of regular points of ∂ e Ω i . As the points of Z ∩ ∂ e Ω i are regular for Ω i then, by another application of [Wa12, Corollary 8.47], they are regular for Ω i as well. Therefore, Ω i is regular for H and H * , and E ⊂ ∂ e Ω i . Let this gives ω i (H) ≤ 1 2 ω i (E) and hence ω i (G) > 0. Similarly, by the maximum principle, this gives Moreover, by the maximum principle, and since Ω i is a regular domain, Thus, Note that ω 1 ≪ ω 1 on G by the maximum principle and since ω 1 (G) > 0, it is not hard to show using the Lebesgue decomposition theorem that there is G 1 ⊂ G of full ω 1 -measure upon which we also have ω 1 ≪ ω 1 . Hence ω 1 (G 1 ) > 0, which implies ω 2 (G 1 ) > 0. The same reasoning gives us a set G 2 ⊂ G 1 upon which ω 2 ≪ ω 2 ≪ ω 2 . Thus, ω 2 ≪ ω 1 ≪ ω 2 on G 2 .
The notion of halving metric space was first introduced by Korey [Kor98] and plays an important role in the proofs of our theorems.  Proof. Set ω := ωx Ω and let us assume that there exists E ⊂ ∂Ω such that ω(E) > 0 and ω(F ) = ω(E)/2 for every F ⊂ E. Given s ∈ R and v ∈ S n , we define the half-space H s,v := {ξ ∈ R n+1 :ξ · v ≥ s}. By the mean value theorem and our assumption, the map s → ω(H s,v ∩ E) is not continuous for any v ∈ S n . In particular, for any v ∈ S n , there exists s v , such that ω(∂H sv,v ∩ E) > 0. Set now V v := ∂H sv,v , which is an n-dimensional plane. Define S := {(y ′ , y n , τ ) ∈ S n : y ′ = 0}, which is a contradiction.
Every result related to the heat equation we have stated so far has a dual for the adjoint heat equation H * u = 0 obtained by reversing the sign of the time variable. Therefore, we can define the associated fundamental solution Γ * , the Green function G * Ω , the adjoint caloric measure denoted by ωp * , the associated parabolic capacity Cap * and so forth. A solution of H * u = 0 is called adjoint caloric. Remark that, by [Wa12, Theorem 6.10], we have that Γ(x,ȳ) = Γ * (ȳ,x) and G Ω (x,ȳ) = G * Ω (ȳ,x). For the regularity of the ∂ * e Ω for H * and the corresponding capacity density conditions, it is pretty clear that we should just take time-forwards cylinders and the so-called co-heat balls, which are defined as the heat balls using the adjoint heat kernel.

HAUSDORFF AND TANGENT MEASURES
For a proof see [He17, Lemma 3.2]. Note that this was originally stated under the additional hypothesis that E is Euclidean d-rectifiable but it is easy to see it is redundant. Moreover, by [He17,Lemma 3.8], it holds that there exists c 1 > 0 and c 2 > 0 depending only on d, such that on The Hausdorff dimension of a Borel measure ω is defined by This definition is related with the concept of pointwise dimension of ω atx ∈ supp ω. More specifically, if log µ(C r (x)) log r and d µ (x) = lim sup r→0 log µ(C r (x)) log r denote the lower and upper pointwise dimension respectively, we can argue as in [BW06] to show that dim(ω) = ess sup{d µ (x) :x ∈ supp µ}.
If d µ (x) = d µ (x), we denote the common value by d µ (x). Observe that for ℓ = 1/2 these functions correspond to Lipschitz functions with respect to the parabolic norm · or the equivalent norm |x ′ − y ′ | + |t − s| 1/2 . For ℓ = 1, those are just the usual Lipschitz functions with respect to the Euclidean norm.

Definition 4.2.
We say that Γ ⊂ R n+1 is an admissible d-dimensional graph if there exists a vector field f : R d → R n−d+1 such that, possibly after a rotation in space and a translation, If f ∈ C m (R d ; R n−d+1 ), 1 ≤ m ≤ ∞, we say that Γ f is an admissible d-dimensional C mgraph. If f is an affine map, then Γ f = V is a plane that contains a line parallel to the timeaxis and we call it an admissible d-dimensional plane. In fact, after rotation in space and a translation, we can always assume By the co-area formula [He17,Theorem D], if E is Euclidean d-rectifiable, then (4.2) σ = cH d+1 p | E . In fact, using the Rademacher theorem in [Or19], one can show that this theorem is true for parabollically rectifiable sets as well, that is, for sets that are exhausted, up to a set of H d+1 p -measure zero, by admissible Lip 1, 1 2 -graphs with 1 2 -derivative in time in BMO. 4 If µ and ν are Radon measures on R n+1 , we define the distance between µ and ν in the parabolic ball C r by where the supremum is taken over all functions f ∈ Lip (1, 1 2 ) (R n+1 ) which are supported in C r and satisfy Lip(f ) ≤ 1. If a sequence of Radon measures µ j converges weakly to a Radon measure µ, we use the notation µ j ⇀ µ.
