Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

Maximal regularity with weights for parabolic problems with inhomogeneous boundary

conditions

Journal of Evolution Equations

Citation (APA)

Lindemulder, N. (2019). Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions. Journal of Evolution Equations, 20 (2020)(1), 59–108. https://doi.org/10.1007/s00028-019-00515-7

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1424-3199/20/010059-50, published online June 6, 2019 https://doi.org/10.1007/s00028-019-00515-7

Journal of Evolution Equations

Maximal regularity with weights for parabolic problems with

inhomogeneous boundary conditions

Nick Lindemulder

Abstract. In this paper, we establish weighted Lq–Lp-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the initial-boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel–Lizorkin and Besov type, whose trace theory is also subject of study.

1. Introduction

This paper is concerned with weighted maximal Lq–Lp-regularity for vector-valued parabolic initial-boundary value problems of the form

∂tu(x, t) + A(x, D, t)u(x, t) = f (x, t), x ∈ O, t ∈ J,

Bj(x, D, t)u(x, t) = gj(x, t), x∈ ∂O, t ∈ J, j = 1, . . . , n, u(x, 0) = u0(x), x∈ O.

(1)

Here, J is a finite time interval,O ⊂ Rdis a smooth domain with a compact bound-ary∂O and the coefficients of the differential operator A and the boundary operators B1, . . . , Bn are B(X)-valued, where X is a UMD Banach space. One could for

in-stance take X = CN_{, describing a system of N initial-boundary value problems.}

Our structural assumptions onA, B1, . . . , Bn are an ellipticity condition and a

con-dition of Lopatinskii–Shapiro type. For homogeneous boundary data (i.e., gj = 0, j = 1, . . . , n), these problems include linearizations of reaction–diffusion systems and of phase field models with Dirichlet, Neumann and Robin conditions. However, if one wants to use linearization techniques to treat such problems with nonlinear boundary conditions, it is crucial to have a sharp theory for the fully inhomogeneous problem.

Mathematics Subject Classification: Primary 35K50, 46E35, 46E40; Secondary 42B15, 42B25 Keywords: Anisotropic spaces, Besov, Bessel potential, Inhomogeneous boundary conditions, Maximal regularity, Mixed-norms, Parabolic initial-boundary value problems, Sobolev, Traces, Triebel–Lizorkin, Vector-valued, Weights.

The author is supported by the Vidi subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).

During the last 25 years, the theory of maximal regularity turned out to be an im-portant tool in the theory of nonlinear PDEs. Maximal regularity means that there is an isomorphism between the data and the solution of the problem in suitable function spaces. Having established maximal regularity for the linearized problem, the non-linear problem can be treated with tools as the contraction principle and the implicit function theorem. Let us mention [7,15] for approaches in spaces of continuous func-tions, [1,45] for approaches in Hölder spaces and [3,5,13,14,24,53,55] for approaches in Lp-spaces (with p∈ (1, ∞)).

As an application of his operator-valued Fourier multiplier theorem, Weis [65] characterized maximal Lp-regularity for abstract Cauchy problems in UMD Banach spaces in terms of anR-boundedness condition on the operator under consideration. A second approach to the maximal Lp-regularity problem is via the operator sum method, as initiated by Da Prato and Grisvard [16] and extended by Dore and Venni [23] and Kalton & Weis [37]. For more details on these approaches and for more information on (the history of) the maximal Lp-regularity problem in general, we refer to [17,39].

In the maximal Lq–Lp-regularity approach to (1), one is looking for solutions u in the “maximal regularity space”

Wq1(J; Lp(O; X)) ∩ Lq(J; W2np (O; X)). (2)

To be more precise, problem (1) is said to enjoy the property of maximal Lq–Lp -regularity if there exists a (necessarily unique) space of initial-boundary dataDi_.b.⊂ Lq(J; Lp(∂O; X))n× Lp(O; X) such that for every f ∈ Lq(J; Lp(O; X)) it holds that (1) has a unique solution u in (2) if and only if(g = (g1, . . . , gn), u0) ∈ Di_.b.. In

this situation, there exists a Banach norm onDi.b., unique up to equivalence, with Di.b.→ Lq(J; Lp(∂O; X))n⊕ Lp(O; X),

which makes the associated solution operator a topological linear isomorphism be-tween the data space Lq(J; Lp(O; X))⊕Di.b.and the solution space Wq1(J; Lp(O; X))

∩ Lq_{(J; W}2n

p (O; X)). The maximal Lq–Lp-regularity problem for (1) consists of

es-tablishing maximal Lq–Lp-regularity for (1) and explicitly determining the space Di.b..

The maximal Lq–Lp-regularity problem for (1) was solved by Denk, Hieber & Prüss [18], who used operator sum methods in combination with tools from vector-valued harmonic analysis. Earlier works on this problem are [40] (q = p) and [64] ( p ≤ q) for scalar-valued second-order problems with Dirichlet and Neumann boundary conditions. Later, the results of [18] for the case that q = p have been extended by Meyries & Schnaubelt [48] to the setting of temporal power weightsv_μ(t) = tμ,μ ∈ [0, q −1); also see [47]. Works in which maximal Lq–Lp-regularity of other problems with inhomogeneous boundary conditions are studied, include [20–22,24,48] (the case q= p) and [50,61] (the case q= p).

