Limited regularity of solutions to fractional heat and Schr\"odinger equations

When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$: $r^+Pu(x,t)+\partial_tu(x,t)=f(x,t)$ on $\Omega \times \,]0,T[\,$, $u(x,t)=0$ for $x\notin\Omega $, $u(x,0)=0$, is known to be solvable in relatively low-order Sobolev or H\"older spaces. We now show that in contrast with differential operator cases, the regularity of $u$ in $x$ at $\partial\Omega $ when $f$ is very smooth cannot in general be improved beyond a certain estimate. An improvement requires the vanishing of a Neumann boundary value. --- There is a similar result for the Schr\"odinger Dirichlet problem $r^+Pv(x)+Vv(x)=g(x)$ on $\Omega $, $v(x)=0$ for $x\notin \Omega $, with $V(x)\in C^\infty $. The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a $dist(x,\partial\Omega)^a$ singularity.


Introduction
The main purpose of the paper is to investigate limitations on the regularity of solutions to nonlocal parabolic Dirichlet problems for x in a bounded smooth subset Ω of R n and t in an interval I = ]0, T [ : where P is the fractional Laplacian (−∆) a on R n , 0 < a < 1, or a pseudodifferential generalization (that can be x-dependent and non-symmetric); here r + denotes restriction to Ω. For the stationary Dirichlet problem (0.2) r + P v(x) = g (x) in Ω, supp u ⊂ Ω, it is known from works of Ros-Oton, Serra, Grubb [RS14,RS16,G15,G14], that the solution v bears a singularity at ∂Ω like d(x) a , where d(x) = dist(x, ∂Ω), but that v/d a is steadily more regular, the more regular g is. One has for example: g ∈ C σ (Ω) =⇒ v/d a ∈ C a+σ (Ω) for σ > 0 with σ, a + σ / ∈ N, (0.3) [RS14,RS16] for small σ (allowing low regularity of ∂Ω), and in [G14] for all σ, and (0.4) is shown in [G15], drawing on early work of Hörmander.
It is natural to ask whether the nonstationary results can be lifted to higher regularities in x like in the stationary case: Will solutions have a C ∞ -property if f is C ∞ ? or e.g. a higher Hölder regularity, when f belongs to a higher Hölder space? Such rules holds for differential operator heat problems, and for interior regularity [G18], but, perhaps surprisingly, they do not hold up to the boundary in the present nonlocal cases.
A first counterexample to the C ∞ -lifting was given in [G18b], derived from a certain irregularity of the eigenfunctions of the Dirichlet realization of P . In the present study we show more precisely how the regularity of the solution is limited to u ∈ D a,C (P ) with respect to x (not D a+δ,C (P ) with δ > 0), unless the boundary value of u/d a vanishes.
The heat equation result is based on an analysis of the solutions of the resolvent equation for λ = 0, (0.10) (r + P − λ)v = g in Ω, supp v ⊂ Ω, and a precise description of the spaces D r,C (P ).
We also study Schrödinger Dirichlet problems (0.11) (r + P + V )v = g in Ω, supp v ⊂ Ω, with a C ∞ -potential V , and find a related limitation on the smoothness of solutions, when V does not vanish on ∂Ω.
About the operators P : The fractional Laplacian (−∆) a on R n , 0 < a < 1, can be described as a pseudodifferential operator (ψdo) or as a singular integral operator: The operators we shall study are the following generalization of (0.12) to a large class of ψdo's: The classical strongly elliptic ψdo's P = Op(p(x, ξ)) of order 2a, with symbol p(x, ξ) ∼ j∈N 0 p j (x, ξ) being even: is a second-order strongly elliptic differential operator.) The singular integral definition (0.13) can also be generalized, by replacement of the kernel function |y| −n−2a by other positive functions K(y) homogeneous of degree −n − 2a and even, i.e. K(−y) = K(y), and with possibly less smoothness, or nonhomogeneous but estimated above in terms of |y| −n−2a (see e.g. the survey [R16]); this gives translationinvariant symmetric operators. These are operators defining stable Lévy processes. The case where K is homogeneous and C ∞ for y = 0 is a special case of our ψdo's, with symbol p(ξ) = F K(y).