If µ and ν are Radon measures in R n+1 , we define where the supremum is taken over all functions f ∈ Lip (1, 1 2 ) (R n+1 ) which are supported in C r and satisfy Lip(f ) ≤ 1. By density, it is enough to consider the supremum in the class of C ∞ c (C r ) functions such that Lip(f ) ≤ 1. We also define F r (µ) := F r (µ,0). A standard argument shows that As it is easy to see that where (·) + stands for the positive part of a function, we infer that

Definition 4.4 (d-cone). A set of Radon measures M is a d-cone if c T0 ,r [µ]
∈ M for all µ ∈ M, c, r > 0. Given a d-cone M, the set {µ ∈ M : F 1 (µ) = 1} is referred to as its basis. We say that M has closed (resp. compact) basis if its basis is closed (resp. compact) with respect to the weak topology of the space of Radon measures.
The next lemma collects some of the relevant properties of F r and d r (·, M). For more details, see [KPT09, Section 2] and the references therein.
Lemma 4. 5. Let µ, ν be Radon measures in R n+1 ,ξ ∈ R n+1 and r > 0. The following properties hold: Definition 4.6. We say that ν is a tangent measure of µ at a pointx ∈ R n+1 if ν is a nonzero Radon measure on R n+1 and there are sequences c i > 0 and r i ց 0 so that c i Tx ,r i [µ] converges weakly to ν as i → ∞ and write ν ∈ Tan(µ,x).

Remark 4.7.
A Besicovitch covering theorem for parabolic balls in R n+1 was proved in [It18, Theorem 1.1]. This is an important tool for parabolic geometric measure theory. In particular, one can show that Radon measures in R n+1 satisfy the Lebesgue density theorems and the Lebesgue differentiation theorems with respect to parabolic balls, as reported in the next lemma.
Proof. The proof follows from the argument in [Mat95, Corollary 2.14] using the Besicovitch theorem for parabolic balls in [It18].
Remark 4.9. Once we have made the appropriate modifications in the definitions of the blow-up mappings to reflect the parabolic dilation, all the theorems related to tangent measures that are required to obtain our results hold with the same proofs as in the Euclidean setting.
(2) If, additionally, µ is asymptotically doubling atx ∈ R n+1 , then for any ν ∈ Tan(µ,x) there exists c 1 > 0 and a sequence {r i } ı≥1 decreasing to 0 such that In this case,0 ∈ supp ν for all ν ∈ Tan(µ,x). As a result of Lemma 4.8 we obtain the following localization property of tangent measures.
Proof. With Lemma 4.8 at our disposal, we just follow the proof of [DeL08, Lemma 3.12].
(2) Tan(ν,ȳ) ⊂ Tan(µ,x) for allȳ ∈ supp ν.  Proof. By the parabolic Rademacher theorem in [Or19], one can show that a locally parabolic Lipschitz vector field ψ : where ǫx(r) → 0 as r → 0. In fact, that theorem is only stated for d = n and globally parabolic Lipschitz functions (see e.g. [ Observe that H d+1 p | Γ j (C r (x)) ≈ r −d−1 for anyx ∈ Γ j and r > 0 (the implicit constant depends on the Lipschitz character of Γ j ). Thus, Lemma 4.13 implies that if ν ∈ Tan(H d+1 p | Γ j ,x), there exists a positive constant cx and an admissible d-dimensional plane Vx passing through the origin, such that ν = cxH d+1 p | Vx . Thus, as E is Euclidean drectifiable, it holds that σ = cH d+1 p | E and the result follows.

NODAL SET OF CALORIC FUNCTIONS
Given a caloric function h, we denote by Σ h := {h = 0} the nodal set of h and by σ h the associated surface measure to deduce that Σ h has locally finite H n -measure and Σ h ∩ |Dh| −1 (0) is a (euclidean) (n − 1)-rectifiable set. In particular, for any cylinder C r centered on Σ h , the set Σ h ∩ C r can be decomposed into a union of an n-dimensional C 1 -submanifold C r ∩ Σ h ∩ {|Dh| > 0} with finite n-dimensional Hausdorff measure and a closed set C r ∩ Σ h ∩ {|Dh| = 0} of Hausdorff dimension not larger than n − 1.
For our purposes, we need a finer study of this decomposition in terms of admissible graphs and so we define the regular and the singular set of Σ h by Additionally, we set to be the space-regular and the time-regular sets respectively.
Proof. The proof is an easy application of the implicit function theorem. Indeed, if |∇h(ȳ)| > 0, we can assume without loss of generality that ∂ n h(ȳ) = 0. Thus, we can find a cylinder C ρ (ȳ) centered atȳ in which ∂ n h = 0 and a smooth function ϕ : R n → R such that We remark that, despite the implicit function theorem defines ϕ just locally, we can extend it to a C ∞ -function defined in the whole R n (see e.g. [St70,Theorem 5,p. 181]).
In general, we cannot expect to express R t locally as an admissible graphs (see also the examples after the next lemma). However, we can still make some general consideration about its geometry and we show that its dimension is lower than that of R x . Lemma 5.2 (Structure of the time-regular set). If h : R n+1 → R is a non-zero caloric function, then for everyȳ ∈ R t there existsρ =ρ(ȳ) > 0 and a smooth (n−1)-dimensional C ∞ -graph Σȳ such that R t ∩ Cρ(ȳ) ⊂ Σȳ ∩ Cρ(ȳ). In particular, σ R t = 0.
