TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES

Transcription

1 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES MARTIN MEYRIES AND MARK VERAAR Abstract. In this aer we characterize trace saces of vector-valued Triebel-Lizorkin, Besov, Bessel-otential and Sobolev saces, equied with anisotroic ower weights. One of our main tools is a new Sobolev embedding result for weighted function saces. In the main art of the aer we consider intersections of function saces with mixed regularity. We rove mixed derivative embeddings with microscoic smoothing and obtain results on traces of intersected saces. This rovides an effective method to characterize trace saces for evolution equations and boundary value roblems. We illustrate this by means of a new result on maximal L -L q - regularity for the linearized two-hase Stefan roblem with Gibbs-Thomson correction. 1. Introduction In recent years the L -L q -maximal regularity aroach to arabolic PDEs has attracted much attention. In the influential works [13, 3, 57] a new theory of maximal L -regularity was founded and many classes of examles are shown to have this roerty. Maximal regularity means that there is an isomorhism between the data and the solution of the linear roblem in suitable function saces. Having established such a shar regularity result one can treat quasilinear roblems by quite simle tools, like the contraction rincile and the imlicit function theorem see [5, 12, 39] and references therein). Due to scaling invariance of PDEs one often requires q for the underlying function sace L L q ) see e.g. [2] and [11, Section 3]), where is the integrability in time and q is for the sace variable. In the L -L q -aroach to linear and quasilinear arabolic roblems with nonhomogeneous boundary conditions it is essential to know the recise temoral and satial trace saces of the unknowns. In this way different tyes and scales of function saces meet and come naturally into lay. For examle, in the L -L q -aroach to the heat equation one looks for strong solutions in the arabolic Sobolev sace H 1, R + ; L q R d )) L R + ; H 2,q R d )), whose temoral trace sace at t = is well-known to be the Besov sace B 2 2/ q, R d ). More recently, it turned out that the satial trace sace at the coordinate x d = is the intersection sace 1.1) F,q 1 1/2q) R + ; L q R d 1 )) L R + ; Bq,q 2 1/q R d 1 )), where F s,q denotes a Triebel-Lizorkin sace. The satial trace sace 1.1) was obtained in [55, 56] for q and more general cases were considered in [14, 27]. We conclude that the L -L q -aroach for already such a basic examle as the heat equation with inhomogeneous boundary conditions involves three scales of function saces. In the case of free boundary roblems or, more generally, for arabolic boundary value roblems of relaxation tye see [15]), a second unkown is involved, which only lives on the boundary. 2 Mathematics Subject Classification. 46E35, 46E4, 8A22. Key words and hrases. Weighted function saces, ower weights, vector-valued function saces, anisotroic function saces, Besov saces, Triebel-Lizorkin saces, Bessel-otential saces, Sobolev saces, Sobolev embeddings, traces, Newton olygon embeddings, Stefan roblem with Gibbs-Thomson correction. The first author was suorted by the roject ME 3848/1-1 of the Deutsche Forschungsgemeinschaft DFG). The second author was suorted by a VENI subsidy of the Netherlands Organisation for Scientific Research NWO). 1

2 2 MARTIN MEYRIES AND MARK VERAAR For instance, for the transformed and linearized two-hase Stefan roblem with Gibbs-Thomson correction, the otimal sace for the boundary unknown is 1.2) F 3/2 1/2q),q R + ; L q R d 1 )) F,q 1 1/2q) R + ; H 2,q R d 1 )) L R + ; Bq,q 4 1/q R d 1 )), see [15, 28] or Lemma 9.1 below. To treat the corresonding roblem with nontrivial initial values one now has to determine the recise temoral trace sace at t = of this trile intersection sace. Moreover, if more than one boundary condition is involved, then mixed derivative embeddings are imortant to determine the otimal regularities of all boundary inhomogeneities, see [15, 16, 18, 28, 34, 35] and Section 7. Stochastic arabolic equations and Volterra integral equations see [37, 58, 59]) are other scenarios in which intersection saces, even in an abstract form, come naturally into lay. For an oerator A with a bounded H -calculus on a sace X = L q with q 2 it is shown in [37] that the athwise otimal regularity in the context of stochastic maximal L -regularity is H s, R + ; X) L R + ; DA 1/2 s )), s [, 1/2). In many situations, e.g., when boundary conditions are involved, the fractional ower domain DA 1/2 s ) is only a closed subsace of a function sace as above. This is our motivation to study intersection saces in an abstract form. In a next ste it is natural to introduce temoral ower weights wt) = t γ for the intersection saces. Indeed, in many cases maximal regularity roerties of arabolic roblems are indeendent of the weight see [35, 41]). Moreover, the weights yield flexibility for the initial regularity and thus a scale of hase saces where the solution semiflow acts. This can be used to show a smoothing effect of the arabolic equation and comactness of the semiflow, which is an imortant roerty for the investigation of the long-time behavior of solutions see [41, Remark 3.3] for a discussion). In this article we can to a large extent overcome the roblem of mixed regularity scales and study trace saces and mixed derivative embeddings for a general class of intersection saces on the line with ower weights. Our results allow to characterize the regularity of the initial values in the temorally weighted L -L q -aroach to arabolic roblems with general boundary conditions, as treated in [15, 28] see also Remark 9.4). We demonstrate the scoe of our methods by roving maximal L -L q -regularity for the fully inhomogeneous linearized and transformed two-hase Stefan roblem with Gibbs-Thomson correction. As it turns out, the weights are not only useful for the alications. Sobolev embeddings for weighted Triebel-Lizorkin saces F,q, s being indeendent of the microscoic arameter q see Section 3), are in fact the main technical tool in our investigations. Let us describe the results in more detail. We start with the trace saces at the hyerlane {x, ) : x R d 1 } of the ower weighted vector-valued function saces 1.3) W k, R d, w; X), H s, R d, w; X), B s,qr d, w; X), F s,qr d, w; X), which denote Sobolev, Bessel-otential, Besov and Triebel-Lizorkin saces see Section 2.1 for the definitions). The next theorem is our main result. The roof will be given below in Proosition 4.4, Theorem 4.7 and Corollary 4.8. Theorem 1.1. Let X be a Banach sace, 1, ), q [1, ], wx, t) = t γ with γ > 1 and s >. Then for the trace oerator tr at {x, ) : x R d 1 } it holds s tr B,qR s d, w; X) ) = B,q R d 1 ; X), tr F,qR s d, w; X) ) = B Assume in addition that γ < 1 and m N. Then m tr W m, R d, w; X) ) = B, R d 1 ; X), tr H s, R d, w; X) ) = B s, s, R d 1 ; X). R d 1 ; X). Here the notation tr AR d, w; X) ) = BR d 1 ; X) means that tr : AR d, w; X)) BR d 1 ; X) is continuous and surjective, and that it has a continuous right-inverse. The recise definition of the trace oerator tr is given in Section 4.1.