Let us mention some of the results. The general assumption is: Hypothesis 0.1. For some a > −1, P = Op(p(x, ξ)) is a classical strongly elliptic ψdo of order 2a on R n , with even symbol p(x, ξ), cf. (0.14), and possibly with a smoothing term R added (continuous from E ′ (R n ) to C ∞ (R n )).
Firstly, we show in Section 2 an exact representation of the elements of the domain spaces D r,H p (P ) (0.5) and D r,C (P ) (0.6) in terms of a component supported in Ω with smoothness 2a + r and a component pulled back from the boundary ∂Ω by Poisson-like operators applied to weighted boundary values γ a j v, this is a development of results from [G15,G14].
Based on this, we show in Sections 3 resp. 4: Theorem 0.2. Let 0 < a < 1, and let V ∈ C ∞ (Ω). When g ∈ C a (Ω), the solutions of (0.11) are in D a,C (P ). Let V = 0 on an open subset Σ of ∂Ω, and let v be a solution of (0.11). If there is a δ > 0 such that v ∈ D a+δ,C (P ) with g ∈ C a+δ (Ω), then γ a 0 v vanishes on Σ. Here γ a 0 v can be regarded as the Neumann boundary value of v.

Preliminaries
The domain spaces for homogeneous Dirichlet problems were described in many scales of spaces in [G15] and [G14]; we shall here just focus on two scales, namely the Besselpotential scale H s p (1 < p < ∞) (the main subject of [G15]), which serves to show general estimates in L p Sobolev spaces, and the Hölder-Zygmund scale C s * (included in [G14] in a systematic way) leading to optimal Hölder estimates. We shall go through the various concepts somewhat rapidly, since they have already been explained in previous papers; the reader may consult e.g. [G15,G14] if more details are needed.
We recall that the standard Sobolev spaces W s,p (R n ), 1 < p < ∞ and s ≥ 0, have a different character according to whether s is integer or not. Namely, for s integer, they consist of L p -functions with derivatives in L p up to order s, hence coincide with the Bessel-potential spaces H s p (R n ), defined for s ∈ R by Here F is the Fourier transformû(ξ) = F u(ξ) = R n e −ix·ξ u(x) dx, and the function ξ equals (|ξ| 2 + 1) 1 2 . For noninteger s, the W s,p -spaces coincide with the Besov spaces, defined e.g. as follows: For 0 < s < 2, |f (x) + f (y) − 2f ((x + y)/2)| p |x + y| n+ps dxdy < ∞; The Bessel-potential spaces are important because they are most directly related to L p (R n ); the Besov spaces have other convenient properties, and are needed for boundary value problems in an H s p -context, because they are the correct range spaces for trace maps γ j u = (∂ j n u)| x n =0 : surjectively and with a continuous right inverse; see e.g. the overview in the introduction to [G90]. For p = 2, the two scales are identical, but for p = 2 they are related by strict inclusions: H s p ⊂ B s p when p > 2, H s p ⊃ B s p when p < 2. The following subsets of R n will be considered: R n ± = {x ∈ R n | x n ≷ 0} (where (x 1 , . . . , x n−1 ) = x ′ ), and bounded C ∞ -subsets Ω with boundary ∂Ω, and their complements. Restriction from R n to R n ± (or from R n to Ω resp. ∁Ω) is denoted r ± , extension by zero from R n ± to R n (or from Ω resp. ∁Ω to R n ) is denoted e ± . Restriction from R n + or Ω to ∂R n + resp. ∂Ω is denoted γ 0 . We denote by d(x) a function of the form d(x) = dist(x, ∂Ω) for x ∈ Ω, x near ∂Ω, extended to a smooth positive function on Ω; d(x) = x n in the case of R n + . Along with the spaces H s p (R n ) defined in (1.1), we have the two scales of spaces associated with Ω for s ∈ R: here supp u denotes the support of u. The definition is also used with Ω = R n + . H s p (Ω) is in other texts often denoted H s p (Ω) or H s p (Ω), andḢ s p (Ω) may be indicated with a ring, zero or twiddle; the current notation stems from Hörmander [H85], Appendix B2. There are similar spaces with B s p . We recall that H s p (Ω) andḢ −s p ′ (Ω) are dual spaces with respect to a sesquilinear duality extending the L 2 (Ω)-scalar product; 1 p + 1 p ′ = 1.

