Removable singularities for solutions of the fractional Heat Equation in time varying domains

In this paper we study removable singularities for solutions of the fractional heat equation in time varying domains. We introduce associated capacities and we study some of its metric and geometric properties.


Introduction
In the present paper we study removable singularities for solutions of the fractional heat equation in time varying domains.Our main motivation comes from the paper [MPrTo] where we studied removable singularities for regular (1, 1/2)−Lipschitz solutions of the heat equation in time varying domains.Our ambient space is R N +1 with a generic point denoted as x = (x, t) ∈ R N +1 , where x ∈ R N and t ∈ R. For a smooth function f depending on (x, t) ∈ R N +1 , the heat equation is just (1.1) Θ(f ) = ∆f + ∂ t f = 0, with fundamental solution W (x, t) = (4πt) −N/2 exp(−|x| 2 /4t) for t > 0 and 0 if t ≤ 0. We set 0 < s < 1 and let Θ s denote the fractional heat operator, Θ s = (−∆) s + ∂ t .Then, for a smooth function f depending on (x, t) ∈ R N +1 , Θ s (f ) = (−∆) s f + ∂ t f = 0 is just the fractional heat equation.The pseudodifferential operator (−∆) s = (−∆ x ) s , 0 < s < 1, is the fractional Laplacian with respect to the x variable.It may be defined through its Fourier transform, (−∆) s f (ξ) = |ξ| 2s f (ξ), or by its representation where c(n, s) is a normalization constant (see [La] and [S] for its basic properties).The standard Laplace operator, −∆, is recovered when taking the limit s → 1 (see [DPaVa,Section 4]), but there is a big difference between the local operator −∆, that appears in the classical heat equation and represents Brownian motion, and the non-local family (−∆) s , 0 < s < 1, which are generators of Lévy processes in Stochastic PDEs (see [A] i [Be]).Given x = (x, t) and ȳ = (y, u), with x, y ∈ R N , t, u ∈ R, we consider the s-parabolic distance in R N +1 defined by dist p (x, ȳ) = max |x − y|, |t − u| 1/2s .L.P. was supported by PID2020-114167GB-I00 (MINECO, Spain) and J.M. was supported by PID2020-112881GB-I00 (MINECO, Spain).
Sometimes we also write |x − ȳ| p instead of dist p (x, ȳ).Notice that for s = 1/2, dist p is comparable to the usual distance.We denote by B p (x, r) an s-parabolic ball (i.e., in the distance dist p ) centered at x with radius r.By an s-parabolic cube Q of side length ℓ, we mean a set of the form where I 1 , . . ., I N are intervals in R with length ℓ, and I N +1 is another interval with length ℓ 2s .We write ℓ(Q) = ℓ.
We say that a Borel measure µ in R N +1 has upper parabolic growth of degree N + 2s − 1, 0 < s < 1, if there exists some constant C such that (1.2) µ(B p (x, r)) ≤ Cr N +2s−1 for all x ∈ R N +1 , r > 0.
Given a compact set E ⊂ R N +1 , we define its 1/2−fractional caloric capacity by , where D ′ is the space of distributions in R N +1 and P (x, t) denotes the fundamental solution of the 1/2−fractional heat equation in R N +1 , that is (1.4) The fractional heat equation can be solved (by applying Fourier transform) for all 0 < s ≤ 1 by means of the fundamental solution P s (x, t), which is the inverse transform of the function e −|ξ| 2s t .It is worth mentioning that only in the particular cases s = 1 and s = 1/2 the kernel is known to be explicit.In the first case we get the Gaussian kernel for the standard heat equation W (x, t) = (4πt) −N/2 e −|x| 2 /4t and in the fractional case s = 1/2 we have (1.4).For 0 < s < 1, there is no explicit fundamental solution for the s−fractional heat equation.In the 1960s, the probabilists Blumenthal and Getoor [BG,Theorem 2.1] generalized in a precise way the power-like tail behaviour observed in the case s = 1/2 to the other values of 0 < s < 1, that is, they established that the fundamental solution, P s (x, t), of the fractional heat equation, for 0 < s < 1, satisfies N+2s 2 when t > 0 (and 0 for t ≤ 0).Here a ≈ b means that a/b is uniformly bounded above and below by a constant.Notice the marked difference with the Gaussian kernel W (x, t) of the heat equation (1.1) (case s = 1).The behaviour as x tends to infinity of P s is power-like while W has exponential spatial decay.
In the paper, we will also introduce and study some s-fractional caloric capacities γ s Θ , 0 < s < 1, associated with the kernels P s of the s-fractional heat equation.