3 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES 3 Remark 1.2. i) The weight wx, t) = t γ can be seen as a ower of the distant function to the hyerlane {x, ) : x R d 1 } where the trace is taken. As a result, the exonent γ aears in the trace regularity s. Of course, for γ = one recovers the unweighted case. The situation of the theorem serves as a model case for saces F,qΩ, s w; X), where Ω R d is a smooth bounded domain and wx) = distx, Ω) γ, see [29, 33]. ii) In the unweighted case γ =, the trace saces of 1.3) are well-known at least for X = C, see [45, 49, 5] and the references therein. Recently, also the general vector-valued case was investigated and the trace saces were characterized in [42, 44]. Results for radial weights wx) = x γ are obtained for scalar Sobolev saces in [1], and in the general case for 1.3) in [24]. In the radial case the weight exonent does not aear in the trace regularity. Traces of Sobolev-Slobodetskii and Bessel-otential saces with weights equal to a ower of the distance to the boundary are intensively studied, see the references given in [5, Sections 2.9.2, 3.1.1]. In articular, in [22, Théorème 7.1] a artial result for Besov saces is obtained. iii) The additional condition γ 1, 1) for the W - and H-saces holds if and only if w belongs to the Muckenhout class A see Section 2.1). To determine the trace sace of a Besov sace in Theorem 1.1 we follow the arguments of [42] for the unweighted case. The result is in fact a consequence of the descrition of the B-saces by the decomosition method and elementary Fourier analytic L -estimates for the trace of functions with comact Fourier suort. Our aroach to the trace sace of an F -sace is surrisingly simle and is based on Sobolev embeddings and the trace sace of a Besov sace. In Section 3 we derive the embeddings for weighted saces with anisotroic ower weights are derived from our results in [36], where radial weights were considered. The trace sace of a W - and H-sace follows as in [42] from a sandwich argument, based on the embedding 1.4) F s,1r d, w; X) H s, R d, w; X) F s, R d, w; X), and analogously with H relaced by W see Section 2.1). Here the second embedding is valid only if w satisfies a local A -condition, which results in γ < 1. The first embedding it true for all w A. We note at this oint that H s, R d ; X) = F s,2r d ; X), i.e., the Littlewood-Paley decomosition for L R d ; X), is valid if and only if X can be renormed as a Hilbert sace. It is known that the trace roblem is intimately related to Hardy s inequality. As another consequence of the weighted Sobolev embeddings with fixed we rovide in Theorem 3.5 a microscoic imrovement of Hardy s inequality in the target sace. This is another hint for the canonical use of weights and Sobolev embeddings for trace considerations. The interretation of Hardy s inequality as a Sobolev inequality with weights is well-known, see [51, Section 16.4]. The second and main art of our aer is on intersections of saces with mixed regularity e.g. H α, R; X) L R; DA α )). To investigate such saces we first show an oerator-valued Fourier multilier theorem for B- and F -saces with A -weights and give a characterization of these saces in terms of differences, see Sections 5 and 6. Then we use the multilier theorem to rove mixed derivative embeddings for abstract intersection saces in Theorem 7.1. For B- and H-saces, embeddings of this tye are well-known and widely used in the context of boundary value roblems with inhomogeneous symbols, see [15, 16, 18, 28, 34, 35]. In the case when the inner saces are F -saces, we rove a new smoothing effect with resect to the microscoic arameter. A secial case of Theorem 7.3 is the following. Theorem 1.3. Let d, m N,, 1, ), q [1, ], s 1, s 2 R, α 1, α 2 > and θ, 1). Let r 1 = s 1 + θα 1 and r 2 = s θ)α 2. Then F,q s1+α1 R d ; F s2, R m ) ) F,q s1 R d ; F s2+α2, R m ) ) F,q r1 R d ; F r2,1 Rm ) ), The results remains true if one relaces F by B in the outer scale. In view of the monotonic roerties of the F -saces 2.7) and 1.4), the essential oint of this result is the fact that on the

4 4 MARTIN MEYRIES AND MARK VERAAR left-hand side in the inner scale one has the large saces F s2, and F s2+α2, and on the righthand side the small saces F r2,1, hence there is a microscoic imrovement. Using 1.4), the embeddings aly in articular to the intersection saces of tye 1.2). An analogous microscoic imrovement for the outer regularity scale does not hold, see Proosition 7.5. Our trace result for abstract weighted intersection saces on the line is as follows. For the notion of sectorial oerators and the real interolation saces D A θ, ) we refer to Section 2.2. Theorem 1.4. Let 1, ) and wt) = t γ with γ 1, 1), suose that s R and α > satisfy s < < s + α, and let A be a sectorial oerator on a Banach sace X with sectral angle φ A < min{ π 2, π α } and r. Let θ = r + s + α. Then for the trace oerator tr at t = it holds tr F s+α,q R, w; DA r )) F s,qr, w; DA r+α )) ) = D A θ, ), q [1, ], tr B s+α, R, w; DA r )) B,R, s w; DA r+α )) ) = D A θ, ), tr H s+α, R, w; DA r )) H s, R, w; DA r+α )) ) = D A θ, ), tr W s+α, R, w; DA r )) H s, R, w; DA r+α )) ) = D A θ, ). Remark 1.5. i) If s > 1, then the right-inverse of tr can be chosen such that it only deends on A. In the other cases the extension oerator in our roof deends on s, α and A. As shown in Lemma 8.1, tr is mas continuously to D A θ, ) for all γ > 1. ii) The essential oint is the indeendence of the trace sace of the F -intersection saces of the microscoic arameter q [1, ]. It follows from a sandwich argument that the trace sace of saces with mixed regularity inside the F -scale is also equal to D A θ, ). More recisely, let A s+α and B s be function saces such that F,1 s+α A s+α F, s+α, see e.g. 1.4). Then the theorem shows that F s,1 B s F s,, tr A s+α R, w; DA r )) BR, s w; DA r+α )) ) = D A θ, ). In secial situations, real interolation allows to relace e.g. DA r+α ) by D A r + α, ), see the roof of [34, Theorem 4.2]. Moreover, in combination with Theorem 1.3 one obtains indeendence in the outer and the inner regularity scale in the context of traces. iii) As mentioned before, the intersection saces and their variants arise in the maximal L - L q -regularity aroach to deterministic and stochastic arabolic evolution equations, see [14, 15, 16, 18, 28, 35, 37, 41, 55, 58, 59]. iv) The theorem alies in articular to any fractional ower of on X { H r,q R d ), B r q,σr d ), F r q,σr d ) : r R, q 1, ), σ [1, ] }. Since the sectral angle of is equal to zero, α > above may be arbitrary large. v) The result is a generalization of [34, Theorem 4.2] in the weighted case, and of [16, Theorem 4.5] and [59, Theorem 3.6] see also [58, Theorem 3.1.4]) in the unweighted case. In these works only the H- and B-saces were considered. The continuity of tr was roved in [59, Theorem 3.6] under the assumtion that X is a UMD Banach sace and that A is R-sectorial with R-angle not larger than π α. The reason for these stronger assumtions is that the roof in [59] relies on the oerator-valued Fourier multilier result due to [57]. Moreover, the roof uses a result on comlex interolation of H-saces with Dirichlet boundary conditions see [46], and [6] for the vector-valued case), which is not trivial to extend to weighted case. The condition φ A < π 2 is not assumed in [59]. Also for Theorem 1.4 it should not be essential. vi) The traces of anisotroic Besov and Triebel-Lizorkin saces, not necessarily of intersection tye, are studied in [6, 8, 27]. However, there are only artial results on how the saces considered there are related to the intersection saces when X and DA) are function saces over R d as above, see [6, Sections ], [27, Section 5] and [28, Section 5.2]. In articular, the case of an oerator with boundary conditions on a domain is not included there.