It is desirable to have maximal Lq–Lp-regularity for the full range q, p ∈ (1, ∞), as this enables one to treat more nonlinearities. For instance, one often requires large

q and p due to better Sobolev embeddings, and q = p due to scaling invariance of PDEs (see, e.g., [30]). However, for (1) the case q = p is more involved than the case q = p due to the inhomogeneous boundary conditions. This is not only reflected in the proof, but also in the space of initial-boundary data ( [18, Theorem 2.3] versus [18, Theorem 2.2]). Already for the heat equation with Dirichlet boundary conditions, the boundary data g have to be in the intersection space

F1− 1 2 p q,p (J; Lp(∂O)) ∩ Lq(J; B 2₋1 p p,p (∂O)), (3)

which in the case q = p coincides with W1− 1 2 p

p (J; Lp(∂O)) ∩ Lp(J; W 2−1_p

p (∂O));

here F_qs_,p denotes a Triebel–Lizorkin space and W_ps = Bs_p,p a non-integer order Sobolev–Slobodeckii space.

In this paper, we will extend the results of [18,48], concerning the maximal Lq– Lp_{-regularity problem for (1), to the setting of power weights in time and in space}

for the full range q, p ∈ (1, ∞). In contrast to [18,48], we will not only view spaces (2) and (3) as intersection spaces, but also as anisotropic mixed-norm function spaces on J × O and J × ∂O, respectively. Identifications of intersection spaces of type (3) with anisotropic mixed-norm Triebel–Lizorkin spaces have been considered in a previous paper [43], all in a generality including the weighted vector-valued setting. The advantage of these identifications is that they allow us to use weighted vector-valued versions of trace results of Johnsen & Sickel [36]. These trace results will be studied in their own right in the present paper.

The weights we consider are the power weights

vμ(t) = tμ (t ∈ J) and w_γ∂O(x) = dist( · , ∂O)γ (x ∈ O), (4) where μ ∈ (−1, q − 1) and γ ∈ (−1, p − 1). These weights yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary, which is nicely illustrated by the result (see Example3.7) that the corresponding version of (3) becomes

F1− 1 2 p(1+γ ) q,p (J, vμ; Lp(∂O)) ∩ Lq(J, vμ; B 2−1_p(1+γ ) p,p (∂O)).

Note that one requires less regularity of g by increasingγ .

The idea to work in weighted spaces equipped with weights like (4) has already proven to be very useful in several situations. In an abstract semigroup setting, tempo-ral weights were introduced by Clément & Simonett [15] and Prüss & Simonett [54], in the context of maximal continuous regularity and maximal Lp-regularity, respec-tively. Other works on maximal temporally weighted Lp-regularity are [38,41] for quasilinear parabolic evolution equations and [48] for parabolic problems with inho-mogeneous boundary conditions. Concerning the use of spatial weights, we would like to mention [9,46,52] for boundary value problems and [2,10,25,56,62] for problems with boundary noise.

The paper is organized as follows. In Sect.2we discuss the necessary preliminaries, in Sect.3we state the main result of this paper, Theorem3.4, in Sect.4we establish the necessary trace theory, in Sect.5we consider a Sobolev embedding theorem, and in Sect.6we finally prove Theorem3.4.

2. Preliminaries

2.1. Weighted mixed-norm Lebesgue spaces

A weight onRdis a measurable functionw : Rd −→ [0, ∞] that takes its values almost everywhere in(0, ∞). We denote by W(Rd) the set of all weights on Rd. For p ∈ (1, ∞) we denote by Ap = Ap(Rd) the class of all Muckenhoupt Ap-weights,

which are all the locally integrable weights for which the Ap-characteristic[w]Ap is

finite. Here, [w]Ap = sup Q  Q w   Q w−p/p p/p

with the supremum taken over all cubes Q⊂ Rdwith sides parallel to the coordinate axes. We furthermore set A_∞ :=_p∈(1,∞)Ap. For more information on

Mucken-houpt weights we refer to [31].

Important for this paper are the power weights of the formw = dist( · , ∂O)γ, where O is a C∞_{-domain inR}d_{and whereγ ∈ (−1, ∞). If γ ∈ (−1, ∞) and p ∈ (1, ∞),}

then (see [27, Lemma 2.3] or [52, Lemma 2.3]) w_γ∂O:= dist( · , ∂O)γ _{∈ A}

p ⇐⇒ γ ∈ (−1, p − 1). (5)

For the important model problem caseO = Rd₊, we simply writew_γ := w∂R

d

+

γ =

dist( · , ∂Rd₊)γ.

Replacing cubes by rectangles in the definition of the Ap-characteristic[w]Ap ∈

[1, ∞] of a weight w gives rise to the Ar ec

p -characteristic[w]Ar ec

p ∈ [1, ∞] of w.

We denote by Ar ecp = Ar ecp (Rd) the class of all weights with [w]Ar ec

p < ∞. For

γ ∈ (−1, ∞) it holds that wγ ∈ Ar ecp if and only ifγ ∈ (−1, p − 1).

Let d= |d |1= d1+ · · · + dlwithd = (d1, . . . , dl) ∈ (Z≥1)l. The decomposition

Rd_{= R}d1× . . . × Rdl.

is called thed -decomposition ofRd. For x ∈ Rdwe accordingly write x = (x1, . . . , xl)

and xj = (xj,1, . . . , xj_,dj), where xj ∈ Rdj and xj,i ∈ R ( j = 1, . . . , l; i =

1, . . . , dj). We also say that we view Rd as beingd -decomposed. Furthermore, for

each k∈ {1, . . . , l} we define the inclusion map

and the projection map

πk= π[d ;k]: Rd−→ Rdk, x = (x1, . . . , xl) → xk.

Suppose thatRd is d -decomposed as above. Let p = (p1, . . . , pl) ∈ [1, ∞)l

andw = (w1, . . . , wl) ∈

l

j=1W(Rdj). We define the weighted mixed-norm space Lp,d(Rd, w) as the space of all f ∈ L0(Rd) satisfying

|| f ||Lp,d_(Rd_,w):= ˆ Rdl. . . ˆ Rd1| f (x)| p1w 1(x1)dx1 p2/p1 . . . wl(xl)dxl 1/pl < ∞. We equip Lp,d_(Rd_{, w) with the norm || · ||}

Lp,d_(Rd_,w), which turns it into a Banach

space. Given a Banach space X , we denote by Lp,d(Rd, w; X) the associated Bochner space

Lp,d(Rd, w; X) := Lp,d(Rd, w)[X] = { f ∈ L0(Rd; X) : || f ||X ∈ Lp,d(Rd, w)}.