We shall now give a brief description of the main results in the paper.Section 2 includes some estimates of the kernels P s (x, t) (0 < s < 1) and its derivatives, that will be needed in the rest of the paper.The four next sections deal with the case s = 1/2.More concretely, in Section 3 we prove a localization result for s = 1/2, that is, for a distribution ν, we localize the potential P * ν in the L ∞ −norm.The localization method for the Cauchy potential ν * 1/z in the plane is a basic tool developed by A.G. Vitushkin in the theory of rational approximation in the plane.This was later adapted in [P] for Riesz potential ν * x/|x| N in R N and used in problems of C 1 −harmonic approximation.These localization results have also been essential to prove the semiadditivity of analytic capacity and of Lipschitz harmonic capacity, see [To1] and [V] respectively (see also [Pr] for other related capacities).
In Section 4 we study the connection between 1 2 −fractional caloric removability and the capacity γ 1/2 Θ .In particular, we show that a compact set E ⊂ R N +1 is 1 2 −fractional caloric removable if and only if γ 1/2 Θ (E) = 0. We also compare the capacity γ ∞ and we prove that if E has zero N −dimensional Hausdorff measure, i.e., H N (E) = 0, then γ 1/2 Θ (E) = 0 too.In the converse direction, we show that if E has Hausdorff dimension larger than N , then γ 1/2 Θ (E) is positive.Hence, the critical dimension for 1 2 −fractional caloric capacity (and thus for 1 2 −fractional caloric removability) occurs in dimension N , in accordance with the classical case.Also by analogy with the classical case, one should expect that γ is the same as (1.3) but instead of distributions one considers positive measures).However, there is a big obstacle when trying to follow this approach.Namely, the kernel P is not antisymmetric and thus, if ν is such that T ν = P * ν is in L ∞ (R N +1 ), apparently one cannot get any useful information regarding T * ν (T * being the dual operator to T ).This prevents any direct application of the usual T 1 or T b theorems from Calderón-Zygmund theory, which are essential tools in the case of analytic capacity and Lipschitz harmonic capacity.Hence, In Section 4, due to the lack of antisymmetry of our kernel P , we also introduce a new capacity γ 1/2 Θ,+ .We set γ Θ,+ (E) = sup µ(E), the supremum taken over all positive measures supported on Θ,+ .We show that the capacity γ 1/2 Θ,+ can be characterized in terms of the L 2 -norm of T .In Sections 5 and 6, we give some concrete examples of 1 2 −caloric removable and nonremovable sets with Hausdorff dimension N .In particular, in the first section we construct a self similar Cantor set in R N +1 with positive and finite Hausdorff N -dimensional measure which is 1 2 −caloric removable and in the second one we show that on the plane, the capacity γ 1/2 Θ vanishes on vertical segments and is positive on horitzontal ones.These examples allow us to deduce that, in dimension two, neither analytic capacity nor newtonian capacity are comparable to the 1/2−fractional caloric capacity γ 1/2 Θ (although these three capacities have the same critical dimension).
Sections 7 and 8 are devoted to the study of the s−fractional capacities γ s Θ , when 1/2 < s < 1 and 0 < s < 1/2 respectively.In the first case, namely 1/2 < s < 1, we are able to show that the critical dimension for the γ s Θ -capacity is N +2s−1.We have had several technical problems when trying to prove this statement for the case 0 < s < 1/2 but we can show that sets E with zero (N + 2s − 1)−dimensional Hausdorff measure, i.e., H N +2s−1 (E) = 0, have γ s Θ (E) = 0.
Some comments about the notation used in the paper: as usual, the letter C stands for an absolute constant which may change its value at different occurrences.The notation A B means that there is a positive absolute constant C such that A ≤ CB.Also, A ≈ B is equivalent to A B A. The gradient symbol ∇ refers to (∇ x , ∂ t ), with x ∈ R N and t ∈ R.
2. Some preliminary estimates on the kernels P and P s In the next two lemmas we will obtain upper bounds for the kernels P (x, t), P s (x, t) and its derivatives.We will need them later.
Lemma 2.1.For any x = (x, t) ∈ R N +1 , x ∈ R N and t ∈ R, the following holds: Notice that the kernel P is not differentiable with respect to t at the points (x, 0), x ∈ R N .
Proof.The estimate in (a) comes from the explicit expression of P. For the estimates in (b) we compute , and so we get |∇ x P (x)| 1 |x| N +1 .On the other hand, for t = 0 .
The first term in the above inequality is bounded by For the second term in (2.1) let t > t ′ .If t ′ > 0, then If t < 0, |P (x) − P (x ′ )| = 0 and if t > 0 and t ′ < 0, then The following lemma shows some growth properties of our kernel P s .