5 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES 5 vii) It should be ossible to generalize Theorem 1.4 to dimensions d 2. Given the situation of the theorem, using a Fubini argument and the methods of Section 8 one can see that the trace tr at {x, ) : x R d 1 } mas continuously into B F s+α,q R d, w; DA r )) F s,qr d, w; DA r+α )) s+α, R d 1 ; DA r )) L R d 1 ; D A θ, )). We exect that this is indeed the trace sace. It seems elaborate to construct a right-inverse. Theorem 1.4 is shown in Section 8. The roof of the continuity of tr is based on a semigrou reresentation of the real interolation saces D A θ, ), Hardy s inequality, the equivalent norm in terms of differences for the F -saces and weighted Sobolev embeddings see Lemma 8.1). The right-inverse for tr is essentially the semigrou e A, combined with an extension oerator to R. The roof of the desired maing roerties in Lemma 8.3 is once more based on Sobolev embeddings. In the final Section 9 we aly the general results and rove maximal L -L q -regularity in the arameter range 2 +1 < q < 2 for the fully inhomogeneous linearized two-hase Stefan roblem with Gibbs-Thomson correction see [18] and the references therein). The corresonding original roblem is a free boundary roblem which can be used to model the melting of ice. Based on the results of [28] for trivial initial values, our main contribution is here to determine the temoral trace sace of 1.2). The case = q was considered in [16] for the one hase roblem, and in [18] for the two-hase roblem. This aer is organized as follows. Section 2: reliminaries on weighted function saces. Section 3: Sobolev embeddings theorem with an anisotroic ower weight. Section 4: trace sace of weighted function saces Theorem 1.1). Section 5: oerator-valued Fourier multilier theorem for weighted F and B-saces. Section 6: equivalent norms in terms of differences for weighted F and B-saces. Section 7: mixed derivative embeddings for saces of intersection tye Theorem 1.3). Section 8: traces of intersection saces Theorem 1.4). Section 9: linearized two-hase Stefan roblem with Gibbs-Thomson correction. 2. Preliminaries 2.1. Function saces and elementary embeddings. We briefly recall the definitions and embedding roerties of the weighted vector-valued function saces. For details and references we refer to [36, Sections 2 and 3]. For 1, ) the Muckenhout class of weights is denoted by A, and A = >1 A. In the resent work we mainly consider ower weights w on R d, d 1, of the form wx, t) = t γ, x = x, t) R d, x R d 1, t R. Here we have w A if and only if γ 1, 1) see [25, Examle 1.5]). For a Banach sace X, 1, ) and w A the norm of L R d, w; X) is given by ) 1/ f L R d,w;x) := fx) Xwx) dx. R d One further defines L R d, w; X) := L R d ; X). Let S R d ; X) be the Schwartz class of X- valued, smooth raidly decreasing functions on R d, and let S R d ; X) = L S R d ); X) be the sace of X-valued temered distributions. The Fourier transform of a distribution f is denoted by F f or f. Let ΦR d ) be the set of all sequences ϕ k ) k S R d ) such that 2.1) ϕ = ϕ, ϕ 1 ξ) = ϕξ/2) ϕξ), ϕ k ξ) = ϕ 1 2 k+1 ξ), k 2, ξ R d,

6 6 MARTIN MEYRIES AND MARK VERAAR where the Fourier transform of the generating function ϕ satisfies 2.2) ϕξ) 1, ξ R d, ϕξ) = 1 if ξ 1, ϕξ) = if ξ 3 2. For ϕ k ) k ΦR d ) and f S R d ; X) we set S k f = ϕ k f = F 1 ϕ k f). As a consequence of the boundedness of the Hardy-Littlewood maximal function on L R d, w), for w A we have see e.g. [36, Lemma 2.3]) 2.3) su S k f L R d,w;x) C f L R,w;X). d k Now fix a Banach sace X, 1, ), q [1, ] and w A. Then for f S R d ; X) we set f B s,q R d,w;x) := 2 ks S k f ) k lq L R,w;X)), d f F s,q R d,w;x) := 2 ks S k f ) k L R d,w;l X)), q f H s, R d,w;x) := F 1 [1 + 2 ) s/2 f] L R d,w;x). These norms define the Besov sace B s,qr d, w; X), the Triebel-Lizorkin sace F s,qr d, w; X), and the Bessel-otential sace H s, R d, w; X), resectively, which are all Banach saces. Any other ψ k ) k Φ leads to an equivalent norm on the B- and F -saces. If s R + \N, then we set and for m N, W s, R d, w; X) := B s,r d, w; X), f W m, R d,w;x) := α m D α f L R d,w;x), where the derivatives D α are taken in a distributional sense. These norms define the Slobodetskii and the Sobolev saces, resectively. Note that L R d, w; X) = H, R d, w; X) = W, R d, w; X). Each of the above saces embeds continuously into S R d ; X). Conversely, S R d ; X) embeds continuously into each of the above saces, where this is a dense embedding if, q <. Let a Banach sace E be continuously embedded into S R d ; X). The sace E is said to have the Fatou roerty if for all f n ) n E it holds lim f n = f in S R d ; X), n lim inf n f n E < = f E, f E lim inf n f n E. Proosition 2.1. Let X be a Banach sace, s R, 1, ), q [1, ] and w A. Then each of the saces B s,qr d, w; X) and F s,qr d, w; X) has the Fatou roerty. Such roerties were used in [19] for the first time and indeendently in [9]. For a slightly different formulation as ours and a roof we refer to [42, Proosition 2.18] and [44, Proosition 4]. The Fatou roerty is useful as a substitute for the density of S R d ; X) in the B- and F -saces in the imortant case when q =. There are elementary embeddings between the function saces, see [36, Proositions 3.11 and 3.12]. For s R and m N we shall make articular use of 2.4) 2.5) F s,1r d, w; X) H s, R d, w; X) F s, R d, w; X), F m,1r d, w; X) W m, R d, w; X) F m, R d, w; X). The above embeddings into F s, and F m, are valid for w A. In [36, Remark 3.13] it is shown that a local A -condition is necessary for 2.4) and 2.5) to hold. For ower weights wx, t) = t γ as above this condition is equivalent to the usual A -condition. The above embeddings for F s,1r d, w; X) are valid for all w A. For all q [1, ], s R and 1, ) one further has 2.6) B s,min{,q} Rd, w; X) F s,qr d, w; X) B s,max{,q} Rd, w; X), and if 1 q q 1, then 2.7) B s,q R d, w; X) B s,q 1 R d, w; X), F s,q R d, w; X) F s,q 1 R d, w; X).