2.2. Anisotropy

Suppose thatRd isd -decomposed as in Sect.2.1. Given a ∈ (0, ∞)l, we define the(d , a)-anisotropic dilation δ(d ,a)_λ onRdbyλ > 0 to be the mapping δ(d ,a)_λ onRd given by the formula

δ_λ(d ,a)x:= (λa1_x

1, . . . , λalxl), x∈ Rd.

A(d , a)-anisotropic distance function on Rd is a function u : Rd −→ [0, ∞) satisfying

(i) u(x) = 0 if and only if x = 0.

(ii) u(δ_λ(d ,a)x) = λu(x) for all x ∈ Rdandλ > 0.

(iii) There exists a c> 0 such that u(x + y) ≤ c(u(x) + u(y)) for all x, y ∈ Rd. All (d , a)-anisotropic distance functions on Rd are equivalent: Given two (d , a)-anisotropic distance functions u andv on Rd_{, there exist constants m, M > 0 such}

that mu(x) ≤ v(x) ≤ Mu(x) for all x ∈ Rd

In this paper, we will use the(d , a)-anisotropic distance function | · |_d,a: Rd_−→

[0, ∞) given by the formula |x|d,a:= ⎛ ⎝l j=1 |xj|2/aj ⎞ ⎠ 1/2 (x ∈ Rd_). 2.3. Fourier multipliers

Let X be a Banach space. The space of X -valued tempered distributions onRd is defined asS(Rd; X) := L(S(Rd); X); for the theory of vector-valued distributions

we refer to [4] (and [3, Section III.4]). We write L1_(Rd_{; X) := F−1L}1_(Rd_{; X) ⊂} S_(Rd_{; X). To a symbol m ∈ L}∞_(Rd_{; B(X)), we associate the Fourier multiplier}

operator

Tm : L1(Rd; X) −→ L1(Rd; X), f → F−1[m ˆf].

Given p ∈ [1, ∞)l andw ∈ l_j=1A_∞(Rdj), we call m a Fourier multiplier on

Lp,d(Rd, w; X) if Tmrestricts to an operator on L1(Rd; X) ∩ Lp,d(Rd, w; X) which

is bounded with respect to Lp,d(Rd, w; X)-norm. In this case, Tmhas a unique

exten-sion to a bounded linear operator on Lp,d(Rd, w; X) due to denseness of S(Rd; X) in Lp,d(Rd, w; X), which we still denote by Tm. We denote by Mp,d ,w(X) the

set of all Fourier multipliers m ∈ L∞(Rd; B(X)) on Lp,d(Rd, w; X). Equipped with the norm||m||_{Mp,d ,w(X)} := ||Tm||_B(Lp,d_(Rd_,w;X),Mp,d ,w(X) becomes a

Ba-nach algebra (under the natural pointwise operations) for which the natural inclusion

Mp,d ,w(X) → B(Lp,d(Rd, w; X)) is an isometric Banach algebra homomorphism;

see [39] for the unweighted non-mixed-norm setting.

For each a∈ (0, ∞)land N ∈ N, we define M(d ,a)_N as the space of all m∈ CN(Rd) for which ||m||_M(d ,a) N := sup_|α|≤N sup ξ∈Rd (1 + |ξ|,a)a·dα|Dαm(ξ)| < ∞.

We furthermore define RM (X) as the space of all operator-valued symbols m ∈ C1(R\{0}; B(X)) for which we have the R-bound

||m||RM(X):= R



t m[k](t) : t = 0, k = 0, 1< ∞; see, e.g., [17,33] for the notion ofR-boundedness.

If X is a UMD space, p∈ (1, ∞)l, w ∈  l j₌₁Ar ecpj (R dj), l ≥ 2, Ap(Rd), l= 1,

and a∈ (0, ∞)l, then there exists an N ∈ N for which

M(d ,a)_N → Mp,d ,w(X). (6)

If X is a UMD space, p∈ (1, ∞) and w ∈ Ap(R), then

RM (X) → Mp_,w(X). (7)

For these results, we refer to [26] and the references given there. 2.4. Function spaces

For the theory of vector-valued distributions, we refer to [4] (and [3, Section III.4]). For vector-valued function spaces, we refer to [51] (weighted setting) and the refer-ences given therein. Anisotropic spaces can be found in [6,36,42]; for the statements below on weighted anisotropic vector-valued function space, we refer to [42].

Suppose thatRd isd -decomposed as in Sect.2.1. Let X be a Banach space, and let a ∈ (0, ∞)l. For 0 < A < B < ∞, we define d_A,a_,B(Rd) as the set of all sequencesϕ = (ϕn)n∈N ⊂ S(Rd) which are constructed in the following way: given

aϕ0∈ S(Rd) satisfying

0≤ ˆϕ0≤ 1, ˆϕ0(ξ) = 1 if |ξ|d,a≤ A, ˆϕ0(ξ) = 0 if |ξ|d,a≥ B, (ϕn)n≥1 ⊂ S(Rd) is defined via the relations

ˆϕn(ξ) = ˆϕ1(δ₂(d ,a)_−n+1ξ) = ˆϕ0(δ₂(d ,a)−n ξ) − ˆϕ0(δ(d ,a)_2−n+1ξ), ξ ∈ Rd, n ≥ 1.