Lemma 2.2.Let s ∈ (0, 1).For any x = (x, t) ∈ R N +1 , x ∈ R N and t ∈ R, the following holds: (1) Proof.The first property follows from the definition of the kernel.For t ≤ 0, P s (x, t) = 0 and for t > 0, We prove now the second one.Applying Fourier transform to the fractional heat equation with respect to the space variable x, and calling the new variable ξ, we get the equation ∂ t u = −|ξ| 2s u, that allows to solve the initial-value problem in Fourier space by means of the formula u(ξ, t) = u 0 (ξ)e −|ξ| 2s t .Applying the inverse transform, the fractional heat equation can be solved for all 0 < s < 1 by means of the fundamental solution, P s (x, t), which is the inverse transform of the function e −|ξ| 2s t .Hence the kernel P s has Fourier transform P s (ξ, t) = e −|ξ| 2s t .It is well known that for t > 0 (recall that for t ≤ 0, P s (x, t) = 0) it has the self-similar form (2.2) for some positive C ∞ function φ, radially decreasing and satisfying φ(u) ≈ (1 + u 2 ) − N+2s

2
(see [BG] and [Va]).Using (2.2) we deduce that where ψ(z) = (−∆) α φ(z).Since φ(ξ) = e −|ξ| 2s , then ψ(ξ) = |ξ| 2α e −|ξ| 2s and using the expression of the inverse Fourier transform of a radial function (see [G,Section B.5] or [SW, Section IV.I] for a proof) we obtain , which implies the second estimate in the statement of the lemma.Observe that for α = s this is Therefore using that P s is the fundamental solution of the s-fractional heat equation (2.5) , t > 0, and the fact that (which is estimate (2.3) with α = s) we get which is the third estimate in the statement of the lemma.
Next we will estimate the spatial derivative ∇ x P s (x, t).Clearly If we can show that (2.7) φ ′ (u) (1 + u 2 ) − N+1 2 , then we are done because In order to estimate ∇φ(z), we consider the equation for φ that comes from (2.5), that is and (2.6) we deduce (2.7).To prove the last estimate in the statement of the lemma, that is we claim that for t > 0, (2.9) for some function F s (u) ≈ u . (2.9) can be proved using (2.2), that is for some φ with φ(u) ≈ (1 + u 2 ) − N+2s

2
. In fact, the last equality being a definition for F s .Hence and claim (2.9) is proved.Notice that we have Hence, to show (2.8) it is enough to see that for all t ∈ R, Once (2.10) is available we obtain (2.8) easily: To show (2.10), we write We begin with I 1 .Notice that here |r − t| ≈ |t|.Then, If |t| ≤ 1, then If |t| > 1, then we can write Hence I 11 min 1, 1 |t| .Since F s (t) = 0 for t ≤ 0, to estimate I 12 we only need to consider positive t, therefore This finishes the estimate of I 1 .To deal with I 2 , we distinguish two cases, according to whether r has the same sign as t or not.In the first case we write r ∈ Y , and in the second one, r ∈ N .
In the case r ∈ N , with |t|/2 ≤ |r| ≤ 2|t|, it turns out that |r − t| ≈ |t|, and thus Observe that this last expression is very similar to the ones in I 11 and I 12 .Then, by almost the same arguments we deduce that To deal with the case when the sign of s is the same as the one of t (i.e., s ∈ Y ), we take into account that Notice that (2.9) tells us that if |y| = 1, then F s (ξ) = P s (y, ξ).Hence due to property (3) of this lemma and since in this case |t|/2 ≤ |ξ| ≤ 2|t|, it is immediate to check that for this ξ we have . Thus, Finally we will deal with I 3 .

Localization estimates for s = 1/2
The main objective of this section is to show the following localization result.
In the statement the operator ∇ refers to ∇ = (∇ x , ∂ t ).Before proving Theorem 3.1 we need several lemmas and definitions.We say that a C 1 function ϕ is admissible for Recall that with R j , 1 ≤ j ≤ N , being the Riesz transforms with Fourier multiplier ξ j /|ξ|.Then setting Therefore if ϕ satisfies ( Then ϕ is admissible for Q.
Proof.Due to the above explanation, we only have to show that ˆIQ In fact, where the forelast inequality is obtained by integrating on annulus, for example.
Proof.The first condition in lemma 3.2 is clearly fulfilled.To show the second condition in lemma 3.2 we will use the Cauchy-Schwarz inequality and the L 2 −boundedness of the Riesz transforms as follows: Therefore, standard test functions supported on Q are admissible for Q.
It is worth mentioning here that if ϕ is a standard The first condition in (3.1) is clear and for the second one, notice that if g = ∆ x ϕ * x k, with k(x) = 1 |x| N−1 and * x denoting the convolution on the x variable, then taking the Fourier transform with respect to x, we get: (−∆ x ) 1/2 ϕ = cg, for a suitable constant c = 0.Then, integrating on annuli and using that ∆ The following lemma shows an N −growth condition that every distribution ν fulfilling the hypothesis of Theorem 3.1 satisfies.
Proof.Since P is the fundamental solution of Θ 1/2 , we can write and ϕ is admissible for Q, so it satisfies (3.1).
We will say that a distribution ν has N -growth if for any Proof.We will show that the mean of f = P * ϕν on 1 4 Q is bounded by a constant.Hence at many Lebesgue points of f the inequality (3.3) holds.
