INTERPOLATION, EMBEDDINGS AND TRACES OF ANISOTROPIC FRACTIONAL SOBOLEV SPACES WITH TEMPORAL WEIGHTS

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1 INTERPOLATION, EMBEDDINGS AND TRACES OF ANISOTROPIC FRACTIONAL SOBOLEV SPACES WITH TEMPORAL WEIGHTS MARTIN MEYRIES AND ROLAND SCHNAUBELT Abstract. We investigate the roerties of a class of weighted vector-valued L -saces and the corresonding an)isotroic Sobolev-Slobodetskii saces. These saces arise naturally in the context of maximal L -regularity for arabolic initial-boundary value roblems. Our main tools are oerators with a bounded H -calculus, interolation theory, and oerator sums. 1. Introduction and reliminaries In this aer we investigate a class of anisotroic fractional Sobolev saces on sace-time with weights in the time variable. We treat in articular interolation results for such saces, Sobolev tye embeddings as well as temoral and satial trace theorems in a systematic, oerator theoretic way. Our choice of both the class of saces and the studied roerties is motivated by alications to quasilinear arabolic evolution equations with nonlinear boundary conditions which will be treated in subsequent aers focussing on the longterm behavior, cf. [24]. These aers will be based on linearization; i.e., one writes the nonlinear equation as a linear initial-boundary value roblem whose inhomogeneities contain the nonlinear terms. Since the underlying roblem is quasilinear, the linear and the nonlinear arts will be of the same order so that it is crucial to have shar results for the linear art giving otimal regularity. In the comanion aer [25] we establish the necessary theory for linear inhomogeneous initial-boundary value roblems, based on the resent study of the underlying function saces. The main focus of this work is the resence of the temoral weights. To exlain their role, we first recall some features of the known theory for the unweigthed case by means of an examle. For a domain Ω R n with smooth comact boundary Ω and given functions f, g and u, we consider the heat equation t ut, x) ut, x) = ft, x), x Ω, t >, ν ut, x) = gt, x), x Ω, t >, 1.1) u, x) = u x), x Ω, with an inhomogeneous Neumann boundary condition. We want to work in a framework where the boundary conditions can be understood in the sense of traces, and not just weakly. So we choose an L setting and require that the solution u belongs to the sace E 1 := W 1 J; L Ω) ) L J; W 2 Ω) ) 2 Mathematics Subject Classification. 46E35, 47A6. Key words and hrases. Anisotroic fractional Sobolev saces, olynomial weights, interolation, embeddings, traces, bounded H -calculus, oerator sums. This aer is art of a research roject suorted by the Deutsche Forschungsgemeinschaft DFG). 1

2 for some 1, ) and a finite interval J =, T ). To obtain the otimal saces for the data, one needs shar trace results for the sace E 1. It is known that the Neumann trace g of u E 1 belongs to the anisotroic Slobodetskii sace F 1 = W 1/2 1/2 J; L Ω) ) L J; W 1 1/ Ω) ) and that the trace u of u at time t = belongs to W 2 2/ Ω). Observe that F 1 still retains some time regularity! Moreover, one has f L J; L Ω) ) and the comatibility condition ν u = g t= should hold at time t = if g has a trace at t =, which haens for > 3. In fact, roblem 1.1) has a unique solution u E 1 if and only if these conditions hold. We refer to e.g. [6] for these and much more general results. Further, the trace sace W 2 2/ Ω) is the smallest sace in which all solutions of 1.1) in E 1 are continuous on [, T ], and the corresonding nonlinear roblems share this roerty. Corresondingly, the local) semiflows of solutions live in W 2 2/ Ω) and the norm of this sace is the right one to describe their roerties. For instance, it gives the blow-u condition, see e.g. [22]. One often takes a large to simlify the treatment of the nonlinearities say, > n + 2). Thus the norm of W 2 2/ Ω) is far away from the norms one can control by the usual a riori estimates, such as the norms of W2 1 or L. This unleasant regularity ga is even more significant in the Hölder setting. There is a known technique to reduce the necessary regularity for initial conditions of evolution equations. One studies the roblem in the weighted saces L R + ; E) := { u : R + E : t 1 µ u L R + ; E) }, where E is a Banach sace and µ 1/, 1]. The corresonding weighted Sobolev saces are defined similarly, and the fractional Sobolev saces W s R + ; E) and H s R + ; E) are introduced via real and comlex interolation, resectively. Observe that for µ = 1 we recover the unweigthed case and that the saces become larger if we decrease µ. So this tye of weights allows for functions being more singular at t =. We note that the weight t 1 µ) belongs to the class A used in harmonic analysis, see e.g. [3].) This aroach has been carried out in an L setting by Prüss & Simonett in [26] for roblems with homogeneous boundary conditions i.e., g = in 1.1)). In this case one can work entirely in the framework of semigrou theory. In fact, for the generator A of an exonentially stable semigrou one looks for a solution u in for the evolution equation E 1,µ := W 1 R + ; E) L R + ; DA)) u t) + Aut) = ft), t >, u) = u. 1.2) Clearly, it must hold that f L R + ; E). It is known that the initial value u has to belong to the real interolation sace E, DA)) µ 1/,, see e.g. Theorem in [31]. The main result in [26] now says that one has a unique solution u E 1,µ of 1.2) for all f L R + ; E) and u E, DA)) µ 1/, if and only if) A has maximal L -regularity; i.e., 1.2) with u = has a solution u E 1 for each f L R + ; E). The latter roerty is well understood, see [2] and the references therein. As a result, one can reduce the needed initial regularity for 1.2) almost to the base sace, say, E = L Ω). We further oint out that the temoral weights give the scale of hase saces E, DA)) µ 1/,, which are comactly embedded into each other in many cases. Precisely this oint was used in the very recent aer [17] to establish attractivity of equilibria of quasilinear evolution equations based on the results of [26], see also [18]. 2