Pseudodifferential operators.
A pseudodifferential operator (ψdo) P on R n is defined from a symbol p(x, ξ) on R n ×R n by using the Fourier transform F . We refer to textbooks such as Hörmander [H85], Taylor [T81], Grubb [G09] for the rules of calculus (in particular the definition by oscillatory integrals in [H85]). The symbols p of order m ∈ R we shall use are assumed to be classical: p has an asymptotic expansion p( P (and p) is said to be strongly elliptic when Re p 0 (x, ξ) ≥ c|ξ| m for |ξ| ≥ 1, with c > 0. These classical ψdo's of order m are continuous from H s p (R n ) to H s−m p (R n ) for all s ∈ R. For a complete theory one adds to these operators the smoothing operators (mapping any H s p (R n ) into t H t p (R n )), regarded as operators of order −∞. (For example, (−∆) a fits into the calculus when it is written as Op is a C ∞ -function that equals 1 for |ξ| ≤ 1 2 and 0 for |ξ| ≥ 1; the second term is smoothing.) Remark 1.1. The operators we consider in this paper are moreover assumed to be even, cf. (0.14), for simplicity. In comparison, P of order 2a satisfies the a-transmission condition introduced by Hörmander [H66, H85,G15] relative to a given smooth set Ω ⊂ R n when at all points x ∈ ∂Ω, with interior normal denoted ν(x). The evenness means that this is satisfied for any choice of Ω. The results in the following hold also when one only assumes that the a-transmission condition is satified relative to the particular domain Ω considered.
For our description of the solution spaces for (0.2) we must introduce order-reducing operators. There is a simple definition of operators Ξ t ± on R n , t ∈ R, they preserve support in R n ± , respectively, because the symbols extend as holomorphic functions of ξ n into C ∓ , respectively; C ± = {z ∈ C | Im z ≷ 0}. (The functions ( ξ ′ ±iξ n ) t satisfy only part of the estimates (1.6) with m = t, but the ψdo definition can be applied anyway.) There is a more refined choice Λ t ± [G90, G15], with symbols λ t ± (ξ) that do satisfy all the required estimates, and where λ t + = λ t − . These symbols likewise have holomorphic extensions in ξ n to the complex halfspaces C ∓ , so that the operators preserve support in R n ± , respectively. Operators with that property are called "plus" resp. "minus" operators. There is also a pseudodifferential definition Λ (t) ± adapted to the situation of a smooth domain Ω, cf. [G15].
It is elementary to see by the definition of the spaces H s p (R n ) in terms of Fourier transformation, that the operators define homeomorphisms for all s: The special interest is that the "plus"/"minus" operators also define homeomorphisms related to R n + and Ω, for all s ∈ R: with similar rules for Λ t ± . Moreover, the operators Ξ t + and r + Ξ t − e + identify with each other's adjoints over R n + , because of the support preserving properties. There is a similar statement for Λ t + and r + Λ t − e + , and for Λ (t) + and r + Λ (t) − e + relative to the set Ω.

The a-transmision spaces.
The special a-transmission spaces were introduced by Hörmander [H66] for p = 2, cf. the account in [G15] with the definition for general p: Recall also from [G15] Sect. 5 that there is a hierarchy: H The great interest of these spaces is that they allow an exact description of the solution spaces for the Dirichlet problem (0.2), and are independent of P . The following result comes from [G15]: (Ω), and the mapping r + P : v → g is Fredholm between these spaces. It follows that the Dirichlet domain D r, Proof. The first statement is taken directly from Th. 4.4 of [G15], with µ 0 = a, m = 2a; the factorization index is a because of the strong ellipticity, as shown in detail e.g. in [G16a]. The second statement specializes this to r ≥ 0 (which is all we need in the present paper) where the prerequisite on v can be simplified to v ∈Ḣ a p (Ω), since H a(r+2a) p (Ω) for small ε.

Results in Hölder-Zygmund spaces.