We only need to show that P * ϕν is integrable on 2Q.In fact, we will prove a stronger statement, namely that P * ϕν is in L p (2Q) for each 1 < p < N +1 N .Indeed, fix any q satisfying N + 1 < q < ∞ and let p be its dual exponent, so that 1 < p < N +1 N .We need to estimate the action of P * ϕν on test functions ψ supported on 2Q in terms of ψ q .We clearly have We claim that the test function satisfies the inequalities in lemma 3.2 and hence it is and admissible function.Once this is proved we get which completes the proof of the lemma.
To prove the claim, we need to show that h satisfies the inequalities in lemma 3.2, that is Or equivalently (3.5) for 1 ≤ j ≤ N.For the first inequality in (3.5) apply Hölder's inequality to obtain Since ∇ϕ ∞ ≤ ℓ(Q) −1 and by Hölder again, Applying [Du,Theorem 4.12], for example, we deduce that the singular integral with kernel ψ L q (2Q) so the first inequality in (3.5) is proven.
For the second one write Using Hölder, the L q (R N )−boundedness of the Riesz transform and arguing similar to what we have just done for the term A 1 , Analogously, by an argument similar to the one of A 2 .
To estimate term A, we will show that h ) is admissible for 4Q and then apply lemma 3.4.Since the function and by the mean value theorem and lemma 2.1, which is condition (1) in lemma 3.2.To show (2) in lemma 3.2 for the function h we have to prove Applying Hölder's inequality for some q > 1 to be chosen later and using the L q (R N )boundedness of the Riesz transform R j , we have To estimate the last integral, write Using that ∇φ ∞ ℓ(Q) −1 and , for ȳ = 0. Therefore, using the mean value theorem and arguing as above we get Since (3.7) holds, h is admissible for 4Q and by lemma 3.4 we get A 1.
To estimate term B = | ν, (1 − ψ)ϕ(x)P ε x | we will use lemma 3.5, i.e. the fact that there exists x0 ∈ Q such that The analogous inequality holds as well for the regularized potentials appearing in B, uniformly in ε, and therefore I being the set of indices k ≥ −3 such that supp ϕ k ∩ 4Q = ∅ and J denoting the remaining indices ( i.e. k ≥ −3 with ϕ k ≡ 0 on 4Q).Notice that the cardinality of I is bounded by a dimensional constant.
and for k ∈ J, ).We will show now that we can apply lemma 3.2 to g and g k , k ∈ J. Once this is available, lemma 3.4 will give us Notice first that the support of g is contained in a cube λQ for some λ depending only on N .On the other hand, the support of g k is contained in 2 k+2 Q.To apply lemma 3.2 we have to show that (3.9) and for k ∈ J, (3.10) We check first (3.9).Using lemma 2.1, The estimate for ∂ t g L 1 (λQ) is analogous.For the second inequality in (3.9), let 1 < q < ∞ and p be its dual exponent.Apply Hölder's inequality, the fact that Riesz transforms preserve L q (R N ) and argue as in the estimates of the integrals of A 1 and A 2 (using (3.8)) to obtain: To show the first inequality in (3.10) we have to prove Notice that (3.11) comes from a gradient estimate and lemma 2.1.Hence N , we consider different cases.Write x = (x, t), ȳ = (y, u) and x0 = (x 0 , t 0 ).If t − u > 0 and t 0 − u > 0 a gradient estimate together with lemma 2.1 gives us Notice that this last case only happens in a set of measure smaller or equal than Putting this estimates together we get So the first inequality in (3.10) holds.Moreover which is the second inequality in (3.10).
Take x ∈ ( 3 2 Q) c (x = (x, t)).Then consider which, by corollary 3.3, implies that f is admissible for Q and we may apply lemma 3.4 to obtain |(P ε * ϕν)(x)| ≤ C also in this case.

Capacities and removable singularities
Given a compact set E ⊂ R N +1 , we define , where the supremum is taken over all distributions ν supported on E such that (4.2) We call γ 1/2 Θ (E) the 1 2 -fractional caloric capacity of E. We also define the capacity γ 1/2 Θ,+ (E), in the same way as in (4.1), but with the supremum restricted to all positive measures ν supported on E satisfying (4.2).Clearly, and Proof.Just by the previous definitions, γ first notice that we can assume E to be compact.Let ν be a distribution supported on E satisfying (4.2) and let {A i } i∈I be a collection of sets in R N +1 which cover E, and such that For each i ∈ I, let B i be an open ball centered in By the compactness of E we can assume I to be finite.By means of the Harvey-Polking lemma [HPo,Lemma 3.1], we can construct C ∞ functions ϕ i , i ∈ I, satisfying: • i∈I ϕ i = 1 in i∈I B i , Hence, by corollary 3.3 and lemma 3.4 Since this holds for any distribution ν supported on E satisfying (4.2), γ To prove the second assertion in the lemma, let E ⊂ R N +1 be a Borel set satisfying dim H (E) = s > N .We may assume E to be bounded and we can apply the well known Frostman lemma.Then, it follows that there exists some non-zero positive measure µ supported on E satisfying µ(B(x, r)) ≤ r s for all x ∈ R N +1 and all r > 0.