Proof. If R t = ∅ there is nothing to prove. Fixȳ = (y, s) ∈ R t , and note that since ∂ t h(ȳ) = 0, by the implicit function theorem, we can find ρ = ρ(ȳ) > 0 such that C ρ (ȳ) ∩ Σ h agrees (possibly after a rotation in space) with the graph (x, ϕ(x)) of a C ∞ -function ϕ : R n → R inside C ρ (ȳ). Moreover, since Hh = 0 the latter implies that ∆h(ȳ) = 0. Without loss of generality, we assume that ∂ 2 n h(ȳ) = 0, where ∂ n stands for the partial derivative in x n variable.
Example 5.3. a) Let us consider the caloric polynomial h 1 (x 1 , x 2 , t) = x 2 1 + x 2 2 + 4t. The set Σ h 1 is a rotational paraboloid around the time-axis. Its time-regular set R t is the singleton {0} and its space-regular set is , whose nodal set is a parabolic cylinder. We have that S = ∅, which gives that particular, a line), in contrast to what we have for the function h 1 of the previous example.

CALORIC MEASURE ASSOCIATED WITH A CALORIC FUNCTION
We say that a function f (x, t) is (parabolic) homogeneous of degree m ∈ R if for any We set Θ to be the space of caloric functions such that h(0) = 0. We denote by F (d) the space of homogeneous caloric polynomials of degree d. We remark that every caloric polynomial can be written as the sum of homogeneous caloric polynomials. Indeed, assuming that h(x, t) = α,ℓ c α,ℓ x α t ℓ for c α,ℓ ∈ R and that c α,ℓ = 0 if |α| + 2ℓ > d, we can write Observe that h j is a homogeneous polynomial of degree j, which implies that Hh j is a homogeneous polynomial of degree j − 2. So We denote by P (d) the set of caloric polynomials of degree d vanishing at the origin. Analogously, we define the set Θ * of adjoint caloric functions such that h(0) = 0 and by F * (d) and P * (d) the associated spaces of polynomials.
Lemma 6.1. Let h be a caloric function in R n+1 and let h + and h − indicate the positive and negative parts of h respectively. There exists a unique Radon measure ω h supported on {h = 0} such that Proof. Leth ± be the extension by zero of h ± in the complement of {h ± > 0}. Then, since h is continuous in R n+1 , h ± → 0 continuously on {h = 0}. Thus, as h ± is caloric in {h ± > 0}, we have thath ± is a subcaloric function in R n+1 and by [Wa12, Theorem 6.28], there exists a unique Radon measure ωh ± such that h H * ϕ = 0, and so, ωh + = ωh − . Thus, the first two equalities in (6.1) hold, while the last one follows by adding instead of subtracting the two identities above.
Given a caloric function h, by the discussion in Section 5, Σ h is smooth away from a euclidean (n − 1)-rectifiable set, and so, the set Ω ± = {h ± > 0} is a set of locally finite perimeter in R n+1 (see Definition 5.1 and Theorem 5.23 in [EG15]). Hence, by [Mag12,Theorem 18.11], for a.e. t ∈ R, its horizontal section Ω ± t is a set of locally finite perimeter in R n . In fact, more is true. By Lemma 5.1, around σ-a.e. any point, we have that Σ h is given by an admissible Lipschitz graph, while the rest of the points lie on an (n − 1)rectifiable set. So, for a.e. t, Σ h t is also locally Lipschitz and thus, its measure theoretic boundary (see [EG15,Definition 5.7]) coincides with its topological boundary. Therefore, for a.e. t there is a unique measure theoretic outward unit normal ν ± t to ∂Ω ± t such that we have the generalized Gauss-Green Theorem Moreover, ν ± t coincides with the usual (geometric) outward unit normal on ∂ * Ω ± (see Definitions 5.4 and 5.6, Theorems 5.15 and 5.16, and Lemma 5.5

in [EG15]) .
Lemma 6.2. If h is a caloric function in R n+1 and let σ be the surface measure on Σ h as defined in (5.1), then for any ϕ ∈ C 2,1 c (R n+1 ), Proof. If we apply (6.2) first to h(·, t)∇ϕ(·, t) ∈ C 1 c (R n ; R n ) and then to ϕ(·, t)∇h(·, t) ∈ C 1 c (R n ; R n ) in Ω ± t , for fixed t, and integrate in t, we obtain where we used that h is caloric and where in the last equality we used ν + t = −ν − t and integrated by parts in t using that L n+1 (Σ h ) = 0 and ∂ t h is continuous everywhere in R n+1 . Recall that the points of Σ h where ∇h = 0 are contained in an (n − 1)-rectifiable set and thus, have σ-measure zero. As the tangential component of ∇h on each slice Σ h t is zero, we have that ∇h = ∂ ν + t h and so, ∂ ν + t h = 0 σ-almost everywhere.
Let us recall the parabolic Cauchy estimates for caloric functions. Proposition 6.3 (see e.g. [HL94], Proposition 2.1). Let R > 0 and let h be a caloric function in C R . For r < R and α ∈ Z n + and any positive integer ℓ with |α| + 2ℓ = m, we have Lemma 6. 4. If h is a caloric function in R n+1 and σ is the surface measure on Σ h as defined in (5.1), then we have that where the associated Poisson kernel k h = − ∂h ∂ν + t is positive σ-a.e. and in L ∞ loc (σ). In particular, ω h ≪ σ.