7 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES Sectorial oerators and real interolation. Below we only recall some standard definitions and results on sectorial oerators. For a detailed exosition we refer to [3, 13, 32, 5]. Let A be a densely defined oerator on a Banach sace X. Then A is called sectorial on Σ θ = {λ C\{} : arg λ < θ}, θ, π), if Σ θ is contained in the resolvent set of A and if there is a constant C > such that λλ + A) 1 L X) C. The infimum of all θ with this roerty is denoted by φ A and is called the sectral angle of A. The oerator A generates an analytic C -semigrou if and only if it is sectorial with φ A < π 2. For α R the fractional ower A α of a sectorial oerator A is defined by the extended Dunford calculus as in [13, Section 2]. If α < π φ A, then A α is also sectorial with φ A α α φ A see [13, Theorem 2.3]). For s, 1) and [1, ] the real interolation functor is denoted by, ) s,. For an invertible sectorial oerator A we write D A k + θ, ) = {x DA k ) : A k x X, DA)) θ, }, k N, θ, 1), [1, ]. For our uroses it is further convenient to define D A k, ) = {x X : A 1/2 x D A k 1/2, )}, k N. Here 1/2 may be relaced by any other θ, 1). With this notation, the reiteration theorem and [5, Theorem ] show that for all θ, α > the oerator A α is an isomorhism D A θ + α, ) D A θ, ) and DA θ+α ) DA θ ), resectively. If A is invertible and generates an analytic C -semigrou, then it follows from [5, Theorem ] that ) 1/ 2.8) x σ 1 θ) Ae σa x dσ X σ defines an equivalent norm on D A θ, ) for θ, 1). 3. Sobolev embeddings for anisotroic ower weights In this section we rove Sobolev embeddings for function saces with anisotroic weights. In the case of Triebel-Lizorkin saces these embeddings are indeendent of the microscoic arameter q [1, ]. In [36, Theorems 1.1 and 1.2] we have characterized such embeddings results for saces with radial ower weights in terms of the arameters. The flexibility in both the microscoic arameter q [1, ] and the weights, combined with elementary embeddings 2.4) and 2.5), are key elements in our investigations of trace saces. From [36, Theorems 1.1 and 1.2] one obtains the following sufficient conditions for Sobolev embeddings with radial weights. Theorem 3.1. Let X be Banach sace, 1 < 1 <, q, q 1 [1, ], s > s 1 and w x) = x γ, w 1 x) = x γ1 with γ, γ 1 > d. Suose that Then one has the continuous embedding γ 1 1 γ and s d + γ = s 1 d + γ ) F s,q R d, w ; X) F s1 1,q 1 R d, w 1 ; X). Suose in addition that q q 1. Then 3.2) B s,q R d, w ; X) B s1 1,q 1 R d, w 1 ; X). Our arguments for the corresonding embeddings for anisotroic weights are based on the following inequality of Plancherel-Polya-Nikol skij tye, see [36, Proosition 4.1]. Lemma 3.2. Let X be a Banach sace and let 1 <, 1. Let γ, γ 1 > d and w x) = x γ and w 1 x) = x γ1. Suose γ 1 1 γ and d + γ 1 1 < d + γ.

8 8 MARTIN MEYRIES AND MARK VERAAR Let R > and let f : R d X be a function with su f {ξ R d : ξ < R}. Then there is a constant C, indeendent of f and R, such that where δ = d+γ d+γ1 1 >. f L 1 R d,w 1;X) CR δ f L R d,w ;X), We rove the following comlement of Theorem 3.1. In the scalar case X = C, the embeddings for the Besov saces and their otimality can be deduced from [25, Proosition 2.8]. Theorem 3.3. Let X be a Banach sace, 1 < 1 <, q, q 1 [1, ], s > s 1 and w x, t) = t γ, w 1 x, t) = t γ1 with γ, γ 1 > 1. Suose that 3.3) Then one has the continuous embedding γ 1 1 γ and s d + γ = s 1 d + γ ) F s,q R d, w ; X) F s1 1,q 1 R d, w 1 ; X). Suose in addition that q q 1. Then 3.5) B s,q R d, w ; X) B s1 1,q 1 R d, w 1 ; X). Proof. Ste 1. We first show 3.5). By 2.7), it suffices to consider the case q := q = q 1. We write s 1 +δ = s, where δ = δ +δ with δ = γ+1 γ1+1 1 and δ = d 1 d 1 1. Let f B s,q R d, w ; X) and set f n = S n f. Note that each f n is a smooth bounded function on R d, and that su f n { ξ 3 2 n 1 }. Let F t be the Fourier transform with resect to t R. It follows from Ste 2 of the roof of [42, Theorem 4.9] that for each fixed x R d 1 the function F t f n x, )) is suorted in { λ < 3 2 n 1 }. We use Lemma 3.2 and Minkowski s inequality or the triangle inequality for Bochner integrals in L 1 / R,w)) to estimate f n L 1 Rd,w 1;X) = f n x, t) 1 t γ1 dt dx ) 1/ 1 R d 1 R ) ) 1/ C2 δ n f n x 1/ 1, t) t γ dt dx R d 1 R ) / C2 δ n 1 t ) f n x, t) 1 dx γ 1/. dt R R d 1 As above, for fixed t R the function F x f n, t)) is suorted in { ξ < 3 2 n 1 }. Again Lemma 3.2 gives ) / 1 t f n x, t) 1 dx γ dt) 1/ ) C2 δ n f n x, t) dx 1/ t γ dt R R d 1 R R d 1 Therefore = C2 δ n f n L Rd,w ;X). 3.6) f n L 1 R d,w 1;X) C2 δn f n L R d,w ;X), n N, where C does not deend on f and n. Using s 1 + δ = s, we find that f B s 1 1,qR d,w 1;X) = 2 s1n f n L 1 R d,w 1;X)) n l q C 2 sn f n L R d,w ;X)) n l q = C f B s,qr d,w ;X). This roves 3.5). Ste 2. Using [36, Proosition 5.1], now the embedding 3.4) can be deduced from 3.5). In fact, literally in the same way as in the roof of [36, Theorem 1.2] we obtain 3.4) for all f F s1 1,q 1 R d, w; X) F s,q R d, w; X). The general case follows from a Fatou argument, which we sketch here for the convenience of the reader.