Thus, by Lemma 2.1, we deduce that for all x ∈ R N +1 Therefore, We say that a compact set E ⊂ R N +1 is removable for bounded 1 2 -fractional caloric functions (or 1 2 −fractional caloric removable) if any bounded function f : R N +1 → R satisfying the 1 2 −fractional heat equation in R N +1 \ E, also satisfies the heat equation in the whole space (in the distributional sense).Since E is compact there exists some (open) cube such that E ⊂ Q and Θ 1/2 (f ) ≡ 0 in Q.Consider the distribution ν = Θ 1/2 (f ).Since ν does not vanish identically in Q, there exists some C ∞ function ϕ supported on Q such that ν, ϕ > 0. Now take g = P * (ϕν).By theorem 3.1, and thus, since supp(ϕν) From the preceding lemmas, it is clear that, for any compact set Thus the critical Hausdorff dimension for 1 2 −fractional caloric removability (and for γ Θ ) is N .We consider the operator T ν = P * ν, defined over distributions ν in R N +1 .When µ is a finite measure, one can easily check that T µ(x) is defined for µ-a.e.x ∈ R N +1 by the integral For ε > 0, we also consider the truncated operator whenever the integral makes sense, and for a function f ∈ L 1 loc (µ), we write We also denote We say that T µ is bounded in L 2 (µ) if the operators T µ,ε are bounded in L 2 (µ) uniformly on ε > 0.
Given E ⊂ R N +1 , we define the capacity , where the supremum is taken over all positive measures µ supported on E such that Here T * is dual of T .That is, Notice that by definition, γ In the next theorem we characterize γ 1/2 Θ,+ (E) in terms of the positive measures supported on E such that T µ is bounded in L 2 (µ).
with the implicit constant in the above estimate independent of E.
The arguments to show that γ 1/2 Θ,+ (E) ≈ S are standard.Indeed, let µ be a positive measure supported on E such that γ Θ,+ (E) ≤ 2µ(E), T µ L ∞ (R N+1 ) ≤ 1 and T * µ L ∞ (R N+1 ) ≤ 1.By a Cotlar type inequality analogous to the one in [MaP,Lemma 5.4], say, one deduces that To obtain the boundedness of the operator T µ in L 2 (µ) we will use a suitable T 1 theorem with respect to a measure which may be non-doubling (see for example [To2,Th 3.21]).Since T µ and T * µ are bounded in L ∞ , as a consequence of the result in [HyMar], to apply the T 1 theorem in our case it is enough to check that the weak boundedness property is satisfied for balls with thin boundaries, that is, < T µ χ B , χ B >≤ Cµ(2B), if B is a ball with thin boundary.A ball of radius r(B) is said to have thin boundary if (4.5) µ{x : dist(x, ∂B) ≤ tr(B)} ≤ tµ(2B).
Let's consider a C ∞ function φ with compact support in 2B such that φ ≡ 1 on B. Then Using Theorem 3.1 one can see that the first term in the right hand side is bounded by Cµ(B).To get a bound of the second term we will use that B has a thin boundary.Using the boundedness of K, property (a) in Lemma 2.1, Given j and x / ∈ B such that dist(x, ∂B) ∼ 2 −j r(B), since µ is a measure with N -growth, one has Therefore, by 4.5 Consequently, the weak boundedness property is satified and by the T 1−theorem it follows that T µ is bounded in L 2 (µ), with T µ L 2 (µ)→L 2 (µ) 1.So we deduce that S γ Θ,+ (E).
To prove the converse estimate, let µ be a positive measure supported on E be such that T µ L 2 (µ)→L 2 (µ) ≤ 1 and S ≤ 2µ(E).From the L 2 (µ) boundedness of T µ , one deduces that T and T * are bounded from the space of finite signed measures M (R N +1 ) to L 1,∞ (µ).That is, there exists some constant C > 0 such that for any measure ν ∈ M (R N +1 ), any ε > 0, and any λ > 0, and the same holds replacing T ε by T * ε .The proof of this fact is analogous to the one of Theorem 2.16 in [To2].Then, by a well known dualization of these estimates (essentially due to Davie and Øksendal) and an application of Cotlar's inequality, one deduces that there exists some function h : See Theorem 4.6 and Lemma 4.7 from [To2] for the analogous arguments in the case of analytic capacity and also Lemma 4.2 from [MaP] for the precise vectorial version of the dualization of the weak (1, 1) estimates required in our situation, for example.So we have

The existence of removable sets with positive Hausdorff measure H N
In this section we will construct a self-similar Cantor set E ⊂ R N +1 with positive and finite H N measure which is 1 2 −caloric removable.To obtain this result we need the following theorem, showing that certain distributions are actually measures.