In [G14] (that was written after [G15]), the results were extended to many other scales of spaces, such as Besov spaces B s p,q for 1 ≤ p, q ≤ ∞, and Triebel-Lizorkin spaces F s p,q (Ω), for s > a − 1, (where the * 's can be omitted if s, s − a / ∈ N 0 ), and there are inclusions as described for H a(s) (Ω)-spaces in (1.13)-(1.14): There is a result similar to that of Theorem 1.2 with H s p -spaces replaced by C s * -spaces: (Ω), and the mapping r + P : v → g is Fredholm between these spaces. It follows that the Dirichlet domain D r,C (P ) defined as D r, (Ω).
For r ∈ R + \ N, this is the domain defined in (0.6); and when a + r and 2a + r are noninteger, C a(r+2a) * (Ω) identifies with C a(r+2a) (Ω) = Λ (−a) + e + C r+a (Ω) defined in terms of ordinary Hölder spaces. It is sometimes an advantage to keep the C * -notation, since one does not have to make exceptions for integer indices all the time.
In applications of the above results it is important to get a better understanding of what the a-transmission spaces consist of. Such an analysis was carried out for the scale H a(r+2a) p in [G15] and for C a(r+2a) * in [G14], and in the next section we take it up again with more explicit results relative to the set Ω.

Analysis of a-transmission spaces
2.1 Decompositions in terms of the first trace.
In the following, we shall give a characterization of the a-transmission spaces showing an exact decomposition of the elements in the case of a general domain Ω; it is just a more detailed development of the decomposition principle described in Th. 5.4 of [G15]. For clarity, we begin with a decomposition with just one trace involved.
In the proofs we shall use a localization with particularly convenient coordinates, described in detail in [G16] Rem. 4.3 and Lem. 4.4 and recalled in [G18a] Rem. 4.3, which we also recall here: Remark 2.1. Ω has a finite cover by bounded open sets U 0 , . . . , ; as usual we write ∂R n + = R n−1 . For any such cover there exists an associated partition of unity, namely a family of functions ̺ i ∈ C ∞ 0 (U i ) taking values in [0, 1] such that i=0,...,I ̺ i is 1 on a neighborhood of Ω.
When P is a ψdo on R n , its application to functions supported in U i carries over to functions on V i as a ψdo P (i) defined by When u is a function on U i , we usually denote the resulting function u • κ −1 i on V by u. We shall use a convenient system of coordinate charts as described in [G16], Remark 4.3: Here ∂Ω is covered with coordinate charts κ ′ i : . . , I, and the κ i will be defined on certain subsets of a tubular neighborhood Σ . . , ν n (x ′ )) is the interior normal to ∂Ω at x ′ ∈ ∂Ω, and r is taken so small that the mapping x ′ + tν(x ′ ) → (x ′ , t) is a diffeomorphism from Σ r to ∂Ω× ] − r, r[ . For each i, κ i is defined as the mapping κ i : These charts are supplied with a chart consisting of the identity mapping on an open set U 0 containing Ω \ Σ r,+ , with U 0 ⊂ Ω, to get a full cover of Ω. Note that the normal ν(x ′ ) at x ′ ∈ ∂Ω is carried over to the normal (0, 1) at is the geodesic into Ω orthogonal to ∂Ω at x ′ (with respect to the Euclidean metric on R n ), and there is a positive r ′ ≤ r such that for 0 < t < r ′ , the distance d(x) between x = x ′ + tν(x ′ ) and ∂Ω equals t. Then t plays the role of d in the definition of expansions and boundary values of u ∈ E a (Ω) in [G15] where the u j are constant in t for t < r ′ ; this serves to define the boundary values (denoted γ a,j u in [G15]). By comparison of (2.3) with t a times the Taylor expansion of u/t a in t, we also have: (2.5) γ a 0 u = Γ(a + 1)γ 0 (u/t a ), γ a 1 u = Γ(a + 2)γ 1 (u/t a ) = Γ(a + 2)γ 0 (∂ t (u/t a )), etc.
We first recall (and reprove) a result from [G15] for the case where the domain is R n + : Lemma 2.2. Let a > −1.