3 Unfortunately, it seems to be imossible obtain shar Sobolev) regularity for inhomogeneous boundary value roblems such as 1.1) in a ure semigrou framework. Instead one has to restrict to a PDE setting and reduce equations like 1.1) to model roblems on fulland half-saces by means of local charts and the freezing of coefficients. Nevertheless we see below that semigrou and oerator theory can still lay a crucial role in the roofs. In this way, vector-valued arabolic initial-boundary value roblems with inhomogeneous boundary conditions were treated by Denk, Hieber, Prüss & Zacher in the aers [5], [6] and [7] without temoral weights. In the resent and the following aer [25] we extend their results to the case of the weights t 1 µ), focussing here on the function saces themselves. Our setting also covers linearizations of roblems arising from free boundary roblems, see e.g. [12] and [18]. In Section 2 we start with basic roerties of the weighted saces. The crucial oint is that the oerator t is maximally accretive and has a bounded H -calculus on L R + ; E) if E is of class HT ), see Lemma 2.6 and Theorem 2.7. Using a theorem by Yagi, one can conclude that the saces H s R + ; E) are the domains of fractional owers of 1 t, see Proosition 2.9. This fact also allows us to establish the natural interolation roerties for the scales of W s and H s in Lemma 2.8. These results are comlemented by several roositions on extension, density, intrinsic norms, Sobolev embeddings and a Poincaré inequality. The roofs rely on interolation theory, Hardy s inequality and results by Grisvard, [13]. In view of the above considerations it is clear that we have to study the maing roerties of temoral and satial derivatives and traces in anisotroic saces like E 1,µ. Here saces like W s+α R+ ; W r Ω; E) ) W s R+ ; W r+β Ω; E) ), s, r, α, β, naturally aear, as well as analogous Bessel otential saces H. Of course, one also needs Sobolev tye embeddings. By localization and local charts, we reduce these results to the model cases of full- and half-saces. There we use that the related H saces are the domains of the oerator L = 1 t ) α + 1 ) β/2 having bounded imaginary owers, see Lemma 3.1. Then Sobolevskii s mixed derivative theorem and interolation arguments imly the fundamental embedding result Proosition 3.2 which allows to interolate between time and sace regularity. We can conclude the desired maing roerties of satial derivatives in Lemma 3.4. Another crucial ingredient is Lemma 4.1 on time traces of semigrou orbits in weighted saces. Emloying these tools, we then establish our main Theorem 4.2 on the time traces of the anisotroic fractional saces. Similarly, we deduce the final Theorem 4.5 on satial traces. It turns out that our main results are natural extensions of the unweighted case µ = 1, and thus there is no disadvantage when working in a weighted framework in the context of arabolic roblems. The basic strategy of our roofs of the trace theorems is similar to that in e.g. Section 3 of [6] where the unweighted case was treated, see also [12] or [32]. The more recent aer [4] gives a comrehensive account of unweighted anisotroic saces. However, for the weighted case we first had to establish the underlying theory contained in Sections 2 and 3 and in Lemma 4.1. As a fundamental difference to the unweighted case, our weighted saces are not invariant under the right shift and have no suitable extension to the full time interval R. This behavior reflects the desired fact that our weights vanish at t = and allow for stronger singularities at t =. Moreover, some roofs in the literature cannot simly be generalized to the weighted case e.g., those of the temoral trace theorems in [12] or [32] which use fractional evolution equations). 3

4 We also oint out that the multilication oerator Φ µ u = t 1 µ u does not give an isomorhism from W 1 R + ; E) to W 1 R + ; E), since Φ µ 1) / L, 1). As a consequence, one cannot simly reduce the results to the unweighted case by isomorhy. For other tyes of weights this aroach would be ossible, for instance for exonential weights e ρt which do not change the behavior at t =. This was done in the recent aer [8] where interolation and trace theorems are shown for the exonentially weighted saces. Of course, this aer focusses on different results.) Isotroic weighted saces have been studied in detail in the literature, see e.g. [13], [19] and [31]. In this work we strive for a self contained and systematic treatment of the results needed for evolution equations. This should be of some interest also in the known unweighted case µ = 1. To kee the length within reasonable bounds, we omitted some arguments or calculations which are either routine or similar to arallel cases discussed here or in the literature. Most of the omitted details are resented in the Ph.D. thesis [24] to which we then refer. We now indicate some concets and results from oerator theory that we use frequently. Details can be found at the references given below. The real and comlex interolation functors, ) θ,q and [, ] θ, reectively, are a fundamental tool in our investigations. For the relevant theory the reader is referred to [23] and [31]. If A is a sectorial oerator on a Banach sace E, θ, 1) and q [1, ], we set D A θ, ) := E, DA) ). The theory of strongly continuous θ,q semigrous is resented in e.g. [11]. Throughout we assume that the Banach sace E is of class HT or, equivalently, is a UMD sace), which means that the Hilbert transform is bounded on L 2 R; E), see Sections III of [2]. We note that Hilbert saces are of class HT, as well as the reflexive Lebesgue and fractional) Sobolev saces. The definition and roerties of the bounded H -calculus of a sectorial oerator A are discussed in [5] or [2]. However, in the sequel we almost never use the calculus itself, but rather some theorems requiring it as an assumtion. An imortant consequence of the bounded H -calculus is that A admits bounded imaginary owers, see e.g. Sections 2.3 and 2.4 of [5] or Section III.4.7 of [2]. Finally, we recall the Dore-Venni theorem from [1] in a version of [27] and combined with the mixed derivative theorem due to Sobolevskii, [29]. Proosition 1.1. Let E be a Banach sace of class HT and suose that the oerators A, B on E are resolvent commuting and admit bounded imaginary owers with ower angles satisfying θ A + θ B < π. Assume further that A or B is invertible. Then A + B is invertible, admits bounded imaginary owers with angle not larger than θ A + θ B and the oerator A α B 1 α A + B) 1 is bounded on E for all α [, 1]. We write a b for some quantities a, b if there is a generic ositive constant C with a Cb. If X, Y are Banach saces, BX, Y ) is the sace of bounded linear oerators between them, and we ut BX) := BX, X). 2. Basic roerties of the weighted saces Throughout, we consider a time interval J = R + :=, ) or J =, T ) for some T > and a Banach sace E of class HT, and we let 1, ), µ 1/, 1]. 4

5 We study the Banach sace L J; E) := {u : J E : t 1 µ u L J; E)} with the norm ) 1/ u LJ;E) := t 1 µ u LJ;E) = t 1 µ) ut) E dt. Occasionally, we use these saces also with µ relaced by some α > 1/. It clearly holds L, T ; E) L, T ; E) and L, T ; E) L τ, T ; E) for all τ, T ), but L R + ; E) L R + ; E) for µ 1/, 1). For k N, we define the corresonding weighted Sobolev sace W k J; E) = H k J; E) := { u W k 1,loc J; E) : uj) L J; E), j {,..., k} } where W = H = L by definition), which is a Banach sace endowed with the norm k 1/ u W k J;E) = u H k J;E) := u j) L J;E)). For s R + \N, we write s = [s] + s with [s] N and s, 1). The weighted Slobodetskii and Bessel otential saces WJ; s E) := W J; [s] E), W [s]+1 J; E) ) s, Hs, J; E) := [ W J; [s] E), W [s]+1 J; E) ] s are introduced by means of real and comlex interolation, resectively. In view of Theorem 4.2/2 of [28] and Satz 3.21 of [33], this definition is consistent with the unweighted case; i.e., we have W s = W,1 s and Hs = H,1 s for all s. The general roerties of real and comlex interolation saces see [23] or [31]) imly that one has the scale of dense embeddings W s 1 d H s 2 d W s 3 J j= d H s 4, s 1 > s 2 > s 3 > s 4, 2.1) for all fixed 1, ) and µ 1/, 1]. In the sequel we will often use that B W k J; E) ) B W k+1 J; E) ) B W s J; E) ) B H s J; E) ), where k N and s k, k+1), to deduce assertions for exonents s from the integer case. Before giving further definitions, we state a first basic roerty which can easily be roved using Hölder s inequality. Lemma 2.1. Let J =, T ) be finite or infinite, 1, ) and µ 1/, 1]. We then have 1 L J; E) L q,loc J; E) for all 1 q < 1 µ + 1/. It thus holds W k J; E) W k 1,loc J; E) for all k N so that the trace u uj) ) is continuous from W k J; E) to E for all j {,..., k 1}. So we can define W k J; E) = H k J; E) := { u W k J; E) : u j) ) =, j {,..., k 1} } for k N which are Banach saces with the norm of W k. For convenience we further set W J; E) = H J; E) := L J; E). For a number s = [s] + s R + \N, we again define the corresonding fractional order saces by interolation; i.e., we ut W s J; E) := W [s] J; E), W [s]+1 J; E) ) s,, 5