Theorem 5.1.Let E ⊂ R N +1 be a compact set with H N (E) < ∞ and let ν be a distribution supported on E such that P * ν ∞ ≤ 1.Then ν is a signed measure, absolutely continuous with respect to H N +1 | E and there exists a Borel function f : In what follows, we say that a distribution ν in R N +1 has N −dimensional growth, if for any cube Q ⊂ R N +1 and any Theorem 5.1 is a consequence of Lemma 3.4 and the following result.
Lemma 5.2.Let E ⊂ R N +1 be a compact set with H N (E) < ∞.Let ν be a distribution supported on E wich has N -dimensional growth.Then ν is a signed measure, absolutely continuous with respect to H N | E and there exists a Borel function f : Proof.First we will show that ν is a signed measure.By the Riesz representation theorem, it is enough to show that, for any C ∞ function ψ with compact support, where C(E) is some constant depending on E.
To prove (5.1), we fix ε > 0 and we consider a family of open cubes Since E is compact, we can assume that I ε is finite.By standard arguments, we can find a family of non-negative functions ϕ i , i ∈ I ε , such that • each ϕ i is supported on 2Q i and cϕ i is admissible for 2Q i , for some absolute constant c > 0, • i∈Iε ϕ i = 1 on i∈Iε Q i , and in particular on E. Indeed, to construct the family of functions ϕ i we can cover each cube Q i by a bounded number (depending on n) and then apply the usual Harvey-Polking lemma ( [HPo,Lemma 3.1] to the family of cubes {R i,j }.
We write For each i ∈ I ε , consider the function where ∇ = (∇ x , ∂ t ).We claim that c η i is admissible for 2Q i , for some absolute constant c > 0.
To check this, just note that ϕ i ψ is supported on 2Q i and satisfies Hence, .
So, by corollary 3.3, the claim above holds and, consequently, by the assumptions in the lemma, From the preceding estimate, we deduce that which gives (5.1) and proves that ν is a finite signed measure, as wished.
It remains to show that there exists some Borel function f : To this end, let g be the density of ν with respect to its variation |ν|, so that ν = g |ν| with g(x) = ±1 for |ν|-a.e.x ∈ R N +1 .We will show that (5.2) lim sup This implies that |ν| = f H N | E for some non-negative function f 1.This fact is well known (see Theorem 6.9 from [Ma]).So to complete the proof of the lemma it suffices to show (5.2) (since then we will have ν = g f H N | E with |g f | 1).Notice that, by the Lebesgue differentiation theorem, Let x ∈ E be a Lebesgue point for |ν| with |g(x)| = 1, let ε > 0 to be chosen below, and let r 0 > 0 be small enough such that, for 0 Suppose first that and let ϕ x,r be some non-negative C ∞ function supported on B(x, 2r) which equals 1 on B(x, r) such that c ϕ x,r is admissible for the smallest cube Q containing B(x, 2r), so that ˆϕx,r dν r N .
To get rid of the doubling assumption (5.3), notice that for |ν|-a.e.x ∈ R N +1 there exists a sequence of balls B(x, r k ), with r k → 0, satisfying (5.3) (we say that the balls B(x, r k ) are |ν|-doubling).Further, we may assume that r k = 2 h k , for some h k ∈ N. The proof of this fact is analogous to the one of Lemma 2.8 in [To2].So for such a point x, by the arguments above, we know that there exists some k 0 > 0 such that assuming also that x is a |ν|-Lebesgue point for the density g.Given an arbitrary r ∈ (0, r k 0 ), let j be the smallest integer r ≤ 2 j , and let 2 k be the smallest j ≤ k such that the ball B(x, 2 k ) is |ν|-doubling (i.e., (5.3) holds for this ball).Observe that 2 k ≤ r k 0 .Then, taking into account that the balls B(x, 2 h ) are non-doubling for j ≤ h < k and applying (5.4) for r = 2 k , we obtain Hence, (5.2) holds and we are done.
Next we will construct a self-similar Cantor set E ⊂ R N +1 with positive and finite measure H N which is removable for the 1 2 −fractional heat equation.Our example is inspired by the typical planar 1/4 Cantor set in the setting of analytic capacity (see [Ga] or [To2,p. 35], for example).
We construct the Cantor set E as follows: we let E 0 be the unit cube, i.e.
such that they are all contained in Q 0 and each one contains a vertex of Q 0 .
We proceed inductively: In each generation k, we replace each cube and located in the same relative position to Q k−1 j as the cubes Q 1 1 , . . ., Q 1 2 N+1 with respect to Q 0 .Notice that in each generation k there are 2 k(N +1) closed cubes with side length 2 k .We denote by E k the union of all these cubes from the k-th generation.By construction, k .Taking into account that, for each k ≥ 0, Further, H N | E coincides, modulo a constant factor, with the probability measure µ supported on E which gives the same measure to all the cubes Theorem 5.3.The Cantor set E defined in (5.5) is removable for 1 2 − fractional caloric functions.