Proof. Let
A be a compact subset of Σ h . Note that since σ and ω h are Radon measures, it holds that ω h (A) < ∞ and σ(A) < ∞. By Urysohn's lemma, we can find a sequence of decreasing functions {ϕ j } ∞ j=1 ⊂ C ∞ c (R n+1 ), 0 ≤ ϕ j ≤ 1, so that ϕ j = 1 on A and ϕ j → χ A pointwisely. Let us denote K j := supp ϕ j and observe that K j+1 ⊂ K j for all j ≥ 1. Then, by (6.3), we have Note that ϕ j has compact support K j and h is smooth in R n+1 by caloricity, hence it is locally Lipschitz in R n+1 (in the euclidean norm). Integrating by parts in t we get Since ∂ t h is bounded in K 1 and K j ⊂ K 1 for any j ≥ 1, by dominated convergence, we have As σ(A) < ∞ and Σ h is Euclidean n-rectifiable, (4.2) entails H n+1 p (A) < ∞ and consequently L n+1 (A) ≈ H n+2 p (A) = 0. Thus, the second integral on the right-hand side of (6.8) is zero, inferring that If C ρ is a cylinder of radius ρ ≈ diam K 1 which is centered at Σ h and satisfies K 1 ⊂ C ρ/2 , by (6.5) we have that since h ∈ L ∞ (C ρ ) and σ is locally finite. Thus, by (6.9) and the dominated convergence theorem, we conclude (6.6) for compact subsets of Σ h . Recall that ω h and σ are Radon measures and so, the result for Borel sets follows from inner regularity (see [Mat95, Definition 1.5]).

CALORIC POLYNOMIAL MEASURES
Lemma 7.1. If h ∈ Θ and ω h the associated caloric measure, then ). An application of the chain rule gives which proves the first part of the statement. If we further assume that h ∈ F (k), we obtain , and (7.2) follows.
We recall that we use the notation C r := C r (0) for cylinders centered at the origin. A consequence of the previous lemma is the following corollary.
Corollary 7.2. If h ∈ F (k) and r 1 , r 2 > 0 it holds that In particular, for any r > 0 and M ≥ 1, Proof. By Lemma 4.5 and Lemma 7.1 we have that and so, (7.4) readily follows. By Lemma 4.5-(2) and (7.4), it is straightforward to see that (7.5) holds. Proof. Let us recall that Σ h = R x ∪ R t ∪ S (see section 5) and that the Poisson kernel is given by −∂ νt h. Therefore, by Lemma 5.1, for anyx ∈ R x there is a sufficiently small neighborhood ofx in which Σ h agrees with an admissible smooth graph. Since h vanishes on Σ h , the component of ∇h which is tangential to Σ h t is the zero vector and we have ∇h = ∂ νt h. So, ∂ νt h(x) = 0 for anyx ∈ R x and thusx ∈ supp ω h showing that In light of [HL94, Theorem 1.1] and Lemma 5.2, for ρ > 0 small enough, it holds that H n−1 (R t ∪ S) ∩ C ρ < ∞, and so, R t ∪ S has empty interior in the relative topology of Σ h . Hence,0 which finishes the proof.
Lemma 7. 5. The d-cones F (k) and P(k) have compact basis for all k. Moreover, for h ∈ P(k) and r > 0 we have Proof. In light of Lemma 7.4, the proof is a routine adaptation of Lemmas 5.5 and 5.6, and Corollary 5.7 in [AM19].
In our argument we need the following formula for the expansion of caloric functions by Han and Lin, which we report for the reader's convenience.
We apply the previous lemma to show that the first non-zero term in the expansion h m of h determines the density of ω h at0.
where the implicit constants depend on n, m and c 0 .
Proof. Let ρ 0 and c 0 be as in Lemma 7.7 and let r < ρ 0 /2. We apply Lemma 4.5-(2), (7.8), and (7.5), and we have that Arguing similarly but using the converse inequalities of the ones we used above, we infer that where the last inequality holds because0 ∈ supp ω hm . Lemma 7.9. Let h ∈ Θ and h j be as in (7.6). If m ≥ 1 is the smallest integer such that h m ≡ 0, then Tan(ω h ,0) = {cω hm : c > 0}. Proof. Let R > 0 and r < 1/2. By Theorem 7.6 and h j = 0 for j < m, we have that Givenȳ ∈ C R , we have that δ r (ȳ) ∈ C rR , so we combine (7.14) with the homogeneity of h m in order to get where the last term converges to 0 as r → 0. In particular, this shows that r −m h • T −1 0,r converges to h m uniformly on compact subsets of R n+1 .
The definition of caloric polynomial measure together with Lemma (7.1), gives us that In order to finish the proof of the lemma, it suffices to use (7.15) and the fact that ω r −m h•T −1 0,r converges weakly to ω hm by Lemma 7.3. Indeed, ω h has positive lower and finite upper (n + m)-density at0 by Lemma 7.8, so we can apply Lemma 4.10 to conclude that every measure in Tan(ω h ,0) is of the form cω hm for some c > 0.
The next result is an important application of the previous lemma.
Proof. It is enough to argue as in [AM19, Lemma 5.9] and invoke Lemma 7.9.
Lemma 7.11. Let h ∈ P (d) and assume that a kℓ x k t ℓ .