6 H s J; E) := [ W [s] J; E), W [s]+1 J; E) ] s, obtaining as before a scale of function saces We further have W s 1 d H s 2 d W s 3 d H s 4, s 1 > s 2 > s 3 > s ) W s J; E) W s J; E) and H s J; E) H s J; E) for all s. Hardy s inequality is a fundamental tool for the investigation of the L -saces. It states that t t α 1 ϕτ) dτ) dt α 1/) t 1 α ϕt)) dt 2.3) for all α 1/, ) and measurable nonnegative functions ϕ : R + R, see Theorem 33 in [15]. For the saces with vanishing initial values we deduce the following weighted inequality. Lemma 2.2. Let J =, T ) be finite or infinite, 1, ), µ 1/, 1] and s. Then t 1 µ s) ut) E dt C,s u W s J;E) if u WJ; s E), J and this remains true if one relaces W s J; E) by H s J; E). Proof. The case s = is trivial and the case s = 1 is a consequence of Hardy s inequality with ϕ = u E. The assertion for s = k N follows by induction, so that we have W k J; E) L J, t 1 µ k) dt; E). 2.4) Theorem of [31] yields that L J, t 1 µ k) dt; E), L J, t 1 µ k+1)) dt; E) ) θ, = L J, t 1 µ θk) dt; E) 2.5) for all θ, 1) and that 2.5) remains true if one relaces, ) θ, by [, ] θ. Interolation of the embedding 2.4) now imlies the assertions for all s. The above inequalities can be used to show that the multilication oerator Φ µ u = t 1 µ u is an isomorhism from the weighted to the unweighted saces, rovided one restricts to vanishing initial values. Without this restriction, Φ µ does not ma roerly since Φ µ 1) / L. Lemma 2.3. Let J =, T ) be finite or infinite, 1, ), µ 1/, 1] and s. Then the ma Φ µ, given by Φ µ u)t) := t 1 µ ut), induces an isomorhism from WJ; s E) to W s J; E) and from HJ; s E) to HJ; s E). The inverse Φ 1 µ is given by Φ 1 µ u)t) = t 1 µ) ut). The lemma is shown for s {, 1} in Proosition 2.2 of [26]. The case of other integers can be treated similarly, see Lemma of [24]. The general case then follows by interolation. We use the above lemma to rove basic density results for the weighted saces. Lemma 2.4. Let J =, T ) be finite or infinite, 1, ), µ 1/, 1] and s. Then the sace C c J\{}; E) is dense in W s J; E) and H s J; E), and C c J; E) is dense in W s J; E) and H s J; E). 6

7 Proof. Due to 2.1) and 2.2) we only have to consider the case s = k N. The receding lemma shows that Φ µ u belongs to W k J; E) for every u W k J; E). As in Theorem of [31] for the scalar-valued case, one sees that C c J\{}; E) is dense in W k J; E) for k N. For any given ε > there thus exists a function ψ C c J\{}; E) with Φ µ u ψ W k J;E) < ε. Consequently, u Φ 1 µ ψ W k J;E) ε, and so Cc J\{}; E) is dense in WJ; k E). For u WJ; k E), we choose ψ 1 Cc J; E) with ψ j) 1 ) = uj) ) for j {,..., k 1}. Since u ψ 1 WJ; k E), by the above considerations there is ψ 2 Cc J \ {}; E) being ε-close to u ψ 1 in WJ; k E) Hence, ψ 1 + ψ 2 Cc J; E) is ε-close to u in WJ; k E). We next construct extension oerators to R + for the weighted saces on finite intervals; i.e., continuous right-inverses of the restriction oerator. For vanishing initial values one can achieve an extension whose norm is bounded indeendently of the length of the underlying interval. This fact is crucial for the treatment of nonlinear evolution equations. For simlicity we consider only s [, 2] in this case. Lemma 2.5. Let J =, T ) be finite and < 1/ < µ 1. Given k N, there is an extension oerator E J from J to R + with E J B WJ; s E), WR s + ; E) ) B HJ; s E), HR s + ; E) ) for all s [, k]. Here we can relace W by W and H by H. There is further an extension oerator E J from J to R + with E J B W s J; E), W s R + ; E) ) B H s J; E), H s R + ; E) ) for all s [, 2], whose oerator norm is indeendent of T. Moreover, E J, E J B L J; E), L R + ; E) ) with oerator norms indeendent of T. Proof. Let k N. We extend u W, k T ; X) to Eu W, k T k ; X) with T k = T + T/2k + 2) in the same way as in, e.g., Theorem 5.19 of [1]. In articular, Eut) only deends on u on T/2, T ) if t T. Take a function ϕ C R + ) which is equal to 1 on, T ) and has its suort in, T k ), and set E J := ϕe with trivial extension for t T k ). It is easy to check that E J is bounded from W, k T ; X) to WR k + ; E), and the general case follows from interolation. For k {, 1, 2} and a function u WJ; k E) we define EJ by the formula { ut), t, T ), E Ju)t) := 3ψ 1 µ u)2t t)1 [T,2T ] t) 2ψ 1 µ u)3t 2t)1 [T, 3 T ]t), t T, 2 2T τ τ 2 where ψτ) = and 1 T 2 I is the characteristic function of an interval I. Using Lemma 2.2, it can be checked by a direct calculation that EJ is bounded on WJ; k E) for all k {, 1, 2}, see Lemma in [24]. Interolation yields the general case. The exlicit reresentations of E J and EJ show that these oerators admit an L -estimate indeendent of T. We now investigate the realization of the derivative t and its fractional owers on the weighted saces. The roerties of these and similar oerators are fundamental for all our further considerations. We start with a generation result. Lemma 2.6. For 1, ), µ 1/, 1] and s, the family of left translations {Λ E t } t, given by Λ E t u)τ) := uτ + t), τ, 7