Proof.We will suppose that E is not removable and we will reach a contradiction.By Theorem 4.2, there exists a distribution ν supported on E such that | ν, 1 | > 0 and where µ is the probability measure supported on E such that µ( +1) for all i, k.It is easy to check that µ (and thus |ν|) has upper growth of degree N .Recall that T denotes the operator T ν = P * ν, defined over distributions ν in R N +1 .Then, arguing as in [MaP,Lemma 5.4], it follows that there exists some constant K such that (5.6) Then we consider the auxiliary operator By the separation condition between the cubes Q k i , the upper growth of |ν|, and the condition (5.6), it follows easily that (5.7) for some fixed constant K ′ .We will contradict the last estimate.To this end, consider a Lebesgue point x0 ∈ E (with respect to µ) of the density f = dν dµ such that f (x 0 ) > 0. The existence of this point is guarantied by the fact that ν(E) > 0. Given ε > 0 small enough to be chosen below, consider a cube Given m ≫ 1, to be fixed below too, it is easy to check that if ε is chosen small enough (depending on m and on f (x 0 )), then the above condition ensures that every cube Notice also that, writing Let z = (z, u) be one of the upper corners of Using the fact that dist(z, Then, by the choice of z, it follows that (5.9) We write Taking into account (5.9) and the fact that, for k such that for all ȳ = (y, s), using also (5.8), we deduce Together with the previous estimate for It is clear that if we choose m big enough and then ε small enough, depending on m, this lower bound contradicts (5.7), as wished.
In this section we will show that, in the plane, the capacity γ 1/2 Θ is not comparable to analytic capacity nor to Newtonian capacity; two classical capacities related to complex analysis and potential theory with critical dimension 1, as γ 1/2 Θ .More precisely, we will construct two sets, the first one will be a compact set E 1 ⊂ C with γ 1/2 Θ (E 1 ) > 0 but vanishing Newtonian capacity.The second example we will be a compact set E 2 ⊂ C with positive analytic capacity but removable for the 1 2 −fractional heat equation.First we recall some definitions.For a compact set E ⊂ C, the analytic capacity of E is defined as γ(E) = sup{|f ′ (∞)|}, where the supremum is taken over all analytic functions f : The Newtonian capacity of a compact set E ⊂ C can be defined as It is well-known that if E ⊂ C is a segment of length ℓ, then γ(E) = ℓ/4 and C(E) = 0 (see [To2,Proposition 1.4] and [Ga,Corollary 3.6] respectively).
We will prove that if ).The result reads as follows.Proposition 6.1.Let L be a positive number.Set

Proof.
a) Let χ E 1 be the characteristic function of E 1 and consider the positive measure where P is the fundamental solution of the 1 2 -fractional heat equation in R 2 , that is Then there exists a distribution S supported on E 2 such that P * S ∞ ≤ 1. Approximating the distribution S by signed measures (take for instance the convolution of S with approximations of the identity in the variable t), we can assume that there exists a signed measure ν supported on the segment for some small ε > 0, satisfying P * ν ∞ ≤ 2. Since P is a non negative kernel, there exists a positive measure µ supported on E ε with µ(E ε ) > 0 such that (6.1) To get a contradiction we will show that (6.1) implies µ(E ε ) = 0. Since, by lemma 3.4, the measure µ has linear growth, given η > 0 we can take c ∈ (−ε, L + ε) such that µ({(0, t) : c ≤ t ≤ L + ε}) < η.
Let • * ,p denote the norm of the parabolic BMO space: where the supremum is taken over all s−parabolic cubes Q ⊂ R N +1 , dm stands for the Lebesgue measure in R N +1 and m Q f is the mean of f with respect to dm.
For a function f : R N +1 → R and α ∈ (0, 1), the α−fractional derivative with respect to t (recall that x ∈ R N and t ∈ R) is defined as Now we introduce the γ s Θ capacities for s ∈ (1/2, 1).Given a compact set E ⊂ R N +1 , we define (7.1) where the supremum is taken over all distributions ν supported on E satisfying Notice that for s = 1/2 we obtain γ 1/2 Θ .The capacity γ s Θ is called the s-fractional caloric capacity of E. We set γ s Θ,+ (E) as in (7.1) but with the supremum restricted to all positive measures ν supported on E satisfying (7.2).Clearly We are now ready to describe the basic relationship between the capacity γ s Θ and Hausdorff content (the d-dimensional Hausdorff content will be denoted by H d ∞ ).
Theorem 7.1.Let s ∈ (1/2, 1).For every Borel set E ⊂ R N +1 , To prove theorem 7.1 we need to study the growth conditions satisfied by positive measures and distributions with properties (7.2).