9 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES 9 For f F s,q R d, w; X) the series f N = N n= f n converges to f in S R d ; X) as N, and it holds f N F s1 1,q 1 R d, w; X) F s,q R d, w; X) for each N by 3.6). Using [36, Proosition 2.4], we further have f N s F,q R d,w;x) C f F s,q R d,w;x), and therefore f N s F 1 1,q 1 R d,w;x) C f F s,q R d,w;x). Now Proosition 2.1 gives f F s1 1,q 1 R d, w; X) and which finishes the roof. Remark 3.4. f s F 1 1,q 1 R d,w;x) lim inf f N s F 1 N 1,q 1 R d,w;x) C f F s,q R d,w;x), i) In the Besov scale Theorem 3.1 has an extension to > 1 see [36, Theorem 1.1]). ii) If d = 1, then Theorems 3.1 and 3.3 coincide. iii) If d 2, then Theorem 3.3 does not have an extension to > 1. Indeed, let f : R d C be a function with sufficiently small Fourier suort and let f r be a dilatation only in the x -variable, i.e., f r x, t) = frx, t) for r >. Then one can reeat the argument in [36, Proosition 4.7] to see that d 1 1 d 1, which imlies 1. iv) Theorem 3.3 is also a model case for saces over a bounded smooth domain Ω R d with weights w j x) = distx, Ω) γj, see [29, 33]. v) If γ < 1, then one can combine the result for the F -saces with the embeddings 2.4) and 2.5) to obtain Sobolev embeddings for weighted H- and W -saces. As an alication we give a very short roof and a generalization in the target sace of a wellknown Hardy inequality. Early versions of Hardy s inequality can be found in [22], [31, Proosition 2.3] and [51, Proosition 5.7] and references therein. The inequality seems to go back to certain inequalities for double integrals in [7]. Theorem 3.5. Let X be a Banach sace, 1, ), q [1, ] and wx, t) = t γ for γ > 1. Suose that < s < and vx, t) = t γ s. Then one has f L R d,v;x) C f F,1 R d,v;x) C f F s,q R d,w;x), f F s,qr d, w; X). In articular, the estimate holds with F s,qr d, w; X) relaced by B s,r d, w; X). If additionally γ < 1, then the estimate holds with F s,qr d, w; X) relaced by H s R d, w; X). Proof. Let w 1 x, t) = t γ s. Then by assumtion γ s > 1 and hence w 1 A. Since f = n S nf, the triangle inequality and Theorem 3.3 yield L f L R d,dx t γ s dt;x) S n f X = f F w,1 R d,w 1;X) C f F s,q R,w;X). d 1) n The final assertion follows from 2.4) and w A if 1 < γ < 1. Remark 3.6. i) The same result as in Theorem 3.5 holds for the radial weights wx) = x γ and vx) = x γ s, where < s < d+γ. Indeed, this is a secial case of Theorem 3.1. This should be comared to [51, Section II.16] and the references given therein. In articular, it is exlained that Hardy s inequalities may be seen as Sobolev embeddings for weighted saces. Our result seems to be a strengthening of this and shows the usefulness of Triebel-Lizorkin saces for this matter. ii) The usual formulation of Hardy s inequality in terms of increments can be derived from Theorem 3.5 and Proosition 6.1 below. iii) Using homogenous versions of Theorems 3.1 and 3.3 one can also derive inequalities of Hardy tye for homogenous norms.

10 1 MARTIN MEYRIES AND MARK VERAAR 4. Traces of function saces with anisotroic ower weights In this section we determine the trace saces of F -, B-, H- and W -saces with anisotroic weights Definition of the trace. In the imortant case q = the Schwartz functions are not dense in the B- and F -saces, and thus one cannot define the trace on these saces as an extension of the trace in the classical sense. For a unified aroach one has to consider a more general definition of a trace. As in [42] we emloy a concet due to Nikol skij [38]. Let f : R d X be strongly measurable. Then g : R d 1 X is called the trace of f on R d 1 {}, if there are f : R d X, [1, ] and δ > such that i) f = f a.e. with resect to the Lebesgue measure on R d ; ii) f, t) L R d 1 ; X) for t < δ; iii) f, ) = g a.e. with resect to the Lebesgue measure on R d 1 ; iv) lim t f, t) g L R d 1 ;X) =. This definition is indeendent of f, and δ see [42, Remark 4.2]), and it coincides with the restriction of f to {x, ) : x R d 1 } if f is continuous on R d 1 δ, δ). If the trace exists in the above sense, then we write trf := g, and obtain in this way a linear oerator tr The trace sace of a Besov sace. The results and roofs in this subsection are comletely analogous to the results for unweighted vector-valued Besov saces obtained in [42]. Since the weight disaears for =, in the roofs below we only consider the case <. We first generalize [42, Lemma 4.5] to the weighted setting. Lemma 4.1. Let X be a Banach sace, 1, ) and wx, t) = t γ with γ > 1. Let f L R d ; X) be such that su f { ξ R} for some R >. Then there is a constant C, indeendent of f and R, such that for all t R we have f, t) L R d 1 ;X) CR f L R,w;X). d Proof. By a scaling and a translation argument it suffices to consider the case R = 1 and t =. Let ψ Cc R d ) be such that ψ 1 on B 1. Then for all x R d 1 we have ) fx, ) = F 1 ψ f)x, ) = fx y, t) t γ ψy, t) t γ dt dy. R d 1 R Thus Hölder s inequality gives, with = 1 and γ = γ 1, fx, ) fx y, ) L R, γ ;X) ψy, ) L R, γ ;X) dy. R d 1 Taking the L -norm with resect to x and using Minkowski s inequality, we obtain fx, ) L R d 1 ;X) f L R d,w;x) ψy, ) L R, γ ;X) dy. R d 1 Since γ > 1, the second factor is finite. The next result is analogous to [42, Proosition 4.4]. Proosition 4.2. Let X be a Banach sace, 1, ) and wx, t) = t γ with γ > 1. Then the trace of f B,1 Rd, w; X) exists, and for any ϕ n ) n ΦR d ) it holds trf = ϕ n f, ), where the convergence is in L R d 1 ; X). n=

11 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES 11 Proof. As in [42], for f B,1 Rd, w; X) and ϕ n ) n ΦR d ) we let f = n= S nf and verify that it satisfies the requirements from the above definition. Note that n= S nf L R d,w;x) <, since f B,1 Rd, w; X) The series converges to f in L R d, w; X), and thus f = f a.e. on R d. By Lemma 4.1 and Ŝ n f {ξ R d : ξ 3 2 n }, for t R we have f, t) L R d 1 ;X) Sn f, t) L R d 1 ;X)) n l 1 C 2 n S n f L R d,w;x)) n l 1 = C f B,1 Rd,w;X) For the last condition we use Lemma 4.1 and the ointwise estimate from [42, Lemma 2.3] to obtain S n f, t) S n f, ) L R d 1 ;X) C 2 n t 2 n M S n f r ) 1/r L /r R d,w;x), where r, ) and M is the Hardy-Littlewood maximal oerator. Choosing r such that w A /r, and using that M is bounded on L q R d, v) for v A q and q 1, ), we get S n f, t) S n f, ) L R d 1 ;X) C 2 n t 2 n S n f L R d,w;x), n N. Now as in [42] it can be shown that lim t f, t) = f, ) in L R d 1 ; X), which finishes the roof. Remark 4.3. For γ > and wx, t) = t γ it follows from Theorem 3.3 that B,1 Rd, w; X) B 1,1 Rd ; X). Thus for these exonents the above result can also be deduced from this embedding and the unweighted case γ =. This argument does not work for γ 1, ). After this rearation we determine the trace sace of a Besov sace in the general case. We follow the arguments of [42, Theorem 4.9]. Proosition 4.4. Let X be a Banach sace, d 2, 1, ), q [1, ], wx, t) = t γ with γ > 1 and s >. Then tr as in Proosition 4.2 is a continuous and surjective oerator s tr : B,qR s d, w; X) B,q R d 1 ; X) There exists a continuous right-inverse ext of tr which is indeendent of s,, q, γ and X. Remark 4.5. For = q 1, ) and w A the result was shown in [22, Théorème 7.1]. Using an atomic aroach, in [24, Theorem 3.5] the trace saces in the case of radial ower weights are determined, also assuming an A -condition. For weights wx, t) = t γ the Fourier analytic arguments from [42] that are extended in the roof below do not require this assumtion. Proof of Proosition 4.4. Ste 1. We show the continuity of tr. Let ϕ n ) n ΦR d ) and φ n ) n ΦR d 1 ). For f S R d ; X) and g S R d 1 ; X) we write S n f = ϕ n f, T n g = φ n g, n, S 1, T 1. Take f B s,qr d, w; X), and write s = s >. In [42] it is shown that trf s B,qR d 1 ;X) = S n fx, ) s B,qR d 1 ;X) C 2 ls S n+l 1 fx, ) L R ;X)) l d 1 l q, n= where the constant C does not deend on f. Note that su F S n+l 1 f) { ξ 2 n+l }. Alying Lemma 4.1, we get trf s B,qR d 1 ;X) C 2 ls n+l) 2 l Sn+l 1 f L R,w;X)) d l q n= n=.