8 forms a strongly continuous contraction semigrou on WR s + ; E) and on HR s + ; E). Its generator is the derivative t with domain W s+1 R + ; E) and H s+1 R + ; E), resectively. Proof. I) We write Λ t = Λ E t for simlicity. For each t the oerator Λ t mas L R + ; E) into itself and is contractive because of Λ t u L R + ;E) τ + t ) 1 µ) uτ + t ) E dτ u L R + ;E). Analoguously one shows that Λ t is a contractive ma on WR k + ; E) for all k N. By interolation, this fact carries over to WR s + ; E) and HR s + ; E) for all s. It is further clear that {Λ t } t forms a semigrou of oerators on these saces. Lemma 2.4 says that D = Cc [, ); E) is dense in all the saces above. The left translations act strongly continuous on D with the su-norm, and they thus form C -semigrous on WR s + ; E) and HR s + ; E). II) We denote the generator of {Λ t } on WR s + ; E) by A. We first let s = k N. To show A t, we take u DA) WR k + ; E). Then u j) L 1,loc [, ); E) for all j {,, k} by Lemma 2.1, and for a, b R + with a < b it holds b a 1 h u j) τ + h) u k) τ) ) dτ = 1 h b+h b u j) τ) dτ 1 h a+h a u j) τ) dτ. As h, the right hand side converges to u j) b) u j) a) for almost all a, b R +. The integrand on the left hand side tends to Au j) in L R + ; E), and thus in L 1 a, b; E). Hence, the left hand side converges to b a Auj) τ) dτ. We infer that u W k+1 1,loc R +; E), with u j+1) = Au j), so that DA) W k+1 R + ; E) and t DA) = A. Since A generates a contraction semigrou, we know that 1 A is invertible. It is further easy to see that 1 t is injective on W k+1 R + ; E). Consequently, 1 t = 1 A, which yields t = A. This fact imlies that 1 t is an isomorhism from W k+1 R + ; E) to WR k + ; E). By interolation, we can relace here k N by any s, and W by H. Finally, let s k, k+1). The first art and Proosition II.2.3 of [11] imly that DA) = {u W k+1 R + ; E) : u WR s + ; E)} and Au = u. Using that 1 t is an isomorhism from W s+1 R + ; E) to WR s + ; E), we conclude that DA) = W s+1 R + ; E). The same arguments work with W relaced by H. Since W 1 R + ; E) is the domain of the generator of {Λ E t } t by the above lemma, standard interolation theory yields u W s R +;E) u L R + ;E) + t s 1 Λ E t u u L R + ;E) dt for s, 1), see Proosition 5.7 of [23]. Substitution and localization then imly that u W s J;E) u LJ;E) + [u] W s J;E) [u] W s J;E) := T t with τ 1 µ) ut) uτ) E t τ) 1+s dτ dt 2.6) for finite or infinite J =, T ), s, 1) and u W s J; E), see Proosition of [24]. Using the generation roerty of t and the transference rincile, we show that t with domain W 1 R + ; E) ossesses a bounded H -calculus on L R + ; E). Here it is for the first time essential that E is of class HT. As shown in [26], the realization of t with domain 8

9 W 1 R + ; E) also admits a bounded H -calculus on L R + ; E) and it is in articular sectorial), although t with domain W 1 R + ; E) does not generate a semigrou on L R + ; E). Theorem 2.7. Let < 1/ < µ 1. Then the oerators t with domain W 1 R + ; E) and t with domain W 1 R + ; E) ossess a bounded H -calculus on L R + ; E) with H -angle π/2. Proof. The assertion on t is roved in Theorem 4.5 of [26]. To treat the oerator t, we emloy the vector-valued transference rincile from [16]. We first introduce vector-valued extensions of oerators. Let Ω, ν) be a measure sace and S be a bounded, ositive oerator on L Ω, ν). For simle functions u of the form u = N i=1 u ix i with x i E and simle u i : Ω C, we define S E u ) := N i=1 Su i) )x i. The oerator S E extends uniquely to L Ω, ν; E) with S E BLΩ,ν;E)) = S BLΩ,ν)), see Lemma 1.14 of [2]. We consider the left translation Λ E t, t, on L R + ; E) = L R +, t 1 µ) dt; E). Clearly, Λ E t = Λ C t ) E is the vector-valued extension of the scalar left translation Λ C t. Due to Lemma 2.6, the family {Λ C t } t forms a C -semigrou of ositive contractions on L R + ), and t with domain WR 1 + ; E) generates the vector-valued extension {Λ E t } t on L R + ; E). Moreover, t is injective on this sace. Theorem 6 of [16] now yields that t admits a bounded H - calculus with angle π/2. Theorem 2.7 imlies that 1+ t : WR 1 + ; E) L R + ; E) is isomorhic. The restriction of this oerator to finite J is still injective, and its bounded inverse is given by the restriction of the inverse for J = R +. By induction and interolation we deduce that 1+ t : W s+1 J; E) WJ; s E) is isomorhic for all s and J. For s = [s] + s with [s] N and s [, 1), we thus obtain the imortant and exected) equality WJ; s E) = { u W J; [s] E) : u [s]) WJ; s E) }, 2.7) where the natural norms are equivalent with constants indeendent of J. In Lemma 2.6 we have seen that 1 t is an isomorhism from W s+1 R + ; E) to WR s + ; E) for all s. For J = R +, it follows that W s J; E) = { u W [s] J; E) : u [s]) W s J; E) }. 2.8) By means of extension and restriction, one can extend this identity to finite J. Observe that the formula for E J given in the roof of Lemma 2.5 imly that if u is contained the sace on right hand side, then its extension belongs to the analogous sace with J = R +.) These characterizations remain valid if one relaces the W -saces by the H-saces. We next rove general interolation roerties of the weighted saces. Lemma 2.8. Let J =, T ) be finite or infinite, 1, ), µ 1/, 1], s 1 < s 2 and θ, 1). Set s = 1 θ)s 1 + θs 2. Then it holds [ H s 1 J; E), H s 2 J; E) ] θ = Hs J; E). 2.9) If also s / N, we have H s 1 J; E), H s 2 J; E) ) θ, = W J; s E). 2.1) Moreover, if s 1, s 2, s / N then [ W s 1 J; E), W s 2 J; E) ] θ = W J; s E), W s 1 J; E), W s 2 J; E) ) θ, = W s J; E). 2.11) 9

10 If F d E is a Banach sace of class HT, then it holds for τ that H τ J; E), HJ; τ F ) ) θ, = ) Hτ J; E, F )θ,, 2.12) [ H τ J; E), HJ; τ F ) ] θ = ) Hτ J; [E, F ]θ. 2.13) All these assertions remain true if one relaces the W - and H-saces by W - and H-saces, resectively, where the constants of the norm equivalences do not deend on T if s 2 2. Proof. I) First, let J = R +. Throughout, we consider the oerator A = 1 t on L R + ; Z) with DA) = W 1 R + ; Z) for any Banach sace Z of class HT, and we ut X := L R + ; E). By Theorem 2.7, the oerator A ossesses a bounded H -calculus on X so that A admits bounded imaginary owers see 2.15) of [5]). A theorem by Yagi now imlies that DA α ) = [X, DA)] α = H α R + ; E) for all α, 1), see Theorem of [31]. For α 1, formula 2.8) further yields DA α ) = {u DA [α] ) : A [α] u DA α [α] )} = {u H [α] R + ; E) : u [α]) H α [α] R + ; E)} = H α R + ; E). 2.14) Emloying this equation and again Yagi s theorem, we obtain 2.9) by comuting [ H s 1 R + ; E), H s 2 R + ; E) ] θ = [ DA s 1 ), DA s 2 ) ] θ = DAs ) = HR s + ; E). II) We continue with 2.1). The oerator A s 1 induces an isomorhism H s 1 J; E), H s 2 J; E) ) θ, = DA s 1 ), DA s 2 ) ) θ, X, DA τ ) ) θ,, where τ = s 2 s 1. We first assume that θτ / N, and ut k = [θτ] N. It follows from reiteration see e.g. Theorem of [31]) that X, DA τ ) ) θ, = DA k ), DA τ ) ) σ,, with σ = τθ k τ k, 1), and that the oerator Ak induces an isomorhism DA k ), DA τ ) ) σ, X, DA τ k ) ) σ, = X, DA) ) στ k), = W στ k) R + ; E), where στ k) = τθ k, 1). On the other hand, A s 1+k) induces an isomorhism W τθ k R + ; E) W s 2 s 1 )θ+s 1 R + ; E) = WR s + ; E). Hence, 2.1) holds if θs 2 s 1 ) / N. III) Next we derive 2.11). We take an integer k > s 2. We can write W s j R + ; E) = X, H k R + ; E) ) s j /k, thanks to art II) and k s j/k = s j / N. Now reiteration yields W s 1 J; E), W s 2 J; E) ) θ, = X, H k R + ; E) ) s/k, = W s R + ; E), using again art II) and that k s/k / N. The other equation in 2.11) is shown similarly, emloying Remark of [31]. The remaining case θs 2 s 1 ) N in Ste II can now be treated by an reiteration argument with exonents θ ± ε. d E be a Banach sace of class HT and τ. Due to 2.14), the oerator IV) Now let F A τ is an isomorhism H τ R + ; E), HR τ + ; F ) ) θ, L R + ; E), L R + ; F ) ) θ,. 1