Let K 0 denote the Poisson operator from R n−1 to R n + with symbol ( ξ ′ + iξ n ) −1 . When s > a + 1/p, the elements of H a(s) p In fact, the elements of H a(s) p (R n + ) are parametrized as (2.7) u = e + 1 Γ(a+1) x a n K 0 ϕ + w, where ϕ runs through B s−a−1/p p (R n−1 ) and w runs through H (a+1)(s) p (R n + ); here ϕ equals γ a 0 u. Proof. In detail, K 0 is the elementary Poisson operator of order 0 in the Boutet de Monvel calculus (cf. e.g. [B71,G96,G09]): which solves the Dirichlet problem (1 − ∆)u = 0 on R n + , γ 0 u = ϕ on R n−1 . (An extension by 0 for x n < 0 is sometimes tacitly understood.) Define K a 0 by x a n e + K 0 ϕ; by the last expression, it is a right inverse of γ a 0 : u → Γ(a+1)γ 0 (u/x a n ). (These calculations played an important role in [G15], cf. (2.5) and the proofs of Cor. 5.3 and Th. 5.4 there; the constant called c µ in (5.16) is written explicitly here, equal to 1/Γ(µ + 1).) When u ∈ H a(s) p (R n + ), then γ a 0 u ∈ B s−a−1/p p (R n−1 ) (cf. [G15] Th. 5.1), and w = u − K a 0 γ a 0 u has γ a 0 w = 0. By the mapping properties of the Poisson operator K 0 known from [G90], e + K 0 γ a 0 u lies in e + H s−a p (R n + ). Then the last expression for K a 0 in (2.9) shows that K a 0 γ a 0 u ∈ x a n e + H There is a Poisson operator K (0) of order 0 from ∂Ω to Ω (in the Boutet de Monvel calculus) with principal symbol ( ξ ′ + iξ n ) −1 in local coordinates at the boundary, such that K (0) is a right inverse of γ 0 , and the following holds: When s > a + 1/p, the elements of H a(s) p (Ω) have a decomposition In fact, the elements of H a(s) p (Ω) are parametrized as where ϕ runs through B s−a−1/p p (∂Ω) and w runs through H (a+1)(s) p (Ω). The latter space is equal toḢ s p (Ω) when s − a ∈ ]1/p, 1 + 1/p[ . When u is written in this way, γ a 0 u equals ϕ. (Ω) and satisfies Proof. We use the local coordinates κ i : U i → V i , i = 0, 1, . . . , I, described in Remark 2.1, with an associated partition of unity {̺ i } i≤I . We can moreover choose nonnegative functions ψ k i ∈ C ∞ 0 (U i ), k = 1, 2, 3, such that ψ 1 i ̺ i = ̺ i , i.e., ψ 1 i is 1 on supp ̺ i , and similarly ψ 2 ̺ i is a ψdo of order −∞, so it maps z into C ∞ (R n ); moreover, its symbol in local coordinates is holomorphic for Im ξ n < 0, so it preserves support in Ω. Hence the terms in the second sum in the right-hand side of (2.13) are iṅ (Ω) for all t (and absorbed in the w-term in the final formula). Henceforth we can focus on the first sum where u i is compactly supported in the set U i and belongs to H a(s) p (Ω).
where γ a 0 1 Γ(a+1) x a n K 0 = I. Multiplication by ψ 2 i or ψ 3 i does not alter u i ; this gives the representation (where we denote γ 0 ψ k i = ψ k i,0 ) Here e + ψ 3 i x a n K 0 γ a 0 (ψ 2 i u i ) ∈ H a(s) p (R n + ), since it is compactly supported and is the sum of e + x a n K 0 γ a 0 (ψ 2 i u i ) ∈ H a(s) p (R n + ) and (1 − ψ 3 i )e + x a n K 0 γ a 0 (ψ 2 i u i ) ∈ E a (R n + ), using that (1 − ψ 3 i )K 0 ψ 2 i,0 is a Poisson operator of order −∞. Then also ψ 3 i w i is in H a(s) p (R n + ), and since its first boundary value γ a 0 (ψ 3 the operator induced by ψ 3 i K 0 ψ 2 i,0 in the original coordinates. Again γ a 0 1 Γ(a+1) d aK i 0 γ a 0 u i = γ a 0 u i . Finally we find by summation over i the formula here γ 0 K (0) = I and γ a 0 K a (0) = I. This ends the proof.

Systems of traces.