12 12 MARTIN MEYRIES AND MARK VERAAR C 2 2 ns n+l)s l S n+l 1 f L R,w;X)) d l q n= C f B s,q R d,w;x). Ste 2. We define the right-inverse ext as in [49, Section 2.7.2] and [42]. We take ρ n ) n ΦR) s such that ρ n ) = 2 n for all n and set for g B,q R d 1 ; X) ext gx, t) := 2 n ρ n t)t n gx ) in S R d ; X). n= This formula does not deend on the arameters. Moreover, ext g is well-defined in L R d, w; X) and hence in the sense of distributions) since 2 n ρ n T n g L R d,w;x) = 2 n ρ n L R, t γ dt) T n g L R d 1 ;X) n= n= n C2 nγ+1) g L R d 1 ;X) C g B s,q, R d 1 ;X) where we used γ > 1 and s >. Since w A we can find r, min{, q}) such that s,q w A /r. Let g B R d 1 ; X). It follows from [36, Proosition 2.4] that S l ext g L R d,w;x) S l 2 l+j) ρ l+j T l+j g)) L R d,w;x) C j= 1,,1 j= 1,,1 From ρ k = 2 k 1 ρ 1 2 k 1 ) we obtain that ρ l+j L R, γ ;X) = 2 leads to S l ext g L R d,w;x) C l+j 1) 2 Therefore, ext g B s,q R d,w;x) C j= 1,,1 2 l+j) ρ l+j L R, γ ) T l+j g L R d 1 ;X). j= 1,,1 l+j 1)1 Tl+j g L R d 1 ;X). ls 2 )q T l+j g q L R n 1 ;X) ) l l q C g B ) ρ 1 L R, γ ), which s,q R d 1 ;X) It follows that ext is continuous as asserted, indeendent of the arameters. Finally, by the choice of ρ n we have tr ext g = g for all g S R d ; X). By [36, Lemma 3.8], this sace is dense in B,qR s d, w; X) if q <, and the identity extends to all g B,qR s d, w; X). For q = we have B, R s d, w; X) B,1 s ε Rd, w; X) for all ε >, and thus ext is also a right-inverse in this case. Remark 4.6. Analogous to Remark 4.3, in case γ > the continuity of the trace can also be obtained by a simle argument from the unweighted case and the embeddings from Theorem 3.3. Indeed, for s > one has trf B s,q R d 1 ;X) C f B s γ C f B s,q R d ;X),q R,w;X), d since s d+γ = s γ d and γ. Similarly, with these arguments one can give a simle argument for the continuity of the extension oerator for γ 1, )..

13 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES The trace sace for Triebel-Lizorkin, Bessel otential and Sobolev saces. We now determine the trace sace of vector-valued F -, H-, and W -saces with an anisotroic ower weight. As known in the unweighted case, for Triebel-Lizorkin saces the trace mas indeendently of the microscoic arameter q [1, ]. In contrast to [42, Lemma 4.15] and [24, Theorem 3.6], our roof is urely Fourier analytic and does not use the atomic aroach. The rather short argument is based on the above result for Besov saces and the Sobolev embeddings for Triebel-Lizorkin saces from Theorem 3.1. Again an A -condition for the weight is not required and it suffices to have w A. In the unweighted situation a roof based on atomic decomositions was given in [42]. In the scalar and unweighted case other methods can be used [49, Theorem 2.7.2]. Theorem 4.7. Let X be a Banach sace, 1, ), q [1, ], wx, t) = t γ with γ > 1 and s >. Then the trace tr is a continuous and surjective oerator tr : F s,qr d, w; X) B s, R d 1 ; X). Moreover, there exists a continuous right-inverse ext which is indeendent of s,, q, γ and X. Proof. Ste 1. For the continuity of tr it suffices to consider q =. Let f F, R s d, w; X), and set s = s ε, γ = γ ε and wx, t) = t γ, where ε > is sufficiently small. Since s d+γ = s d+ γ and γ < γ, we obtain from Theorem 3.3 that Now by Proosition 4.4, trf B f B s, R d, w;x) = f F s, R d, w;x) C f F s, R d,w;x). s, R d 1 ;X) = trf B s 1+ γ, R d 1 ;X) C f B s, R d, w;x) and the result follows if we combine both estimates. Ste 2. For the continuity of the right-inverse it suffices to consider q = 1. Take ext as in s Proosition 4.4, let g B, we infer from Proosition 4.4 Since s d+ γ ext g B s, R d, w;x) C g B = s d+γ R d 1 ; X), and set s = s + ε, γ = γ + ε and wx, t) = t γ. Then s 1+ γ, R d 1 ;X) and γ < γ, Theorem 3.3 imlies = C g B s,. R d 1 ;X) ext g F s,1 R d,w;x) C ext g F s, R d, w;x) = C ext g B s, R d, w;x) and the continuity follows again from the combination of these estimates. The fact that tr ext g = s g for all g B, R d 1 ; X) is clear from Proosition 4.4. As a consequence of the above result we can determine the trace saces of the H- and W -saces under the assumtion that w A. Corollary 4.8. Let X be a Banach sace, 1, ), wx, t) = t γ with γ 1, 1) and s >. Then the trace tr is a continuous and surjective oerator tr : H s, R d, w; X) B s, If m N, then tr is a continuous and surjective oerator tr : W m, R d, w; X) B m, R d 1 ; X). R d 1 ; X). In both cases there is a continuous right-inverse ext which is indeendent of s, m,, γ and X. Proof. This is a consequence of Theorem 4.7 and the embeddings 2.4) and 2.5). Proof of Theorem 1.1. This follows from Proosition 4.4, Theorem 4.7 and Corollary 4.8.