11 ) Theorem of [31] says that the latter sace equals L R+ ; E, F ) θ,. Since A τ mas this sace isomorhically to H τ ) R+ ; E, F ) θ,, we have shown 2.12). The formula 2.13) is shown in the same way. V) Relacing the oerator A = 1 t by A := 1 + t, the same arguments as above show the asserted equalities for the W - and the H-saces. This finishes the case J = R +. The case of a finite interval is then deduced from the half-line case, using the extension oerators E J and EJ from Lemma 2.5 and Theorem of [31] see also Section I.2.3 of [2]). The deendence of the norm equivalence constants on the length of J carries over from the roerties of the extension oerators. We now extend Theorem 2.7 to fractional owers of the derivative acting on the W - and H-scales. To this aim, we note that if a sectorial oerator A has a bounded H -calculus with H -angle φ [, π) and if α, π/φ), then A α has the same roerty with angle less or equal αφ. This fact easily follows from the roerties of the calculus, see Lemma A.3.5 in [24]. Proosition 2.9. Let < 1/ < µ 1, s, α, 2) and ω >. Then the oerators ω t ) α on H s R + ; E) with domain H s+α R + ; E), ω t ) α on W s R + ; E) with domain W s+α R + ; E), s, s + α / N, ω + t ) α on H s R + ; E) with domain H s+α R + ; E), ω + t ) α on W s R + ; E) with domain W s+α R + ; E), s, s + α / N, are invertible and ossess a bounded H -calculus with H -angle less or equal απ/2. Proof. We first consider the case s =. Proosition 2.11 of [5] and Theorem 2.7 imly that the realization of ω t on L R + ; E) with domain WR 1 + ; E) admits a bounded H - calculus with H -angle equal to π/2. As noted above, also ω t ) α ossesses a bounded H -calculus with H -angle less or equal απ/2, because of α, 2). As in 2.14) we see that then Dω t ) α ) = HR α + ; E) holds for all α. Since ω t is invertible and H τ+s R + ; E) = Dω A) τ+s ) holds for all s, τ, the oerator ω t ) s induces an isomorhism from H τ+s R + ; E) to HR τ + ; E). Using that ω t ) s and ω t ) α commute on HR α + ; E), we derive from Proosition 2.11 of [5] that ω t ) α has a bounded H -calculus on HR s + ; E) with angle not larger than απ/2, and that its domain equals H s+α R + ; E). Let s, s + α / N. Interolation of the H-case and Lemma 2.8 show that ω t ) α with R + ; E) has a bounded H -calculus on W s R + ; E) with H -angle less or domain W s+α equal απ/2. The oerator ω + t can be treated in the same way. The W s -saces can be characterized in terms of the kernel of the temoral trace, as the next roosition shows. These results are mainly due to Grisvard [13]. Observe that the limit number for the existence of a trace is s = 1 µ + 1/. Therefore, if µ runs through the interval 1/, 1] this limit number runs through the interval [1/, 1). Of course, for µ = 1 the limit number s = 1/ for the unweighted case is recovered. We do not treat the limit cases s = k + 1 µ + 1/ with k N, see Remarque 4.2 of [13] and Remark of [31] for a discussion. We also do not consider the corresonding characterizations of the H s -saces, which are not needed below. They should be correct, but it seems that their roofs require a much greater effort. 11

12 Proosition 2.1. Let J =, T ) be finite or infinite, 1, ) and µ 1/, 1]. Then for < s < 1 µ + 1/ it holds C c J\{}; E) d W s J; E) and W s J; E) = W s J; E). 2.15) For k + 1 µ + 1/ < s < k µ + 1/) with k N we have where here one may relace W s by H s, and moreover W s J; E) BUC k J; E), 2.16) W s J; E) = { u W s J; E) : u j) ) =, j {,..., k} }. 2.17) If in addition s [, 2], then the embedding constants for do not deend on J. W s J; E) BUC k J; E) and H s J; E) BUC k J; E) Proof. In [13] much of the roosition is roved for the W -saces in the scalar-valued case E = C with J = R +. An insection of the roofs given there shows that they only make use of basic facts, interolation theory and the scalar versions of Lemmas 2.4 and 2.6 above. Thus the results of [13] carry over to the case of a general Banach sace E. Moreover, the case of a finite interval is obtained from the half-line case by extension and restriction. One can relace W by H as asserted because of 2.1) and 2.2). Assertion 2.15) is shown in Théorème 2.1 and Théorème 4.1 of [13]. The embedding 2.16) is roved for k = in Théorème 5.2 of [13], and the general case k N is an immediate consequence in view of 2.8). For s 1, identity 2.17) is shown in Proosition 1.2 and Théorème 4.1 of [13]. For integers s = k + 1 this equality holds by definition. The general case then follows from 2.7), 2.15) and 2.17) for s [s], as well as 2.8). We next establish embeddings of Sobolev tye for W s and H s. Proosition Let J =, T ) be finite, 1 < < q <, µ > 1/ and s > τ. Then WJ; s E) Wq,µJ; τ 1 µ + 1/) E) holds if s 1 µ + 1/) > τ, 2.18) q WJ; s E) Wq τ J; E) holds if s 1 µ + 1/) > τ 1/q. 2.19) These embeddings remain true if one relaces the W -saces by the H-, the W - and the H-saces, resectively. In the two latter cases, restricting to s [, 2], for given T > the embeddings hold with a uniform constant for all < T T. Proof. Throughout this roof, let T > be given. Since the inequality signs in 2.18) and 2.19) are strict, we may assume that s / N. Again we only have to consider the W -case due to 2.1) and 2.2). I) We rove 2.18) for τ =. For s > 1 µ + 1/, Proosition 2.1 yields W s J; E) L J; E) L q,µ J; E) for all q, ), with the asserted behaviour of the embedding constant in the W -case. If s 1 µ+1/, take q, ) as in 2.18). Choose η 1 µ + 1/, 1) and r, ) such that 1 q 1 s/η + s/η r. 12