Observe that

suppˆϕ0⊂ {ξ | |ξ|d,a≤ B} and supp ˆϕn⊂ {ξ | 2n−1A≤ |ξ|d,a ≤ 2nB}, n ≥ 1.

We putd,a(Rd) :=_0<A<B<∞d_A,a_,B(Rd). In case l = 1 we write a(Rd) = d,a_(Rd_{), (R}d_{) =}1_(Rd_),a A,B(R d_{) =}d,a A,B(R d_{), and} A,B(Rd)=1_A,B(Rd).

Toϕ ∈ d,a(Rd), we associate the family of convolution operators (Sn)n∈N = (Snϕ)n_∈N⊂ L(S(Rd; X), OM(Rd; X)) ⊂ L(S(Rd; X)) given by

Snf = Snϕf := ϕn∗ f = F−1[ ˆϕn ˆf] ( f ∈ S(Rd; X)). (8)

Here,OM(Rd; X) denotes the space of slowly increasing X-valued smooth functions

onRd. It holds that f =∞n=0Snf inS(Rd; X), respectively, in S(Rd; X) whenever f ∈ S(Rd; X), respectively, f ∈ S(Rd; X).

Given a ∈ (0, ∞)l, p ∈ [1, ∞)l, q ∈ [1, ∞], s ∈ R, and w ∈ l_j=1A_∞(Rdj),

the Besov space Bs_p,a_,q,d(Rd, w; X) is defined as the space of all f ∈ S(Rd; X) for which

|| f ||Bs_p,a_,q,d(Rd_,w;X):= ||(2nsS_nϕf)n_∈N|| q_(N)[Lp,d_(Rd_,w)](X)< ∞

and the Triebel–Lizorkin space Fs_p,a_,q,d(Rd, w; X) is defined as the space of all f ∈ S(Rd_{; X) for which}

|| f ||Fs_p,a_,q,d(Rd_,w;X):= ||(2nsS_nϕf)n∈N||Lp,d_(Rd_,w)[ q_(N)](X)< ∞.

Up to an equivalence of extended norms onS(Rd; X), || · ||_Bs,a

p,q,d(Rd,w;X)and|| · ||Fs_p,a_,q,d(Rd_,w;X)do not depend on the particular choice ofϕ ∈ d,a(Rd).

Let us note some basic relations between these spaces. Monotonicity of q-spaces yields that, for 1≤ q0≤ q1≤ ∞,

Bs_p,a_,q 0,d(R d_{, w; X) → B}s,a p,q1,d(R d_{, w; X),} F_ps,a_,q 0,d(R d_{, w; X) → F}s,a p,q1,d(R d_{, w; X).} (9)

For > 0 it holds that

Bs_p,a_,∞,d(Rd, w; X) → Bs_p−,a_,1,d (Rd, w; X). (10) Furthermore, Minkowski’s inequality gives

Bs,a_p,min{p 1,...,pl,q},d(R d_{, w; X) → B}s,a p,q,d(R d_{, w; X) → B}s,a p,max{p1,...,pl,q},d(R d_{, w; X).} (11) Let a ∈ (0, ∞)l. A normed spaceE ⊂ S(Rd; X) is called (d , a)-admissible if there exists an N ∈ N such that

m(D) f ∈ E with ||m(D) f ||E ||m||_M(d ,a)

N || f ||E, (m, f ) ∈ OM(R d_{) × E,}

where m(D) f = F−1[m ˆf]. The Besov space Bs_p,a_,q,d(Rd_{, w; X) and the Triebel–}

Lizorkin space Fs_p,a_,q,d(Rd, w; X) are examples of (d , a)-admissible Banach spaces. To eachσ ∈ R, we associate the operators Jσ[d ; j] ∈ L(S(Rd; X)) and Jσd,a ∈ L(S_(Rd_{; X)) given by} J_σ[d ; j]f := F−1[(1 + |π_{[d ; j]}|2)σ/2 ˆf] and J_σd,af := l k=1 J_σ/a[d ;k]_k f. We callJ_σd,athe(d , a)-anisotropic Bessel potential operator of order σ .

LetE → S(Rd; X) be a Banach space. Write

Jn,d := ⎧ ⎨ ⎩α ∈ l  j=1 ι[d ; j]Ndj : |α j| ≤ nj ⎫ ⎬ ⎭, n∈  Z≥1l.

Given n ∈ Z_≥1l, s, a ∈ (0, ∞)l_{, and s ∈ R, we define the Banach spaces} Wn d[E], H s d[E], H s,a d [E] → S(R d_{; X) as follows:} Wn d[E] := { f ∈ S(Rd) : Dαf ∈ E, α ∈ Jn,d}, Hs d[E] := { f ∈ S(R d_{) : J}[d ; j] sj f ∈ E, j = 1, . . . , l}, Hs,a d [E] := { f ∈ S(Rd) : Jsd,af ∈ E},

with the norms || f ||Wn d[E]= α∈Jn,d ||Dαf||E, || f ||_Hs d[E]= l j₌₁ ||Js[d ; j]j f||E, || f ||_Hs,a d [E]= ||J d,a s f||E.

Note thatH_ds[E] → Hs_d,a[E] contractively in case that s = (s/a1, . . . , s/al).