Hence, the function φ := (2C) −1 rψ r is admissible for the functional F r , |H * φ| ≤ r −1 , and we have For anyx ∈ C 2r , it holds that which, in turn, implies that Therefore, if we let for some ε > 0 small enough, we infer that where in the last equality we used (7.4). Note that0 ∈ supp(ω h d ) and thus, by (7.5), we have that 0 < F 1 (ω h d ) < ∞. We conclude the proof arguing as in (7.12) and (7.13).
Lemma 7.12. Let h, r 0 and C 0 be as in Lemma 7.11. There exists ε 0 > 0 and r 1 > 0 such that if d r ω h , F (k) < ε 0 for all r ≥ r 1 , then k = d.
The next lemma is crucial to our purposes: it provides a connectivity result for parabolic cones of Radon measures. The proof translates unchanged to the parabolic setting and we skip it. . Let F and M be parabolic d-cones and assume that F has compact basis. Moreover, suppose that there is ε 0 > 0 such that the following property holds: if µ ∈ M and there exists r 0 > 0 such that d r (µ, F ) ≤ ε 0 for all r ≥ r 0 , then µ ∈ F . If η is a Radon measure andx ∈ supp η are such that Tan(η,x) ⊂ M and Tan(η,x) ∩ F = ∅, then Tan(η,x) ⊂ F .
The following proposition gathers all the results of this section. After proving all the previous lemmas, the proof is analogous to that of [AM19, Proposition II]. We report it anyways, in order to give the reader the precise references inside this section.

BLOW-UPS IN TIME VARYING DOMAINS
The following lemma studies the blow-up of caloric measure and Green's function in a domain that is quasi-regular for H and regular for H * . The main difference compared to the elliptic case (see [AMTV19], [AM19, Lemma 4.12]) is that due to the lack of a proper bound for the time-derivative of the Green's function, one cannot use Rellich-Kondrachov theorem, and so we have to identify the right relative compactness theorem in mixed-norm Sobolev spaces and apply it in our setting.
be an open set which is quasi-regular for H and regular for H * . Let also ω = ωp Ω be the associated caloric measure with pole atp ∈ Ω. Assume that F ⊂ P * Ω ∩ S ′ Ω is compact,ξ j is a sequence in F , and there exists r j → 0 and c j > 0 such that ω j := c j Tξ j ,r j [ω] ⇀ ω ∞ for some non-zero Radon measure ω ∞ . Let us also assume that there exists a subsequence of r j and a constant c > 0 so that If u = G Ω (p, ·) in Ω and u = 0 in R n+1 \ Ω, let us denote u j := G p, T −1 ξ j ,r j (·) and Then, if a ∈ (0, 1/2) is as in Lemma 3.8 and α = a/16 ∈ (0, 1/32), there exists R > 0, a subsequence of {r j } j and a non-negative function u ∞ ∈ L 2 (C αR ) ∩ L 2 (I αR ; W 1,2 (B αR )) such that u j → u ∞ in L 2 (C αR )-norm and weakly in L 2 (I αR ; W 1,2 (B αR )). Moreover u ∞ is adjoint caloric in C αR ∩ {u ∞ > 0}, Proof. For simplicity, we will only prove the lemma assumingξ j ≡ξ for someξ ∈ P * Ω ∩ SΩ, since the proof of the general case is analogous. Let us recall that forξ 0 ∈ R n+1 , r > 0, and a > 0 we denote . and it holds that C ar/2 (ξ 0 ) ⊂ R + a (ξ 0 ; r). As (8.1) is satisfied, by Lemma 3.8 we have that there exists a ∈ (0, 1/2] such that Cap R − a (ξ; r j ) \ Ω cr 2 j , and so, by (3.33), it holds that for allz ∈ C ar j (ξ) ⊂R + a (ξ; r j ). Since ω ∞ = 0, there exists R > 0 such that ω ∞ (C R ) > 0. Without loss of generality we may assume that R = a/8. By passing to a subsequence, if necessary, we may assume thatp ∈ Ω\C ar j (ξ) and combining the latter bound with Lemma 3.21 for ar j /2 and M ′ = 2M/a, we obtain In particular, if we set Ω j := Tξ ,r j (Ω), the previous inequality implies Since G(p, ·) is in C 2,1 (Ω \ {p}) ∩ C(Ω ∪ ∂ * e Ω \ {p}) and Ω is regular for H * , then, for j large, its extension by zero is continuous in C ar j /8 (ξ) and in the vector-valued Sobolev space L 2 I ar j /8 (t); W 1,2 (B ar j /8 (ξ)) . Moreover, by [Wa12,Theorem 7.20], it is a subtemperature of H * and thus adjoint subcaloric in C ar j /8 (ξ). By a simple rescaling argument, the latter implies that u j ∈ L 2 (I a/8 ; W 1,2 (B a/8 )) is a continuous non-negative adjoint subcaloric function in C a/8 and so, if we apply Caccioppoli's inequality and use the bound (8.4), there exists j 0 ≥ 1 such that for j ≥ j 0 , Moreover, once again by (8.4), Therefore, there exists j 1 ≥ j 0 such that sup j≥j 0 I j (h) ≤ 2I j 1 (h), which, in turn, by dominated convergence and the continuity of u j in C a/16 , implies that uniformly in j ≥ j 0 . Consequently, since W 1,2 (B a/16 ) embeds compactly in L 2 (B a/16 ) and, by (8.5) we have that {u j } j≥j 0 is a bounded subset of L 2 I a/16 ; W 1,2 (B a/16 ) , in light of (8.6), we can apply [Sim87, Theorem 5] (see also Theorems 3 and 6 in [Sim87]), to deduce that {u j } j≥j 0 is relatively compact in L 2 (C a/16 ). Hence, there exists a subsequence of {u j } j≥j 0 and a function u ∞ ∈ L 2 (C a/16 ) ∩ L 2 I a/16 ; W 1,2 (B a/16 ) such that u j → u ∞ strongly in L 2 (C a/16 ) and weakly in L 2 I a/16 ; W 1,2 (B a/16 ) .