13 We have already shown that W η J; E) L r,µ J; E). Lemma 2.8 and Theorem of [31] then imly W s J; E) = L J; E), W η J; E) ) s/η, L J; E), L r,µ J; E) ) s/η, L q,µj; E). In the W -case, all embedding constants are uniform in T T for s [, 2]. II) To rove 2.18) for τ >, we set α = 1 /q)1 µ+1/) > for an exonent q > as in 2.18). We fix some ε > and k N such that s ε > τ +α >, k +α < s ε < k +1+α, and κ := k + α + ε is not an integer. Lemma 2.8 now yields Using 2.8) and 2.18) with τ =, we obtain W s J; E) = W κ J; E), W κ+1 J; E) ) s κ,, W κ J; E) W k q,µj; E) and W κ+1 J; E) W k+1 q,µ J; E). Since, ) θ,, ) θ,q for θ, 1), it follows WJ; s E) Wq,µJ; k E), Wq,µ k+1 J; E) ) s κ,q = W q,µ k+s κ J; E) Wq,µJ; τ E). III) For W -saces, the deendence on T for s [, 2] carries over from Lemma 2.8 and 2.18) with τ =. The embedding 2.19) is shown similarly emloying Lemma 2.1, see Proosition in [24]. We finish this section with Poincaré s inequality in the weighted saces. Lemma Let J =, T ) be finite, 1, ), µ 1/, 1] and s [, 1). Then it holds u LJ;E) T u LJ;E), for all u W 1 J; E). u W s J;E) + u H s J;E) T 1 s u W 1 J;E), Proof. Let u WJ; 1 E). Using Hölder s inequality, we estimate T ) t 1 µ) ut) E t1 µ) s 1 µ) s 1 µ u s) E ds t 1 µ) T 1 1 µ))/ u for t J. The first assertion now follows by integration over J. For s [, 1) we have u W s J;E) + u H s J;E) u s W 1 J;E) u 1 s L J;E) by interolation, imlying the second asserted inequality. L J;E) 3. Weighted anisotroic saces Let again E be a Banach sace of class HT, let J =, T ) be finite or infinite, and let further Ω R n be a domain with comact smooth boundary Ω, or Ω {R n, R n +}. For, q 1, ) and r > we denote by H r Ω; E), W r Ω; E), B r,qω; E), the E-valued Sobolev, Slobodetskii and Besov saces, resectively. If Ω = R n we also use this saces with r. For the definitions and roerties of these saces we refer to [3], [28] or [33]. The scalar-valued case is treated extensively in [31]. In articular, we have B,Ω; r E) = W r Ω; E) for all [1, ), r / N. 13

14 The corresonding saces over the boundary Ω of Ω are defined via local charts as in Definition of [31], for instance. We mainly investigate weighted anisotroic saces, i.e., intersections of saces of the form H s J; H r Ω; E) ), W s J; W r Ω; E) ), H s J; W r Ω; E) ), W s J; W r Ω; E) ), 3.1) where s, r. In what follows we refer to t J as time and to x Ω as sace variables. We start with two imortant tools. First, given k N, there is an extension oerator E Ω to R n for functions defined on Ω i.e., we have E Ω u) Ω = u) satisfying E Ω B B r,qω; E), B r,qr n ; E) ) B H r Ω; E), H r R n ; E) ) 3.2) for all, q 1, ) and r [, k]. For integer r [, k], the constructions in the roofs of Theorems 5.21 and 5.22 [1] for the scalar-valued saces literally carries over to the vectorvalued case. The general case r [, k] can be treated via interolation. Alying E Ω ointwise almost everywhere in time and using again interolation, we obtain a satial extension oerator for the anisotroic saces, again denoted by E Ω : E Ω B H s J; H r Ω; E)), H s J; H r R n ; E)) ), s, r [, k]. 3.3) Of course, here a H-sace may be relaced by a W -sace at the first or the second or at both ositions, and this remains true for the H s - and the W s -saces with resect to time. Second, we consider oerators build from the time derivative and the Lalacian whose domains are given by intersections of the saces in 3.1) with Ω = R n and J = R +. This fact will allow us to study the anisotroic saces by means of these oerators. Lemma 3.1. Let E be a Banach sace of class HT, let 1, ), µ 1/, 1], s, r, α, 2) and β >, let ω, ω satisfy ω + ω and set H s H r ) := H s R+ ; H r R n ; E) ), H s H r ) := H s R+ ; H r R n ; E) ), and analogously for the other tyes of saces in 3.1). Let n be the Lalacian on R n. Then the following holds true. a) The ointwise realization of ω n ) β/2 on any of the saces H s H r ), with domain H s H r+β ), H s W r ), with domain H s W r+β ), r, r + β / N, W s H r ), with domain W s H r+β ), W s W r ), with domain W s W r+β ), r, r + β / N, admits a bounded H -calculus with H -angle equal to zero, and it is invertible if ω >. Here one can relace the H s, W s -saces by the H s, W s -saces, resectively. b) The oerator L := ω t ) α + ω n ) β/2 acting on any of the saces H s H r ), with domain H s+α H β ) H s H r+β ), H s W r ), with domain H s+α W β ) H s W r+β ), r, r + β / N, WH s ), r with domain W s+α H β ) WH s r+β ), s, s + α / N, WW s r ), with domain W s+α W β ) WW s r+β ), s, s + α, r, r + β / N, 14

15 is invertible and admits bounded imaginary owers with ower angle not larger than απ/2. This remains true for the oerator L := ω + t ) α +ω n ) β/2 if one relaces the H s, W s -saces by the H s, W s -saces, resectively. c) For τ, 1] it holds DL τ ) = Dω t ) ατ ) Dω n ) βτ/2 ), D L τ, ) = D ω t) ατ, ) D ω n) β/2τ, ), and this remains true if one relaces L by L and ω t by ω + t. Proof. Since E is of class HT, the oerator n with domain HR 2 n ; E) ossesses a bounded H -calculus on L R n ; E) with H -angle equal to zero, see Theorem 5.5 of [5]. As in Proosition 2.9, the fractional ower ω n ) β/2 with domain H β R n ; E) has the same roerty. If ω >, it is invertible in L R n ; E). Let r. Using ω n ) r/2 as an isomorhism between HR r n ; E) and L R n ; E), it then follows from Proosition 2.11 of [5] that ω n ) β/2 with domain H r+β R n ; E) ossesses a bounded H -calculus on HR r n ; E) with H -angle equal to zero. By interolation, this fact remains true if one considers ω n ) β/2 with domain W r+β R n ; E) in W r R n ; E), rovided r, r + β / N. It is straightforward to see that these roerties carry over to the ointwise realizations on the anisotroic saces, cf. Lemma A.3.6 in [24]. Since all the saces under consideration are of class HT, Proosition 2.9 imlies that ω t ) α with domain H s+α H) r admits a bounded H -calculus on HH s ) r with H -angle less or equal απ/2, and on the corresonding saces where H is relaced by W, with the asserted excetions. Using these facts and that the resolvents of ω t ) α and ω n ) β/2 commute, the assertions for L are a consequence of Proosition 1.1. The same arguments show the assertions for L. Finally, c) is a consequence of Lemma 9.5 in [12] and Teorema 5 in [14] see also Lemma 3.1 in [6]). With the hel of the oerators from Lemma 3.1 we establish fundamental embeddings for the anisotroic saces with regularity exonents s, r) for time and sace. Roughly seaking, we rove that the regularities s+α, r) and s, r +β) imly all regularities on the line segment from s + α, r) to s, r + β). Proosition 3.2. Let J =, T ) be finite or infinite, < 1/ < µ 1, and let Ω R n be a domain with comact smooth boundary Ω, or Ω {R n, R n +}. Let further s, r, α, 2), β >, σ [, 1], and set HH s ) r := H s J; H r Ω; E) ), and analogously for the other anisotroic saces. Then it holds H s+α H) r HH s r+β ) H s+σα H r+1 σ)β ), 3.4) and moreover each of the saces H s+α W r ) HW s r+β ), W s+α H) r WH s r+β ), W s+α H) r HW s r+β ), is continuously embedded into W s+σα H r+1 σ)β ) H s+σα 15 W r+1 σ)β ),