E → F implies Wn d[E] → Wdn[F], Hsd[E] → Hds[F], H s,a d [E] → H s,a d [F]. (12) IfE → S(Rd; X) is a (d , a)-admissible Banach space for a given a ∈ (0, ∞)l, then Wn d[E] = H n d[E] = H s,a d [E], s∈ R, n = sa−1∈  Z≥1l, (13) and Hs d[E] = H s,a d [E], s> 0, s = sa−1. (14) Furthermore,

Dα ∈ B(Hs_d,a[E], Hs−a·dα,a

d [E]), s∈ R, α ∈ N

d_{. (15)}

Let a∈ (0, ∞)l, p∈ [1, ∞)l, q∈ [1, ∞], and w ∈l_j=1A_∞(Rdj). For s, s

0∈ R it holds that Bs+s0,a p_,q,d (R d_{, w; X) = H}s,a d [B s0,a p_,q,d(R d_{, w; X)],} Fs+s0,a p,q,d (R d_{, w; X) = H}s_,a d [F s0,a p,q,d(R d_{, w; X)].}

Let X be a Banach space, a∈ (0, ∞)l, p∈ (1, ∞)l,w ∈l_j=1Apj(Rdj), s ∈ R,

s∈ (0, ∞)l and n∈ (N_>0)l. We define Wn_p,d(Rd_{, w; X) := W}n d[Lp,d(Rd, w; X)], H_ps_,d(Rd, w; X) := H_ds[Lp,d(Rd, w; X)], H_ps,a_,d(Rd, w; X) := H_ds,a[Lp,d(Rd, w; X)]. If • E = Wn p,d(R d_{, w; X), n ∈ (Z} ≥1)l, n= sa−1; or • E = Hs_,a p,d(R d_{, w; X); or} • E = Ha p,d(R d_{, w; X), a ∈ (0, 1)}l_{, a= sa}−1_,

then we have the inclusions

Fs_p,a_,1,d(Rd, w; X) → E → Fs_p,a_,∞,d(Rd, w; X). (16) Theorem 2.1. [43] Let X be a Banach space, l = 2, a ∈ (0, ∞)2, p, q ∈ (1, ∞), s> 0, and w ∈ Ap(Rd1) × Aq(Rd2). Then,

F_(p,q),p,ds,a (Rd, w; X) = Fs/a2

q_,p (Rd2, w2; Lp(Rd1, w1; X)) ∩ Lq(Rd2, w2; Fs/a1

p,p (Rd1, w1; X)) (17)

with equivalence of norms.

This intersection representation is actually a corollary of a more general intersection representation in [43]. In the above form, it can also be found in [42, Theorem 5.2.35]. For the case X= C, d1= 1, w = 1, we refer to [19, Proposition 3.23].

3. The main result

3.1. Maximal Lq_μ–L_γp-regularity

In order to give a precise description of the maximal weighted Lq–Lp-regularity approach for (1), letO be either Rd₊or a smooth domain inRdwith a compact boundary ∂O. Furthermore, let X be a Banach space, let

q ∈ (1, ∞), μ ∈ (−1, q − 1) and p ∈ (1, ∞), γ ∈ (−1, p − 1), letv_μandw_γ∂Obe as in (4), put

Up,q

γ,μ:= Wq1(J, vμ; Lp(O, w∂_γO; X)) ∩ Lq(J, vμ; W2np (O, w∂_γO; X)),

(space of solutions u) Fp,q

γ,μ:= Lq(J, vμ; Lp(O, wγ∂O; X)), (space of domain inhomogeneities f )

Bp,q

μ := Lq(J, vμ; Lp(∂O; X)), (boundary space) (18)

and let n, n1, . . . , nn∈ N be natural numbers with nj ≤ 2n−1 for each j ∈ {1, . . . , n}.

Suppose that for eachα ∈ Nd, |α| ≤ 2n,

a_α ∈ D(O × J; B(X)) with a_αDα ∈ B(U_γ,μp,q, F_γ,μp,q) and that for each j ∈ {1, . . . , n} and β ∈ Nd, |β| ≤ nj,

bj,β ∈ D(∂O × J; B(X)) with bj,βtr∂ODβ ∈ B(U_γ,μp,q, B_μp,q),

where the conditions aαDα ∈ B(Uγ,μp,q, Fγ,μp,q) and bj,βtr∂ODβ ∈ B(Uγ,μp,q, Bμp,q) have

to be interpreted in the sense of bounded extension from the space of X -valued com-pactly supported smooth functions. DefineA(D) ∈ B(U_γ,μp,q, F_γ,μp,q) and B1(D), . . . , Bn (D) ∈ B(Up,q γ,μ, Bμp,q) by A(D) := |α|≤2n a_αDα, Bj(D) := |β|≤nj bj,βtr∂ODβ, j = 1, . . . , n. (19)

In the above notation, given f ∈ F_γ,μp,q and g = (g1, . . . , gn) ∈ [B p_,q

μ ]n, one can ask the question whether the initial-boundary value problem

∂tu+ A(D)u = f,

Bj(D)u = gj, j = 1, . . . , n,

trt=0u = u0.

(20)

Definition 3.1. We say that problem (20) enjoys the property of maximal Lq_μ–L_γp -regularity if there exists a (necessarily unique) linear spaceDi_.b.⊂ [B_μp,q]n× Lp(O, w∂O_γ ; X) such that (20) admits a unique solution u∈ U_γ,μp,q if and only if( f, g, u0) ∈ D = Fp,q

γ,μ×Di.b.. In this situation, we callDi.b.the optimal space of initial-boundary data andD the optimal space of data.

Remark 3.2. Let the notations be as above. If problem (20) enjoys the property of maximal Lq_μ–L_γp-regularity, then there exists a unique Banach topology on the space of initial-boundary dataDi.b.such thatDi.b.→ [Bμp,q]n×Lp(O, w_γ∂O; X). Moreover,

ifDi.b.has been equipped with a Banach norm generating such a topology, then the

solution operator S : D = Fp,q

γ,μ⊕ Di.b. −→ U_γ,μp,q, ( f, g, u0) → S ( f, g, u0) = u

is an isomorphism of Banach spaces, or equivalently, ||u||_Up,q

γ,μ || f ||Fγ,μp,q + ||(g, u0)||Di.b., u = S ( f, g, u0), ( f, g, u0) ∈ D.