For higher s, we also have representations in terms of consecutive sets of traces and Poisson operators.
Th. 5.4 of [G15] contains a formula (5.14) including higher-order traces of u. This formula lacks a sum of terms of the form S jk x µ+k n e + K 0 (γ µ j u) with 0 ≤ k < j, the S jk being constant-coefficient ψdo's on R n−1 of order j − k. In fact, we had overlooked that γ j K k = δ jk only holds for j ≤ k (cf. [G15] (1.7)), whereas it produces a nontrivial ψdo on R n−1 when j > k. This is a minor correction that does not change the outcome (5.15) of the theorem. The treatment of higher-order traces in the following theorems gives a more correct formula.
Theorem 2.4. Let a > −1 and let M be a positive integer. (2.14) where the Ψ jk are ψdo's on ∂Ω of order j − k. With (2.17) and, with γ a j u = Γ(a + 1 + j)γ j (u/d a ) (cf. (0.15)), Proof. 1 • . Since d identifies with t as in Remark 2.1 near ∂Ω, it is verified immediately that γ j K (k) = δ jk when j ≤ k. For j > k it is an elementary fact in the Boutet de Monvel calculus that the composition γ j K (k) results in a ψdo on ∂Ω of order j − k. These facts lead to (2.17), where the triangular matrix is clearly invertible, continuous from 0≤j<M B s−j−1/p p (∂Ω) to itself for all s. The stated mapping property follows, since d j K (0) is a Poisson operator of order −j in the Boutet de Monvel calculus.
For 2 • , we note that by definition,

hence lies in H
Here p = ∞, so p ′ = 1. The inequalities s > a + 1/p and s > a + M − 1/p ′ are here replaced by s > a and s > a + M − 1.
We are of course primarily interested in the results for noninteger positive values of the exponents, where the spaces are ordinary Hölder spaces, C s * = C s for s ∈ R + \ N, but the C s * spaces are useful e.g. by having good interpolation properties -and of course by allowing statements without exceptional parameters. Recall moreover from [J96] and [G14] that the identification of spacesĊ s * (Ω) and e + C s * (Ω) takes place for −1 < s < 0, another useful point.

The regularity of solutions of fractional Schrödinger Dirichlet problems
In preparation for the study of heat equation regularity, we shall consider a related problem for the Schrödinger equation, which is of interest in itself. Consider the Dirichlet problem for the fractional Schrödinger equation: where V is a C ∞ -function, and P satisfies Hypothesis 0.1 with 0 < a < 1.
Recall that for the usual Laplacian ∆, it makes no difference in the regularity of Dirichlet solutions whether a C ∞ -function V is added or not; the solutions of −∆u + V u = f in Ω, γ 0 u = 0 on ∂Ω, satisfy for all k ∈ N 0 , 0 < δ < 1: (Cf. e.g. Courant and Hilbert [CH62] p. 349; V just enters as a zero-order term.) In contrast, for noninteger powers of −∆, and operators P satisfying Hypothesis 0.1 with a / ∈ N, the regularity may be considerably restricted in comparison with the case V = 0. This is linked to the fact that the multiplication by a nonzero function V does not fit into the symbol sequence p We first improve the regularity as far as we can by using the known regularity results for the Dirichlet problem (3.2) r + P u = g in Ω, supp u ⊂ Ω.