A change of variables gives
which, if we take limits as j → ∞, entails In order to complete the proof of the lemma, it suffices to choose α = a/16.
To prove Theorem I, we need the following "two-phase" blow-up lemma.
Let Ω ± be two disjoint open sets in R n+1 which are quasi-regular for H and regular for H * , and let ω ± be the respective caloric measures with polesp ± ∈ Ω ± . Assume that F ⊂ P * Ω + ∩ P * Ω − ∩ S ′ Ω + ∩ S ′ Ω − is compact,ξ j is a sequence in F and that Ω + and Ω − satisfy the joint TBCPC. Let us also assume that there exists r j → 0, c j > 0, and a constant c > 0, such that Let u ± := G Ω ± (p ± , ·) on Ω ± , u ± ≡ 0 on R n+1 \ Ω and denote u ± j (x) := c j u ± δ r jx +ξ j r n j . The following properties hold: (a) There exists α ∈ (0, 1/16), R > 0, a subsequence of r j , u ± ∞ ∈ L 2 (C αR ), and u ∞ ∈ L 2 (C αR ), a non-zero adjoint caloric function in C αR , such that u ± j converge in L 2 (C αR )-norm to u ± ∞ and u j : The function u ∞ extends to an adjoint caloric function in R n+1 and it holds that (8.8) u ∞ L 2 (Cr ) a r −n ω + ∞ C M r/a for all r > 0, and for any ϕ ∈ C ∞ c (R n+1 ), (c) We have that Proof. As in the proof of the previous lemma, we assume for simplicity thatξ j ≡ξ ∈ P * Ω + ∩ P * Ω − for all j and ω ∞ (C a/16 ) > 0. Since the domains Ω + and Ω − are disjoint, the subadditivity property for the thermal capacity [Wa12, Theorem 7.45-(b)] entails Cap E(ξ; r 2 j ) \ Ω + + Cap E(ξ; r 2 j ) \ Ω − ≥ Cap E(ξ; r 2 j ) , j ≥ 0, and so, possibly after passing to a subsequence, without loss of generality, we can assume that Moreover, as Ω + and Ω − satisfy the joint TBCPC condition, there exists q subsequence such that Cap E(ξ; r 2 j ) \ Ω − r n j j ≥ 0.
An immediate consequence of ( 8.15) is that In order to prove the converse inclusion, we observe that forx ∈ R n+1 such that u ∞ (x) = 0, (8.13) implies that u + ∞ (x) = u − ∞ (x) = 0. Moreover, by the unique continuation, u ∞ cannot vanish in any open cylinder containingx, so either u + ∞ or u − ∞ must be positive in that cylinder. This in turn implies 8.16) ⊂ supp ω + ∞ . Lemma 6.4 concludes the proof.
Before presenting the blow-up result that we need in order to prove the second part of Theorem I, we recall that the TFCDC condition is invariant under parabolic scaling (see Section 3). Moreover, if Ω is a domain that satisfies the TFCDC as in 3.28,ξ ∈ SΩ, and ρ > 0, then, if we denoteΩ := T ξ,ρ [Ω], we have that SΩ ⊂ ∂ * eΩ . The next lemma lists the main properties of the blow-ups of caloric measure and the adjoint Green's function in a domain satisfying the assumptions of Lemma 8.1 that has the TFCDC instead of just being regular for H * . Our proof is inspired from the one of the corresponding result for harmonic measure in [AMT17] and although it is the essentially the same for items (a)-(c), the proof of (d) is different.
Let Ω ⊂ R n+1 be an open set which is quasi-regular for H and satisfies the TFCDC, and let F ⊂ P * Ω ∩ SΩ be compact and {ξ j } j ⊂ F . We denote by ω the caloric measure for Ω with pole atp ∈ Ω, and assume that there exists c j > 0 and r j → 0 such that ω j = c j Tξ j ,r j [ω] ⇀ ω ∞ for some non-zero Radon measure ω ∞ . If we denote Ω j := Tξ j ,r j (Ω) and assume that there is c > 0 so that (8.17) Cap E(ξ j ; r 2 j ) \ Ω ≥ c r n j , j = 1, 2, . . .
then there is a subsequence of {r j } j≥1 and a closed set Σ ⊂ R n+1 such that  x) on Ω and u ≡ 0 on (Ω) c , and define u j (x) = c j r n j u δ r j (x) +ξ j , then the sequence u j converges uniformly on compact subsets of R n+1 to a non-zero function u ∞ which is continuous in R n+1 , adjoint caloric in Ω ∞ , and vanishes in (Ω ∞ ) c . Moreover if a ∈ (0, 1/2) and M > 1 are the constants obtained in Lemmas 3.8 and 3.15 respectively, then forx ∈ Σ and r > 0, it holds that Proof. For simplicity, we assume K = {ξ}, as the general case can be proved similarly.