7.1.The case of positive measures.We state first two lemmas concerning the case when the distribution ν is a positive measure.But before going to the next lemma, recall that a function f (x, t) defined in R N +1 is Lip(α), 0α < 1, in the t variable if Lemma 7.2.Let s ∈ (1/2, 1) and let µ be a positive measure in R N +1 with upper parabolic growth of degree N + 2s − 1 with constant 1.Then, 1.
Proof.Let x = (x, t), x = (x, u), and x0 = 1 2 (x + x).Then, writing ȳ = (y, s), we split To shorten notation, we write d := |x − x| p = |t − u| 1 2s .Then we have Since by property (3) in lemma 2.2 we deduce that Next we deal with I 2 .Writing B 0 = B p (x 0 , 2d), we have Observe now that by lemma 2.2 The last estimate follows by splitting the integral into parabolic annuli and using the parabolic growth of order N + 2s − 1 of µ, for example.The same estimate holds replacing x by x.Then gathering all the estimates above, the lemma follows. Lemma First we will estimate the term A 1 .For u, t such that |u − t| ≤ 2 2s ℓ(Q) 2s , we write Observe now that for each z = (z, v) ∈ (1 + 2 2s )Q (see property (3) in lemma 2.2) by splitting the last domain of integration into parabolic annuli and using the growth condition of order N + 2s − 1 of µ.Thus, Plugging this into the integral that defines A 1 , we obtain By exactly the same arguments, just writing y in place of x above, we deduce also that Concerning the term B, we write By Lemma 2.2, it follows that for ξ ∈ Q, For u ∈ A k , using lemma 2.2 we get So (7.3) holds in this case.
Proof of (7.3) in Case 2. As in Case 1, we write The terms A ′ 1 and A ′ 2 can be estimated exactly in the same way as the terms A 1 and A 2 in Case 1, so that Concerning B ′ , we have and that, by Lemma 7.2, P s * (φµ)(x, •) is Lip(1 − 1 2s ) in the variable t, we deduce that This concludes the proof of ∂ As in the case s = 1/2 (see (3.1)) recall that (−∆ x ) 1/2 ϕ ≈ N j=1 R j ∂ j ϕ, with R j being the Riesz transforms with Fourier multiplier ξ j /|ξ|.Then, for an s-parabolic cube Therefore if ϕ satisfies (7.5) and then ϕ is admissible for the s-parabolic cube Q, s ∈ (1/2, 1).Let's remark that the analogues of lemmas 3.2 and 3.3 also hold in this context and if ϕ is a standard test function supported on an s−parabolic cube (ν * P s ), we have Hence, due to (7.7), For I 22 , we split the domain of integration in annuli.Write so that 2 i Q is an s-parabolic cube too (notice that if Q is centered at the origin and we consider the parabolic dilation δ λ (x, t) = (λx, λ 2s t), λ > 0, we have 2 i Q = δ 2 i (Q)).Then, using the decay of g given by (7.6), we get (7.8) To estimate the first integral on the right hand side, recall that supp g ⊂ Q 1 × R. Using Hölder's inequality with some exponent q ∈ (0, ∞) to be chosen in a moment and the fact that f ∈ BM O p (together with John-Nirenberg), then we get: For the last integral on the right hand side of (7.8), we write Therefore, Choosing q > N , we get 7.3.Proof of Theorem 7.1.The inequality γ s Θ,+ (E) ≤ γ s Θ (E) comes from the definition of the capacities.For the second inequality we assume E to be compact and let {Q j } j be a covering of E by s-parabolic cubes Q j ⊂ R N +1 with disjoint interiors.By a parabolic version of a well known lemma of Harvey and Polking (see [HPo]), there exist functions g j ∈ C ∞ 0 (2Q j ) satisfying j g j = 1 in a neighborhood of ∪ j Q j , (7.4) and (7.5).
Let ν be a distribution with compact support contained in E such that (7.2) hold.Then, using lemma 7.4

Thus, γ s
Θ (E) ≤ CH N +2s−1 ∞ (E).The second assertion in the theorem follows a standard argument that we reproduce for the reader's convenience.Suppose E ⊂ R N +1 is a Borel set with dim H,p (E) = α > N + 2s − 1.By a parabolic version of Frostman's Lemma (that can be proved by arguments analogous to classical ones replacing the usual dyadic lattice in R N +1 by a parabolic dyadic lattice), there exists a non-zero positive measure µ supported on E such that µ(B(x, r)) ≤ r α for all x ∈ R N +1 and r > 0. We have to show estimates (7.2) with ν replaced by µ.
Here R s j is the Calderón-Zygmund operator with kernel x j |x| N +2s , the symbol * t denotes convolution with respect to the t variable and K is the kernel , where M = 1 2s (for a real number λ, [λ] stands for its integer part).Notice that for s ∈ (0, 1/2), (8.4) (−∆) s ϕ = N j=1 R s j (∂ j ϕ).