14 14 MARTIN MEYRIES AND MARK VERAAR 5. Fourier multiliers In this section we derive an oerator-valued Fourier multilier theorem for weighted Besov and Triebel-Lizorkin saces. For a comact set K R d, let L K Rd, w; X) = {f L R d, w; X) : su f K}. Lemma 5.1. Let X, Y be Banach saces, let 1, ), q [1, ] and w A. Let r, min{, q}) be such that w A /r. Let K, K 1,... R n be comact sets with θ n = diam K n > for all n. Then there is a constant C such that for all M n ) n F L 1 R d ; L X, Y )) and all f n ) n L R d, w; l q X)) with f n L K n R d, w; X) for n N it holds F 1 M n F f n )) n L R d,w;l q Y )) C su 1 + d/r )F 1 M k θ k )) L 1 R d ;L X,Y )) f n ) n L R d,w;l X)). q k Proof. In the roof of [1, Proosition 2.2] and [49, Theorem 1.6.3] it is shown that for x R d and n N one has 5.1) F 1 M n F f n )x) Y C f nx) 1 + d/r )F 1 M n θ n )) L 1 R d ;L X,Y )), where fnx) f = su nx z) X z R d 1+ θ k is of Peetre tye. Moreover, it is shown in the roof of [1, z d/r Lemma 2.1] and [49, Theorem 1.6.2] that fnx) C M f n r X x))1/r for all x and r >, where M is as before the Hardy-Littlewood maximal oerator. Now from [36, Proosition 2.2] we obtain that M f n r X) n L /r R d,w;l q ) C f n ) n L R d,w;l X)). q The result follows from taking L R d, w; l q )-norms in 5.1). For m L R d ; L X, Y )), let the oerator T m : S R d ; X) S R d ; X) be given by T m f = F 1 m f). The following result gives a sufficient condition on m for the extension of T m to weighted Besov and Triebel-Lizorkin saces. Proosition 5.2. Let X, Y be Banach saces, 1, ), q [1, ], w A and s R. Let r, min{, q}) be such that w A /r. Let A {F, B}. Assume m C R d ; L X, Y )) is such that D α m grows at most olynomially at infinity for all α N d and satisfies 5.2) su α d+ d/r +1 su 1 + ξ ) α D α mξ) L X,Y ) = K m <. ξ R d Then T m = F 1 mf extends to a continous oerator from A s,qr d, w; X) to A s,qr d, w; Y ). The oerator norm of T m is bounded by CK m, where C does not deend on m. Remark 5.3. i) In the scalar case Fourier multilier results of the above tye can be found in [49]. ii) An extensive treatment of oerator-valued Fourier multiliers on Besov saces can be found in [4, 21, 26]. iii) The condition on m guarantees that m ˆf and hence T m f is a well-defined element of S R d ; X) for all f S R d ; X), see [43, Section 3]. In the sequel we only consider symbols which satisfy these requirements. One can see that it suffices to have that m C d+ d/r +1 R d ; L X, Y )) and that the above Mihlin condition holds true. iv) Results on oerator-valued Fourier multiliers in Triebel-Lizorkin sace can be found in [1]. Note that in the unweighted situation in [1, Proosition 3.6] the imortant case q = was excluded, due to the fact that S R d ; X) is not dense in the corresonding function sace. Proof of Proosition 5.2. Let us consider the case of F -saces. Let ϕ n ) n ΦR d ) and ϕ 1 =. Since f = n= S nf, for f F,qR s d, w; X) we have F 1 mf f F s,q R d,w;y ) [ F 1 ϕ n mf ] 2 sn F 1 ϕ n+l F f) n L R d,w;l q Y )). l= 1,,1

15 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES 15 For fixed l, let F 1 ϕ n+l F. Then su f n { ξ 2 n+1 }. Since ϕ k 2 k+1 ) = ϕ 1 4 ), in order to aly Lemma 5.1 we have to show that 5.3) su 1 + d/r )F 1 ϕ 1 4 )m2 k+1 )) L1 R d ;L X,Y )) <. k Let j be the smallest even number which satisfies j > d + d r. Using that F 1 : L 1 L is bounded, we get 5.4) 1 + j )F 1 ϕ 1 4 )m2 k+1 )) L R d ;L X;Y )) 1 + ) j/2 )[ ϕ 1 4 )m2 k+1 )] L1 R d ;L X;Y )). By the Leibniz rule, in the above norm the derivatives consists of a finite linear combination of terms of the form g α,β := D α ϕ 1 4 )2 β k+1) D β m)2 k+1 ), where α, β are multiindices with α + β j. Since ϕ 1 4 ) is suorted in the unit ball, it follows from 5.2) that each g α,β is bounded indeendently of k. Hence the right-hand side of 5.4) is bounded by ck m. Hence 1 + d/r )F 1 ϕ 1 4 )m2 k+1 )) L 1 R d ;L X,Y )) ck m 1 + j+ d r ) L 1 R d ) = CK m, and 5.3) follows. The case of a B-sace can be treated in the same way, using Lemma 5.1 on each { ξ 2 k+1 } searately. We record an imortant consequence of the multilier result. For a definition and roerties of the H -calculus of sectorial oerators we refer to [3]. Sectoriality is defined in Section 2.2. Corollary 5.4. Let X be a Banach sace, 1, ), q [1, ], w A and s R. Let A {F, B}. Then the following assertions hold. 1) The realization of t with domain A s+1,q R, w; X) on A s,qr, w; X) is sectorial with sectral angle equal to π 2 and has a bounded H -calculus on each sector Σ θ with θ > π 2. 2) The realization of with domain A s+2,q R, w; X) on A s,qr, w; X) is sectorial with sectral angle equal to zero and has a bounded H -calculus on each sector Σ θ with θ >. Proof. As in [3, Examle 1.2] one can define the holomorhic functional calculus via Fourier transform and show its boundedness by using Proosition 5.2. Note that any symbol m arising in this context is smooth and satisfies su ξ R d 1 + ξ ) α D α mξ) < for any multiindex α. For the exlicit descrition of the domain is a direct consequence of the lifting roerty of the weighted F - and B-saces see [36, Proosition 3.9]). For t one uses the lifting roerty and Proosition Characterization by differences For a weight w and an integer m 1 define m ) m m h fx) = 1) l fx + m l)h), x, h R d. l l= For f L R d, w; X), let further [f] m) F,q s Rd,w;X) = t sq t d with obvious modifications if q =, and h t ) q dt ) 1/q m h f X dh LRd,w), t f m) F s,q Rd,w;X) := f L R d,w;x) + [f] m) F s,q Rd,w;X). We also write f F s,q R d,w;x) for f m) if there is no danger of confusion. Note that if F,q s Rd,w;X) q = 1, then Fubini s theorem yields [f] m) F,q s Rd,w;X) = c d h s d m h f X dh R d L R d,w)

16 16 MARTIN MEYRIES AND MARK VERAAR One can extend a well-known result on the equivalence of norms to the weighted case cf. [44, Proosition 6], [49, Section 2.5.1] and [52, Theorem 6.9]). Proosition 6.1. Let X be a Banach sace, s >, 1, ), q [1, ] and w A. Let m 1 be an integer such that m > s. Then there is a constant C > such that for all f L R d, w; X) 6.1) C 1 f F s,q R d,w;x) f m) F s,q Rd,w;X) C f F s,q Rd,w;X), whenever one of these exressions is finite. To state a similar result for Besov saces, for f L R d, w; X) let [f] m) B,q s Rd,w;X) = t sq t d m h f X dh again with obvious modifications if q =, and Then the following holds true. h t q L R d,w) f m) B s,q Rd,w;X) := f L R d,w;x) + [f] m) B s,q Rd,w;X). dt ) 1/q, t Proosition 6.2. Let X be a Banach sace, s >, 1, ), q [1, ] and w A. Let m N be such that m > s. There is a constant C > such that for all f L R d, w; X) 6.2) C 1 f B s,q R d,w;x) f m) B s,q Rd,w;X) C f B s,q Rd,w;X), whenever one of these exressions is finite. Remark 6.3. Define the L R d, w; X)-modulus of smoothness as In the unweighted case w 1, the norm ω,wf, m t) := su m h f L R,w;X), t >. d h t f m) B s,q Rd,w;X) := f L R d,w;x) + t sq ω,wf, m t) q dt ) 1/q, t defines an equivalent norm on B,qR s d, w; X) if m > s modification if q = ). We do not know if this extends to the weighted setting. However, by Minkowski s inequality one has t d m h f X dh t d m L R d h f L,w) R d,w;x) dh C su m h f L R,w;X). d h t h t h t Therefore, one always has that f m) B s,q Rd,w;X) C f m) B s,q Rd,w;X). 7. Mixed derivative embeddings 7.1. The general case. For fractional owers of sectorial oerators we refer to Section 2.2. Theorem 7.1. Let X be a Banach sace, 1, ), q [1, ], w A, s R and α >. Let A be a sectorial oerator on X with sectral angle φ A < π, and let A {F, B}. Then for all θ [, 1] one has Remark 7.2. A s+α,q R d, w; X) A s,qr d, w; DA)) A s+θα,q R d, w; DA 1 θ )). i) In the literature embeddings as in the above theorem are often roved using the so-called mixed derivative theorem due to [48]. We give a direct roof using Proosition 5.2. Embeddings of this tye are widely used in the context of arabolic evolution equations, in articular when inhomogeneous boundary conditions are considered, see e.g. [15, 16, 18, 28, 35]. ii) For d = 1, corresonding results for intersection saces on the half-line with H- and W - regularity are roved in [16] in the unweighted case, and in [34] for ower weights wt) = t γ with γ [, 1).