16 rovided all the occurring W - and W -saces have a noninteger order of differentiability. Finally, assuming all orders of differentiability to be noninteger, we have W s+α W r ) W s W r+β ) W s+σα W r+1 σ)β ). 3.5) These embeddings remain true if one relaces Ω by its boundary Ω. They are also valid if one relaces all H -, W - saces by H -, W -saces, resectively. Restricting in the latter case to s + α 2, the embedding constants do not deend on the length of J. Proof. Using extensions and restrictions, and emloying that the saces over Ω are defined via local charts, it suffices to consider the case J Ω = R + R n. The deendence of the embedding constants on J carries over from the roerties of the extension oerators. I) For 3.4) we consider the oerators 1 t ) α and 1 n ) β/2 on H s H r ), which were treated in Proosition 2.9 and Lemma 3.1. Note that we have to restrict to α, 2) to obtain sectoriality of 1 t ) α. Due to the invertibility of these oerators and the descrition of their domains, the exression 1 t ) ασ 1 n ) β1 σ)/2 H s H r ) is an equivalent norm on H s+σα H r+1 σ)β ) for all σ, 1). Since the sum L of these oerators is invertible by Lemma 3.1, we obtain the equivalent norm 1 t ) α + 1 n ) β/2 ) H s H r ) on DL) = H s+α H) r HH s r+β ). Hence 3.4) follows directly from Proosition 1.1. The same arguments show that the embeddings H s+α W r ) HW s r+β ) H s+σα W r+1 σ)β ), W s+α H r ) W s H r+β ) W s+σα H r+1 σ)β ), and 3.5) hold, with the asserted excetions. II) We derive the remaining embeddings from 3.4) by suitable interolation arguments, which were indicated in Remark 5.3 of [12] in a more secific situation. We concentrate on ), the other assertions are obtained similarly, see Proosition of [24]. For s + α, r + β / N and sufficiently small ε >, we aly the real interolation functor, ) 1/2, to the embeddings the case W s+α H r ) H s W r+β H s+α1±ε/β) H s+α1±ε/β) H) r HH s r+β±ε ) H s+σα H) r HH s r+β±ε ) H s+ασ±ε/β) H r+1 σ)β±ε ), H r+1 σ)β ). By Lemma 2.8 the terms in the first and in the second line on the right-hand side interolate to H s+σα W r+1 σ)β ) and W s+σα H r+1 σ)β ), resectively. To interolate the left-hand side, we consider the oerator L = 1 t ) α1+ε/β) + 1 ) β+ε)/2, on HH s ) r with domain DL) = H s+α1+ε/β) H) r HH s r+β+ε ). Lemma 3.1 and Proosition 2.9 imly DL β ε)/β+ε) ) = H s+α1 ε/β) H) r HH s r+β ε ), 16

17 and reiteration yields DL β ε)/β+ε) ), DL) ) 1/2, = D ) L 1 + β ε)/β + ε))/2,. Using again Lemma 3.1 and Proosition 2.9, we see that the latter sace equals W s+α H) r HW s r+β ), as required. III) Starting in Ste I with 1 + t ) α instead of 1 t ) α, the same arguments as above show that the asserted embeddings are also true for the H - and W -saces. Remark 3.3. The roof shows that in the embeddings where only Proosition 1.1 was used, the orders of integrability in sace and time do not have to coincide. In fact, the assertions of Lemma 3.1 remain true for the extension of the Lalacian on Hq r R n ; E) to H s J; H r q R n ; E) ), where, q 1, ) and µ 1/, 1]. As in Ste I of the roof above, we then derive H s+α Hq r ) HH s q r+β ) H s+σα Hq r+1 σ)β ). A tyical alication of Proosition 3.2 is the following result on the maing behavior of the satial derivative on anisotroic H-saces. Lemma 3.4. Let E be a Banach sace of class HT, let J =, T ) be finite or finite, and let Ω R n be a domain with comact smooth boundary, or Ω {R n +, R n }. Let further s, r [, 1), α, 2), β 1, < 1/ < µ 1. Then the ointwise realization of xi, i {1,..., n}, is a continuous ma H s+α H) r HH s r+β ) H s+α α/β H) r HH s r+β 1 ). Its oerator norm is indeendent of T if we restrict to s + α 2 and to H -saces. Proof. By extension and restriction it suffices to consider the case J Ω = R + R n. Clearly the oerator xi mas continuously H s+α H) H r H s r+β ) HH s r+β 1 ). Proosition 3.2 further yields the embedding Hence, xi H s+α H) r HH s r+β ) H s+α α/β H r+1 ). is also a continuous ma from H s+α H) r HH s r+β ) to H s+α α/β H). r 4. The temoral and the satial trace We first consider the temoral trace on anisotroic saces. Using integration by arts, one can see that the reresentation σ σ t ) u) = 2 µ) σ 2 µ) τ 1 µ uτ) dτ 2 µ) t 3 µ) τ 1 µ ut) uτ)) dτ dt 4.1) holds true for all σ >, Banach saces X and u W1,loc 1 [, ); X). Let A generate an exonentially stable analytic C -semigrou on X and θ, 1). Then the norm of the sace D A θ, ) is equivalent to the norm given by x D A θ,), = σ 1 θ) Ae σa x X dσ σ, 4.2) see Theorem of [31]. The formula 4.1) is the key to the following abstract trace theorem. 17