The maximal Lq_μ–L_γp-regularity problem for (20) consists of establishing maximal Lq_μ–L_γp-regularity for (20) and explicitly determining the spaceDi.b.together with a

norm as in Remark3.2. As the main result of this paper, Theorem3.4, we will solve the maximal Lq_μ–L_γp-regularity problem for (20) under the assumption that X is a UMD space and under suitable assumptions on the operatorsA(D), B1(D), . . . , Bn(D).

3.2. Assumptions on(A, B1, . . . , Bn)

As in [18,48], we will pose two type of conditions on the operatorsA, B1, . . . , Bn

for which we can solve the maximal Lq_μ–L_γp-regularity problem for (20): smoothness assumptions on the coefficients and structural assumptions.

In order to describe the smoothness assumptions on the coefficients, let q, p ∈ (1, ∞), μ ∈ (−1, q − 1), γ ∈ (−1, p − 1) and put κj,γ := 1 − nj 2n − 1 2np(1 + γ ) ∈ (0, 1), j = 1, . . . , n. (21) (SD) For |α| = 2n we have aα ∈ BUC(O × J; B(X)), and for |α| < 2n we have aα ∈ L∞(O× J; B(X)). If O is unbounded, the limits a_α(∞, t) := lim_|x|→∞a_α(x, t) exist uniformly with respect to t∈ J, |α| = 2n.

(SB) For each j ∈ {1, . . . , m} and |β| ≤ nj, there exist sj,β ∈ [q, ∞) and rj,β ∈

[p, ∞) with κj,γ > 1 sj,β + d− 1 2nrj,β + |β| − nj 2n and μ > q sj,β − 1 such that bj,β ∈ Fsκ_j,βj,γ,p(J; Lrj,β(∂O; B(X))) ∩ Lsj,β(J; B 2nκj,γ r_j,β,p(∂O; B(X))).

IfO = Rd₊, the limits bj,β(∞, t) := lim|x|→∞bj,β(x, t) exist uniformly with

Remark 3.3. For the lower order parts of(A, B1, . . . , Bn), we only need aαDα,|α| <

2n, and bj_,βtr_∂ODβ,|βj| < nj, j = 1, . . . , n, to act as lower order perturbations in

the sense that there existsσ ∈ [2n − 1, 2n) such that a_αDα, respectively, bj,βtr∂ODβ

is bounded from H2nσ

q (J, vμ; Lp(O, w_γ∂O; X)) ∩ Lq(J, vμ; Hσp(O, w∂O_γ ; X))

to Lq(J, v_μ; Lp(O, w∂O_γ ; X))), respectively, Fqκ,pj,γ(J, vμ; Lp(∂O; X)) ∩ Lq(J, vμ; F2np,pκj,γ(∂O; X)). Here, the latter space is the optimal space of boundary data, see the

statement of the main result.

Let us now turn to the two structural assumptions on A, B1, . . . , Bn. For each φ ∈ [0, π), we introduce the conditions (E)φand(LS)_φ.

The condition(E)_φis parameter ellipticity. In order to state it, we denote by the subscript # the principal part of a differential operator: given a differential operator P(D) =_{|γ |≤k}p_γDγ of order k∈ N, P#(D) =



|γ |=kpγDγ.

(E)φ For all t ∈ J, x ∈ O and |ξ| = 1 it holds that σ (A#(x, ξ, t)) ⊂ φ. IfO is

unbounded, then it in addition holds thatσ (A#(∞, ξ, t)) ⊂ C+for all t ∈ J

and|ξ| = 1.

The condition(LS)φis a condition of Lopatinskii–Shapiro type. Before we can state it, we need to introduce some notation. For each x∈ ∂O, we fix an orthogonal matrix O_ν(x)that rotates the outer unit normalν(x) of ∂O at x to (0, . . . , 0, −1) ∈ Rdand define the rotated operators(Aν, Bν) by

Aν(x, D, t) := A(x, OT

ν(x)D, t), Bν(x, D, t) := B(x, Oν(x)T D, t).

(LS)φ For each t ∈ J, x ∈ ∂O, λ ∈ _π−φ andξ ∈ Rd−1with(λ, ξ) = 0 and all h∈ Xn, the ordinary initial value problem

λw(y) + Aν

#(ξ, Dy, t)w(y) = 0, y > 0 Bν

j,#(ξ, Dy, t)w(y)|y=0= hj, j = 1, . . . , n.

has a unique solutionw ∈ C∞([0, ∞); X) with limy_→∞w(y) = 0.

3.3. Statement of the main result

LetO be either Rd₊or a C∞-domain inRd_{with a compact boundary∂O. Let X be a}

Banach space, q, p ∈ (1, ∞), μ ∈ (−1, q−1), γ ∈ (−1, p−1) and n, n1, . . . , nn∈ N

natural numbers with nj ≤ 2n−1 for each j ∈ {1, . . . , n}, and κ1,γ, . . . , κn,γ ∈ (0, 1)

as defined in (21). Put Ip_,q

γ,μ := B

2n(1−1+μ_q )

p,q (O, w∂O_γ ; X), (initial data space)

Gp,q γ,μ, j := Fqκ,pj,γ(J, vμ; Lp(∂O; X)) ∩ Lq(J, vμ; F 2n_κj,γ p,p (∂O; X)), j = 1, . . . , n, Gp,q γ,μ := G1p,μ,γ,q ⊕ . . . ⊕ G p,q

n,μ,γ. (space of boundary data g) (22)

Theorem 3.4. Let the notations be as above. Suppose that X is a UMD space, that

A(D), B1(D), . . . , Bn(D) satisfy the conditions (SD), (SB), (E)_φand(LS)_φfor some φ ∈ (0,π

2), and that κj,γ = 1+μ

q for all j ∈ {1, . . . , n}. Put

Dp,q γ,μ:=  (g, u0) ∈ G_γ,μp,q ⊕ I_γ,μp,q : trt=0gj− Btj=0(D)u0= 0 when κj,γ > 1+ μ q  , whereBt_j=0(D) :=_|β|≤nj bj,β(0, · )tr∂ODβ. Then, problem (20) enjoys the prop-erty of maximal Lq_μ–L_γp-regularity withD_γ,μp,q as the optimal space of initial-boundary data, i.e., problem (20) admits a unique solution u∈ U_γ,μp,q if and only if( f, g, u0) ∈

Fp,q

γ,μ⊕ Dγ,μp,q. Moreover, the corresponding solution operatorS : F_γ,μp,q ⊕ D_γ,μp,q −→ Up_,q

γ,μis an isomorphism of Banach spaces.