Proof of (a): Let R > 0 and observe that0 ∈ ∂Ω j for all j ≥ 1, which entails ∂Ω j ∩ C R (0) = ∅. The Hausdorff distance is a metric on the collection C C R (0) of all closed subsets of C R (0). Since C R (0) is compact, C C R (0) , d H is compact too. Thus, after passing to a subsequence, C R (0) ∩ ∂Ω j → C R (0) ∩ Σ in the Hausdorff distance sense for some Σ ∈ C C R (0) . A standard diagonalization argument concludes the proof of (a).
Proof of (b): We will first prove that there exists a parabolic cylinder C ρ (x ′ ) which is contained in all Ω j , for j large enough. Arguing by contradiction we assume not. Let φ ∈ C ∞ c (R n+1 ) be a non-negative function such that φ dω + ∞ = 0 and supp φ ⊂ C K (0) for some K > 0. Hence, there existsx 0 belonging to C K (0) ∩ supp ω ∞ , and our counterassumption implies that (8.20) We denote byζ j (x) ∈ (Ω j ) c a point which realizes dist p (x, (Ω j ) c ). In particular, since ρ j → 0 and 0 ∈ ∂Ω j for all j, we have that Observe that Ω j satisfies the TFCDC with the same parameters as Ω because of Lemma 3.14. Moreover,ζ j (x) ∈ SΩ j for j big enough because T min (Ω j ) = (T min (Ω) − τ )/r 2 j → −∞ and T max (Ω j ) = (T max (Ω) − τ )/r 2 j → +∞ as j → ∞. In fact,ζ j (x) ∈ ∂ * e Ω j (see the discussion before the statement of this lemma).

(8.21)
Remark that M depends on a and is chosen so that (3.34) holds. Hence, using the γ-Hölder continuity at the boundary for u j in Ω j (that holds with constants not depending on j because of Lemma 3.14), we infer that So, taking the limit in the previous inequalities as j → ∞ we obtain which is a contradiction. Thus, after passing to a subsequence, if necessary, there exists a cylinder C ρ (x ′ ) ⊂ Ω j for j big enough.
For the the construction of Ω ∞ and, once we define Ω ∞ := R n+1 \Ω ∞ , for the remaining part of the proof of (b), we refer to [AMT17, p. 2143]. For future reference, we remark that Ω ∞ is built as (8.22) where D is the collection of all open cylinders C r (x) such that r is rational,x has rational coordinates, C r (x) ⊂ R n+1 \ Σ and C r (x) is contained in all but finitely many Ω j , for a proper subsequence.
Combining the four cases above, after passing to a subsequence, we obtain that u j is equicontinuous in C K for each K ≥ 1. Hence, we can apply Ascoli-Arzelà's Theorem and a standard diagonalization argument to get that u j has a subsequence that converges uniformly on compact subsets of R n+1 to a continuous function u ∞ . Moreover, forx ∈ Σ, r > 0, andȳ ∈ C r (x) ∩ Ω ∞ , we have which gives (8.18). It remains to prove (8.19). To this end, for φ ∈ C ∞ c (R n+1 ) it holds Taking limits as j → ∞ in ( 8.26), in light of the fact that supp ω ∞ ∩ Ω ∞ ⊂ Σ ∩ Ω ∞ by (c) and that u j → u ∞ locally uniformly in R n+1 , we have that ( 8.19) holds. It is straightforward to see that u ∞ is adjoint caloric in Ω ∞ .
The function u ∞ is not identically zero. Indeed, if φ ∈ C ∞ c (R n+1 ) is a non-negative function so that φ dω + ∞ > 0 and plug it in (8.19), we obtain This readily implies that u ∞ ≡ 0 in R n+1 . In order to finish the proof of (d), we claim that u ∞ ≡ 0 on R n+1 \ Ω ∞ . Assume that there isx ∈ R n+1 \ Ω ∞ so that u ∞ (x) > 0. By the continuity of u ∞ , there exists δ > 0 such that u ∞ (ȳ) > u ∞ (x)/2 for allȳ ∈ C δ (x). The local uniform convergence of u j to u implies that u j > u ∞ (x)/4 on C δ (x) for all j big enough. Thus, since u j vanishes outside of Ω j , we have that C δ (x) ⊂ Ω j for j large. In particular, it holds that C δ/2 (x) ⊂ R n+1 \ Σ, so C δ/2 (x) ⊂ Ω ∞ by the construction ( 8.22) and we can conclude thatx ∈ Ω ∞ , which is a contradiction. This proves our claim and finishes the proof of the lemma.
The next lemma studies the two-phase version of the previous result for domains that satisfy the joint TBCPC. Although, this proof is inspired by the one of [AMT17, Lemma 5.9], there are several differences as well. Let us remark that we do not assume the domains to be complementary, which gives us significant freedom in the applications in Section 9, but also requires additional care in the proof.
Then there exists a subsequence of {r j } j≥1 , u ± ∞ , Ω ± ∞ , Ω ± ∞ , and Σ ± such that the properties ∞ stands for the parabolic surface measure on ∂Ω + ∞ and ν t is the measuretheoretic outer unit normal in its time-slices. Proof. Let us assume for brevity thatξ j ≡ξ. Given Ω ± as in the statement, by the same pigeonholing argument as in Lemma 8.2 and in light of the joint TBCPC property, we can find a subsequence of r j such that (8.28) Cap E(ξ; r 2 j ) \ Ω ± r n j , j ≥ 0.