18 Lemma 4.1. Let X be a Banach sace, 1, ), µ 1/, 1], and let the oerator A on X with domain DA) be invertible and admit bounded imaginary owers with ower angle strictly smaller than π/2. Let s, 1 µ + 1/) and α > satisfy s + α 1 µ + 1/, 1). Then the temoral trace tr, i.e., tr u = u t=, mas continuously W s+α R+ ; DA s ) ) W s R+ ; DA s+α ) ) ) D A 2s + α 1 µ + 1/),. 4.3) Moreover, tr is for α 1 µ + 1/, 1] continuous W α R+ ; X ) L R+ ; D A α, ) ) D A α 1 µ + 1/), ), 4.4) and for s, 1 µ + 1/) it is continuous W 1 R+ ; D A s, ) ) W s R+ ; DA) ) D A 1 + s 1 µ + 1/), ). 4.5) Proof. Lemma 11 and 12 in [9] establish the variants of the embeddings 4.4) and 4.5) in the case without weight. The roofs for µ 1/, 1) are similar, using 4.1), the reresentation 2.6) of the weighted Slobodetskii seminorm and Lemma 2.2, and therefore we omit them. Instead, we concentrate on 4.3), taking u from the function sace on the left-hand side there and assuming that s, 1 µ + 1/), α > and s + α 1 µ + 1/, 1). From 4.1) and 4.2) we deduce u) D A 2s+α 1 µ+1/),), = σ 1 2s+α 1 µ+1/)) Ae σa u) X σ 1 dσ + σ 2s+α) σ τ 1 µ Ae σa uτ) dτ) dσ 4.6) σ t σ 2 µ 2s+α) t 3 µ) τ 1 µ Ae σa ut) uτ)) dτ dt) dσ. Recall that Ae σa x X σ 1+θ A θ x X 4.7) holds for all θ, 1) and x X. Emloying Hölder s inequality, 4.7), 2.6), Lemma 2.2 and Proosition 2.1, we estimate the first summand in 4.6) by σ 2s+α) σ τ 1 µ Ae σa uτ) dτ) dσ σ σ τ 1 µ) Ae σa uτ) σ 1 σ 2s+α) dτ dσ τ 1 µ) A s+α uτ) σ 1+s) dτ dσ σ τ 1 µ) uσ) uτ)) DA s+α ) σ τ) 1+s) dτ + σ 1 µ s) uσ) DA s+α ) u W s R +;DA s+α )). ) dσ We further use 4.7), Hardy s inequality 2.3), Hölder s inequality and 2.6) to control the second summand in 4.6) by σ t σ 2 µ 2s+α) t 3 µ) τ 1 µ Ae σa ut) uτ)) dτ dt) dσ σ σ s+α 1 µ+1/)) t 1 t t 2 µ) 18 ) σ τ 1 µ ut) uτ) DA s ) dτ dt) 1 dσ

19 σ σ s+α 1 µ+1/)) σ 2 µ) σ σ τ 1 µ uσ) uτ) DA s ) dτ) 1 dσ Thus 4.3) holds. τ 1 µ) uσ) uτ) DA s ) σ 1+s+α)) dτ dσ [u] W s+α R + ;DA s )). From the above lemma we deduce a general trace theorem in the time variable for the weighted anisotroic saces. Theorem 4.2. Let E be a Banach sace of class HT, let J =, T ) be finite or infinite, and let Ω R n be a bounded domain with smooth boundary, or Ω {R n, R n +}. Assume that r, β >, < 1/ < µ 1 and that k N, s and α, 2) satisfy s < k + 1 µ + 1/ < s + α. Set H s W r ) := H s J; W r Ω; E)), and analogously for the other anisotroic saces. The order of differentiability of each occurring W - and W -saces is required to be noninteger. Then each of the saces H s+α W r ) HW s r+β ), W s+α H) r WH s r+β ), W s+α H) r HW s r+β ), 4.8) is continuously embedded into Moreover, for α 1 it holds BUC k J, B r+β1+s k+1 µ+1/))/α), Ω; E) ). 4.9) WW α r ) L W r+β ) BUC J, B, r+β1 1 µ+1/)/α) Ω; E) ), 4.1) WW 1 r ) WW s r+β ) BUC J, B, r+βµ 1/)/1 s) Ω; E) ). 4.11) All these embeddings remain true if one relaces Ω by its boundary Ω or the H and W - saces by H and W -saces, resectively. In the latter case, the embedding constants are indeendent of T if s + α 2. Proof. I) We first indicate how to reduce the assertions to some basic cases. Again we may assume that J Ω = R + R n. The left translations form a contractive C -semigrou on the five saces under consideration, due to Lemma 2.6. Hence, one can see as in Proosition III of [2] that the asserted embeddings hold if we can show that the temoral trace oerator tr u = u t= mas these function saces continuously into Y := B, r+β1+s k+1 µ+1/))/α) R n ; E), where one has to set s = k = for 4.1) as well as k = and α = 1 s for 4.11). We may assume that s > k µ + 1/. In fact, if this not the case, we take σ, 1) such that s + σα > k µ + 1/. Proosition 3.2 then allows to embed, say, Z := W s+α H) r HW s r+β ) into W s+σα sace Z 1 := W s+α H r ) W s+σα H r+1 σ)β H r+1 σ)β ). For the ) we then have the embedding into 4.9) roved below, which is recisely the assertion for Z. Moreover, we only have to consider the case k = for the saces in 4.8). In fact, if s k we can simly aly t k to these saces and then use the case k =. If s k α, k), then we 19

20 first use Proosition 3.2 and embed, say, Z = W s+α with σ = k + ε s)/α, for some ε, s + α k). Now, t k W ε H r+1 σ)β H r ) H s W r+β s + α [1, 2 µ + 1/], we can embed, say, Z into Z 2 := W s+α ) into W k+ε H r+1 σ)β ) mas Z into W s+α k H) r ) and the trace result for k = will imly the assertion. Similarly, if we have H) r W s+γ H r+1 σ)β ), where σ = γ/α and s + γ s, 1). The trace result for Z 2 will again imly the assertion for Z. Finally, Proosition 3.2 yields the embeddings H s+α W r ) HW s r+β W s+α H r ) H s W r+β ) W s+1 ε)α ) W s+1 ε)α H r+εβ H r+εβ ) W s+εα H r+1 ε)β ), ) W s+εα H r+1 ε)β ) where ε > is chosen such that the exonents of the W -saces are noninteger. If we can show the assertion for the traces on the right hand side, then we also obtain the asserted embeddings for the saces on left hand side. So besides 4.1) and 4.11), it remains to consider the sace W s+α H) r WH s r+β ) for k = and s + α < ) II) We first show the trace result for the case 4.12). To that urose, we aly 4.3) with X = H r sβ/α, A = 1 n ) β/2α, DA) = H r+1 s)β/α, such that DA s ) = H r and DA s+α ) = H r+β. Observe that r sβ/α < is ossible.) Hence tr is continuous W s+α H) r WH s r+β ) ) D A 2s + α 1 µ + 1/), = B r+β1+s 1 µ+1/))/α),, as asserted. To obtain 4.1), we use 4.4) alied to X = B r,, A = 1 n ) β/2α, DA) = B r+β/α,, giving D A α 1 µ + 1/), ) = B, r+β1 1 µ+1/)/α). Finally, we deduce 4.11) from 4.5) with X = B, r sβ/1 s), A = 1 n ) β/21 s), DA) = W r+β, so that D A 1 + s 1 µ + 1/), ) = B r+βµ 1/)/1 s), in this case. Arguing as in the roof of Theorem 4.5 in [8], one should be able to show that the temoral trace is surjective for all of the saces under consideration in the Theorem 4.2. At this oint we only consider a right-inverse for an imortant secial case related to maximal L -regularity. Lemma 4.3. Let E be a Banach sace of class HT, 1, ), µ 1/, 1], m N, and let Ω R n be a domain with comact smooth boundary Ω, or Ω {R n, R n +}. Then there exists a continuous right-inverse of tr which is given by S : B, 2mµ 1/) Ω; E) W 1 R+ ; L Ω; E) ) L R+ ; W 2m Ω; E) ) Su t) = R Ω e t1 n)m E Ω u, t >. Here, E Ω is the extension to R n from 3.2) and R Ω denotes the restriction from R n to Ω. 2