Remark 3.5. The compatibility condition trt=0gj− Bt_j=0(D)u0= 0 in the definition

ofD_γ,μp,q is basically imposed when(gj, u0) → trt=0gj − Bt_j=0(D)u0 makes sense

as a continuous linear operator fromG_{γ,μ, j}p,q ⊕ I_γ,μp,q to some topological vector space V . That it is indeed a well-defined continuous linear operator fromG_{γ,μ, j}p,q ⊕ Iγ,μp,q to L0(∂O; X) when κj_,γ > 1+μ_q can be seen by combining the following two points:

(i) Supposeκj,γ >1+μ_q . Then, the condition(SB) yields bj,β ∈ Fsκjj,β,γ,p(J; Lrj,β(O;

B(X))) with κj,γ > 1+μ_q > _s1_j,β. By [49, Proposition 7.4], Fsκ_j,βj,γ,p(J; Lrj,β(O; B(X))) → BUC(J; Lrj,β(O; B(X))).

Furthermore, it holds that 2n(1 −1+μ_q ) > nj+1+γ_q , so each tr∂ODβ,|β| ≤ nj,

is a continuous linear operator fromI_γ,μp,q to B2n(1− 1+μ

q )−nj−1+γp

p,q (∂O; X) →

Lp(∂O; X) by the trace theory from Sect.4.1. Therefore,Bt_j=0(D) =_|β|≤n

j

bj_,β(0, · )tr_∂ODβ makes sense as a continuous linear operator from I_γ,μp,q to L0(∂O; X).

(ii) Supposeκj,γ > 1+μ_q . The observation that

Gp,q

γ,μ, j → Fqκ_,pj,γ(J, vμ; Lp(∂O; X))

in combination with the trace theory from Sect.4.1yields that trt=0is a

well-defined continuous linear operator fromG_{γ,μ, j}p,q to Lp(∂O; X) → L0(∂O; X). Remark 3.6. The C∞-smoothness on ∂O in Theorem3.4 can actually be reduced to C2n-smoothness, which could be derived from the theorem itself by a suitable coordinate transformation.

Notice the dependence of the space of initial-boundary data on the weight parame-tersμ and γ . For fixed q, p ∈ (1, ∞), we can roughly speaking decrease the required smoothness (or regularity) of g and u0by increasingγ and μ, respectively.

weights make it possible to solve (20) for more initial-boundary data (compared to the unweighted setting). On the other hand, by choosingμ and γ closer to −1 (depending on the initial-boundary data), we can find more information about the behavior of u near the initial-time and near the boundary, respectively.

The dependence on the weight parametersμ and γ is illustrated in the following example of the heat equation with Dirichlet and Neumann boundary conditions: Example 3.7. Let N∈ N and let p, q, γ, μ be as above.

(i) The heat equation with Dirichlet boundary condition: If 2−2_q(1 + μ) = 1_p(1 + γ ), then the problem

∂tu− u = f,

tr_∂Ou = g, u(0) = u0,

has a unique solution u∈ W_q1(J, v_μ; Lp(O, w∂O_γ ; CN))∩Lq(J, v_μ; W_p2(O, w_γ∂O; CN_{)) if and only the data ( f, g, u}

0) satisfy: • f ∈ Lq_{(J, v} μ; Lp(O, wγ∂O; CN)); • g ∈ F1−2 p1(1+γ ) q,p (J, vμ; Lp(∂O; CN)) ∩ Lq(J, vμ; F 2−1_p(1+γ ) p,p (∂O; CN)); • u0∈ B 2−2_q(1+μ) p_,q (O, w_γ∂O; CN); • trt=0g= tr∂Ou0when 2−2_q(1 + μ) > 1_p(1 + γ ).

(ii) The heat equation with Neumann boundary condition: If 1−2_q(1 + μ) = 1_p(1 + γ ), then the problem

∂tu− u = f, ∂νu = g, u(0) = u0,

has a unique solution u ∈ W_q1(J, v_μ; Lp(O, w_γ∂O; CN)) ∩ Lq(J, v_μ; W2_p (O, w∂O

γ ; CN)) if and only the data ( f, g, u0) satisfy:

• f ∈ Lq_{(J, v} μ; Lp(O, wγ∂O; CN)); • g ∈ F12− 1 2 p(1+γ ) q,p (J, vμ; Lp(∂O; CN)) ∩ Lq(J, vμ; F 1−1_p(1+γ ) p,p (∂O; CN)); • u0∈ B 2₋2_q(1+μ) p,q (O, w_γ∂O; CN); • trt₌₀g= tr_∂Ou0when 1−2_q(1 + μ) > 1_p(1 + γ ). 4. Trace theory

In this section, we establish the necessary trace theory for the maximal Lq_μ–L_γp -regularity problem for (20).

4.1. Traces of isotropic spaces

In this subsection, we state trace results for the isotropic spaces, for which we refer to [44] (also see the references there). Note that these are of course special cases of the more general anisotropic mixed-norm spaces, for which trace theory (for the model problem case of a half-space) can be found in the next subsections and in [42].