Proof of (a): Let φ ∈ C ∞ c (R n+1 ). By hypothesis we have that ω + j ⇀ ω + ∞ and ω − j ⇀ cω + ∞ , and thus, by the local uniform convergence of u ± j on R n+1 , we can find a subsequence of {r j } j≥1 such that So, u ∞ is a weakly adjoint caloric function in R n+1 and Lemma 3.1 applies.
Proof of (b): Let us first prove that In order to prove the converse inclusion, we first observe that u ∞ is non-zero since by Lemma 8.3-(d) it holds that u ± ∞ ≡ 0 and vanish outside Ω ± ∞ . By construction, we have that u ± ∞ ≥ 0 on Ω ± ∞ . Let us assume by contradiction that u + (x) = 0 for somex ∈ Ω + ∞ and let E * (x; r) := {(y, s) ∈ R n+1 : (y, t − s) ∈ E(x, r)} be a reflected heat ball so that E * (x; r) ⊂ Ω + ∞ . The mean value theorem for adjoint caloric functions (that readily follows from the mean value theorem for caloric functions [Wa12, Theorem 1.16]) gives that which implies that u ∞ = u + ∞ ≡ 0 on E * (x; r). In particular, u ∞ has infinite order of vanishing at all the points of E * (x; r) in the sense of [Po96,p. 522]. Hence, by the unique continuation principle for globally adjoint caloric functions [Po96, Theorem 1.2], we deduce that u ∞ ≡ 0 on R n+1 , which is a contradiction. The same argument can be applied to . Let us remark that, by construction, R n+1 = Ω ± ∞ ∪ Ω ± ∞ ∪ Σ ± , where the unions are disjoint. Hence (8.29) We claim that the second and third term under parenthesis in the right hand side of ( 8.29) are the empty set.
First, let us show that Hence, by [Po96, Theorem 1.2] we have that u ∞ ≡ 0, which is a contradiction.
So, since u ∞ is globally adjoint caloric and u ∞ (x) = 0, the strong minimum principle [Wa12, Theorem 3.11] entails u ∞ = 0 on C + ε (x). Thus, by the unique continuation we get u ∞ ≡ 0 on R n+1 , which is again a contradiction.
Proof. The proof is identical to the one of [AMT17, Lemma 5.8] and we omit it.
With the results of Section 7 at our disposal, the following lemma can be proved following the strategy of [AM19, Lemma 6.1]. Its proof relies on a corollary of a "connectivity" lemma (see [AM19, Corollary 3.12]), which also holds in the parabolic setting as it does not use the Euclidean structure.
We are now ready to gather all the results we have proved so far in order to prove the first main theorem of the paper.
We will now show Theorem II following the approach in [TV18].
Proof of Theorem II. The proof follows from the ones of Lemmas 8.1 and 8.2, and so, we will only sketch it. Let us assume that ω 1 (E) > 0, which by mutual absolute continuity implies that ω i (E) > 0 for i = 2, 3 as well. By Lemma 3.23 (or rather its proof), we can find open sets Ω i ⊂ Ω i , i = 1, 2, 3, which are regular for H and H * , and a set so that ω i ( E) > 0 for i = 1, 2, 3, and ω Ω 1 , ω Ω 2 , and ω Ω 3 are mutually absolutely continuous on E. So, it is enough to prove the result assuming that Ω i is regular for H and H * for i = 1, 2, 3.
Since G j is compact and H n+1 G j 1 for all j ≥ 1, there is a compact set G such that, after passing to a subsequence, G j → G in the Hausdorff metric and H n+1 G 1.
It is easy to see that χ G j → g weakly in L 2 E(0; 1) for some non-negative function g ∈ L 2 E(0; 1) , and thus, we must have that g = χ G . Repeating the proof of Lemma 8.2 for + = 1 and − = 2, we obtain that u j → u ∞ in L 2 loc (R n+1 ) for some globally adjoint caloric function u ∞ ≡ 0 which vanishes only on a set of zero H n+1 -measure. Taking limits as j → ∞ in (9.2), we obtain G |u ∞ | = lim j→∞ G j |u j | = 0, which, in turn, implies that u ∞ = 0 on a set of positive Lebesgue measure reaching a contradiction. 9.2. Proofs of Theorems III and IV.
The measure ω h is doubling because ω + | E is. Thus, there exists λ ∈ N such that for every k ∈ N and N ∈ N, Let a and M be as in Lemma 8.2 and denote by N a positive integer such that 4a −1 M ≤ 2 N . Inequality (9.5) together with the Cauchy estimates (6.5), gives that, for α ∈ N n , k ∈ N and ℓ > 0 such that |α| + 2ℓ = m, a,M 2 −k(m+n) ω h C 2 k+N (0) where the second inequality follows from the proof of Lemma 8.2. Thus, for m+n−λ > 0 we have that the right hand side in the previous expression converges to 0 as k → ∞, which implies that D α,ℓ h(0) = 0. In particular, h is a polynomial of degree at most d = m+n−λ, which contradicts (9.4). In order to prove (1.10), it suffices to use Lemma 8.3 and argue as in the proof of Theorem I.