For a given f ∈ C ∞ (Ω), let u ∈Ḣ a (Ω) satisfy (3.1). Then u ∈ C a(3a) * (Ω). The conclusion also holds if merely f ∈ C a (Ω). In fact, the solutions with f ∈ C a (Ω) run through C a(3a) * (Ω), with γ a 0 u running through C 2a * (∂Ω). Proof. Let V ∈ C ∞ (Ω) be given, and let f ∈ C ∞ (Ω). By variational theory, the operator Let u ∈Ḣ a (Ω) be a solution of (3.1) with f ∈ C ∞ (Ω). Using the regularity theory for (3.2), we shall improve the knowledge of the regularity of u in a finite number of iterative steps, as in a related situation in [G15a], pf. of Th. 2.3: Recall the well-known general embedding properties for p, p 1 ∈ ]1, ∞[ : We make a finite number of iterative steps to reach the information u ∈ C 0 (Ω), as follows: Define p 0 , p 1 , p 2 , . . . , with p 0 = 2 and q j = n p j for all the relevant j, such that This means that q j = q 0 − ja; we stop the sequence at j 0 the first time we reach a q j 0 ≤ 0. As a first step, we note that u ∈Ḣ a (Ω) ⊂ e + L p 1 (Ω) implies that f − V u ∈ L p 1 (Ω), whence u ∈ H a(2a) p 1 (Ω) by [G15] Th. 4.4 applied to r + P u ∈ L p 1 (Ω). Then u ∈Ḣ a p 1 (Ω) in view of (1.12). In the next step we use the embeddingḢ a p 1 (Ω) ⊂ e + L p 2 (Ω) to conclude in a similar way that u ∈Ḣ a p 2 (Ω), and so on. If q j 0 < 0, we have that n (Ω) ⊂Ċ 0 (Ω). If q j 0 = 0, the corresponding p j 0 would be +∞, and we see at least that u ∈ e + L p (Ω) for any large p; then one step more gives that u ∈Ċ 0 (Ω). The rest of the argumentation relies on Hölder estimates, as [G14], Section 3. By the regularity results there, (Ω). Clearly, only the smoothness f ∈ C a (Ω) is needed for the whole argumentation.
Since u ∈Ċ ∞ (Ω) implies r + P u ∈ C ∞ (Ω), very high regularity of solutions to (3.1) is not completely excluded. But we shall now show that when V is nonvanishing on parts of the boundary, a higher regularity can only hold if γ a 0 u vanishes there. We show this for 0 < a < 1, leaving cases of higher a to the reader.
By comparison with the expansion (3.5) we can again first conclude that u(x ′ , 0) = 0, and next that z(x ′ , 0) = 0, so we find again that γ a 0 u must be 0. This shows 1 • . For 2 • , we just carry the above argumentation through in coordinate patches intersecting the boundary in open subsets Σ ′ of Σ with Σ ′ ⊂ Σ.
Recall from Remark 2.6 that for u ∈ C a(s) * (Ω), γ a 0 u can be regarded as the Neumann boundary value.
Remark 3.4. What is it that happens when s in the parameter 2a+s passes from a to a+δ, δ > 0? Recall that we are dealing with the operator r + P going from E s 1 = C a(2a+s) * (Ω) to E s 0 = C s * (Ω), for s ≥ 0. Here we always have that (with ε active if 2a + s ∈ N) But note also that (cf. (1.18)) and for δ > 0 there are elements of e + d a C ∞ (Ω) not lying in e + C a+δ (Ω). So for s = a + δ, E s 1 contains nontrivial elements ofĊ a (Ω) \ e + C a+δ (Ω). Briefly expressed, The inclusion E s 1 ⊂ E s 0 is necessary in order to define a resolvent acting in E s 0 ; this is not possible for s > a.

The regularity of solutions of fractional heat Dirichlet problems
Corollary 3.3 can now be applied in a discussion of the regularity of solutions of the fractional heat equation.
However, if for some δ > 0, u(x, t) ∈ W 1,1 (I; C a(3a+δ) * (Ω)) and f (x, t) ∈ L 1 (I; C a+δ (Ω)), then γ a 0 u = 0. Proof. First extend u and f by 0 for t < 0, and next extend the resulting functions across t = T by reflection in t. This results in functionsũ resp.f that satisfy the hypotheses of Theorem 4.1, so the conclusions from that theorem carry over.
It should be noted that the regularity theorem of Ros-Oton and Vivas [RV18] shows that for solutions of (4.5), (4.6) f is C a in x and C 1 2 in t =⇒ u/d a is C 2a in x and C 1 in t, if a = 1 2 (with a slightly weaker statement for a = 1 2 , cf. (0.7)); this is consistent with the first assertion in the corollary. But an extension of the upper indices from a to a + δ, resp. 2a to 2a + δ, may possibly need restrictive hypotheses as in the second assertion.
Remark 4.3. Under the hypothesis γ a 0 u = 0, the heat problem can be studied in smaller spaces for more regular f without contradiction, cf. Remark 3.5. Then it is no longer a Dirichlet problem, but a problem with more boundary conditions.