17 TRACES AND EMBEDDINGS OF ANISOTROPIC FUNCTION SPACES 17 Proof of Theorem 7.1. Without loss of generality we may assume that A is invertible. For all α > the realization of 1 ) α/2 on A s,qr d, w; X) is sectorial with sectral angle equal to zero by Corollary 5.4. As in the roof of Corollary 5.4, it is a consequence of the lifting roerty of the underlying saces that the domain of this oerator equals A s+α,q R d, w; X). It is further straightforward to see that the ointwise) realization of A with domain A s,qr d, w; DA)) on A s,qr d, w; X) is sectorial with sectral angle at most φ A. Ste 1. We show that 1 ) α/2 + A is an isomorhic maing A s+α,q R d, w; X) A s,qr d, w; DA)) A s,qr d, w; X). The boundedness of 1 ) α/2 + A is clear. A s,qr d, w; X) and consider the equation For the boundedness of its inverse, let f 7.1) 1 ) α/2 + A)u = f in S R d ; X) One can check that for all β N d the symbol mξ) = A1 + ξ 2 ) α/2 + A) 1 satisfies su 1 + ξ ) β D β mξ) L X) C β <. ξ R d Hence the unique solution u S R d ; DA)) of 7.1) is given by u = F ) α/2 +A) 1 F f. From Proosition 5.2 we obtain that Au A s,q R d,w;x) C f A s,q R d,w;x). Since A is invertible, we conclude u A s,q R d,w;da)) C f A s,q R d,w;x). Moreover, using this estimate and 7.1), it follows that 1 ) α/2 u A s,q R d,w;x) C f A s,q R d,w;x). Hence also u A s+α,q R d,w;x) C f A s,q,w;x). This comletes the roof of the claim. Rd Ste 2. By the invertibility of the oerator, u 1 ) α/2 + A)u A s,q R d,w;x) defines an equivalent norm on A,q s+α R d, w; X) A s,qr d, w; DA)). Further, for all θ, 1) we have that u 1 ) θα/2 A 1 θ u A s,q R d,w;x) defines an equivalent norm on A s+θα,q R d, w; DA 1 θ )). To show the asserted embedding it thus suffices to rove that 1 ) θα/2 A 1 θ 1 ) α/2 + A ) 1 is bounded on A s,qr d, w; X). As above we can rewrite this oerator into the form F 1 mf, where now the symbol m is given by mξ) = 1 + ξ 2 ) θα/2 A 1 θ 1 + ξ 2 ) α/2 + A ) 1. By [32, Proosition ], for all λ 1 and x X we obtain A 1 θ λ + A) 1 x X C λ + A) 1 x 1 θ DA) λ + A) 1 x θ X C λ θ x X. Using this, one can verify the conditions of Proosition 5.2 for the symbol m.

18 18 MARTIN MEYRIES AND MARK VERAAR 7.2. Refined embeddings. In the case where X and DA) are F -saces we can imrove the above embeddings within the inner regularity scale. The next result includes in articular Theorem 1.3. Theorem 7.3. Let X be a Banach sace, d, m N,,, 1, 2 1, ), q [1, ], s 1, s 2 R, α 1, α 2 > and θ, 1) with 1 = θ + 1 θ. 2 1 Let w A on R d and w, w 1, w 2 A on R m with w 2 = w 1 θ w1, θ and let A {F, B}. Let the saces X, Y and F be given by X = A s1+α1,q R d, w; F s2, R m, w ; X) ), Y = A s1,q R d, w; F s2+α2 1, R m, w 1 ; X) ). F = A r1,q R d, w; F r2 2,1 Rm, w 2 ; X) ), with r 1 = s 1 + θα 1 and r 2 = s θ)α 2. Then X Y F, and there is a constant C such that for all f X Y f F C f θ X f 1 θ Y. In articular, note that one can take = 1 = 2 and w = w 1 = w 2 in the above result. Proof. We write x 1 R d, x 2 R m and fx 1, x 2 ). We have ) f F = x 1 2 ns1+θα1) S n fx 1, ) s F 2 +1 θ)α 2 2,1 R m,w 2;X) n l q L R d,w) The Gagliardo-Nirenberg tye inequality for F -saces from [36, Proosition 5.1] imlies that S n fx 1, ) C S s F 2 +1 θ)α 2 2,1 R m,w 2;X) nfx 1, ) θ F s 2, R m,w S ;X) nfx 1, ) 1 θ Therefore, using Hölder s inequality in l q and in L R d, w), we get [ ) θ ) ] 1 θ f F C 2 ns1+α1) S n f s2 F, R m,w ;X) 2 ns1 S n f s F 2 +α 2 1, R m,w 1;X) ) C 2 ns1+α1) S n f s F 2, R m,w ;X) n ) 2 ns1 S n f s F 2 +α 2 1, R m,w 1;X) θ L R d,w;l q ) n 1 θ L R d,w;l q ). F s 2 +α 2 1, R m,w 1;X) = C f θ X f 1 θ Y.. n L R d,w;l q ) This shows the asserted inequality. The embedding follows from Young s inequality. The the case of A = B is treated in the same way. Remark 7.4. i) The above result imroves Theorem 7.1 not only with resect to the microscoic arameter in the inner scale. It also gives a multilicative estimate with owers corresonding to θ instead of an additive estimate, to which Young s inequality with ε can alied. ii) For w A one can use the elementary embeddings 2.4) and 2.5) between H-, W - and F -saces to obtain a variety of embeddings as above with ossibly different regularities for the inner saces. For instance, it follows that for all, q 1, ) and s R B s+α, R; L q R d ; X) ) B, s R; B β q,q R d ; X) ) B s+θα R; H 1 θ)β,q R d ; X) ) for an arbitrary Banach sace X. iii) Combining the theorem with real interolation techniques as in [16, 28, 34] yields mixed derivative embeddings with differing saces also for the outer regularities. Taking into account embeddings based on tye and cotye of the underlying saces as in [54, Proosition 3.1] gives even more flexibility.,