The following notation will be convenient: ∂ Bs

p,q,γ(∂O; X) := B s−1+γ_p

p,q (∂O; X) and ∂ Fsp,q,γ(∂O; X) := F s−1+γ_p

p,p (∂O; X).

Proposition 4.1. Let X be a Banach space,O ⊂ Rd_eitherRd

+or a C∞-domain inRd with a compact boundary∂O, A ∈ {B, F}, p ∈ [1, ∞), q ∈ [1, ∞], γ ∈ (−1, ∞) and s>1+γ_p . Then

S(Rd_{; X) −→ S(∂O; X), f → f} |∂O,

uniquely extends to a retraction tr_∂OfromA_ps_,q(Rd, w_γ∂O; X) onto ∂A_ps_,q,γ(∂O; X). There is a universal coretraction in the sense that there exists an operator ext_∂O ∈ L(S_{(∂O; X), S}_(Rd_{; X)) (independent of A , p, q, γ, s) which restricts to a} core-traction for the operator tr_∂O ∈ B(A_ps_,q(Rd, w∂O_γ ; X), ∂A_ps_,q,γ(∂O; X)). The same statements hold true withRdreplaced byO.

Remark 4.2. Recall thatS(Rd; X) is dense in Aps_,q(Rd, w∂O_γ ; X) for q < ∞ but not

for q= ∞. For q = ∞ uniqueness of the extension follows from the trivial embedding As

p,∞(Rd, wγ∂O; X) → Bps−,1(Rd, wγ∂O; X),  > 0.

Corollary 4.3. Let X be a Banach space,O ⊂ RdeitherRd₊or a C∞-domain inRd with a compact boundary∂O, p ∈ (1, ∞), γ ∈ (−1, p − 1), n ∈ N_>0and s>1+γ_p . Then

S(Rd_{; X) −→ S(∂O; X), f → f} |∂O, uniquely extends to retractions tr_∂Ofrom W_pn(Rd, w∂O_γ ; X) onto Fn−

1+γ

p

p,p (∂O; X) and from Ws_p(Rd, w∂O_γ ; X) onto Fs−

1+γ

p

p,p (∂O; X). The same statement holds true with Rd replaced byO.

4.2. Traces of intersection spaces

For the maximal Lqμ–Lγp-regularity problem for (20), we need to determine the temporal and spatial trace spaces of Sobolev and Bessel potential spaces of intersection type. As the temporal trace spaces can be obtained from the trace results in [50], we will focus on the spatial traces.

By the trace theory of the previous subsection, the trace operator tr_∂O can be de-fined pointwise in time on the intersection spaces in the following theorem. It will be convenient to use the notation tr_∂O[E] = F to say that tr_∂Ois a retraction fromE onto F.

Theorem 4.4. LetO be either Rd₊or a C∞-domain inRdwith a compact boundary ∂O. Let X be a Banach space, Y a UMD Banach space, p, q ∈ (1, ∞), μ ∈ (−1, q−1) andγ ∈ (−1, p − 1). If n, m ∈ Z_>0and r, s ∈ (0, ∞) with s >1+γ_p , then

tr_∂O 

W_qn(J, v_μ; Lp(O, w∂O_γ ; X)) ∩ Lq(J, v_μ; W_pm(O, w∂O_γ ; X))  = Fn−mn 1+γ p q_,p (J, vμ; Lp(∂O; X)) ∩ Lq(J, vμ; F m−1+γ_p p_,p (∂O; X)) (23) and tr_∂O 

Hqr(J, vμ; Lp(O, w∂O_γ ; Y )) ∩ Lq(J, vμ; Hsp(O, w∂O_γ ; Y ))

 = Fr−rs1+γp q,p (J, vμ; Lp(∂O; Y )) ∩ Lq(J, vμ; F s₋1+γ_p p,p (∂O; Y )). (24)

The main idea behind the proof of Theorem4.4is, as in [60], to exploit the inde-pendence of the trace space of a Triebel–Lizorkin space on its microscopic parameter. As in [60], our approach does not require any restrictions on the Banach space X .

The UMD restriction on Y comes from the localization procedure for Bessel po-tential spaces used in the proof, which can be omitted in the caseO = Rd₊. This localization procedure for Bessel potential spaces could be replaced by a localiza-tion procedure for weighted anisotropic mixed-norm Triebel–Lizorkin spaces, which would not require any restrictions on the Banach space Y . However, we have chosen to avoid this as localization of such Triebel–Lizorkin spaces has not been considered in the literature before, while we do not need that generality anyway. For localization in the scalar-valued isotropic non-mixed-norm case, we refer to [44].

Proof of Theorem4.4. By standard techniques of localization, it suffices to consider the caseO = Rd₊with boundary∂O = Rd−1. Moreover, using a standard restriction argument, we may turn to the corresponding trace problem on the full spaceO × J = Rd_{× R.}

From the natural identifications

W_qn_,μ(L_γp) ∩ Lq_μ(W_pm_,γ) = W_(p,q),(d,1)(m,n) (Rd+1, (w_γ, v_μ); X) and

H_qr_,μ(L_γp) ∩ Lq_μ(H_ps_,γ) = H_(p,q),(d,1)(s,r) (Rd+1, (w_γ, v_μ); Y ), (16) and Corollary4.9, it follows that

tr[Wqn_,μ(L_γp) ∩ Lq_μ(Wpm_,γ)] = F 1−_m11+γ_p ,  1 m,n1 (p,q),p,(d−1,1) (Rd, (1, vμ); X) and tr[H_qr_,μ(L_γp) ∩ Lq_μ(H_ps_,γ)] = F1− 1 s 1+γ p ,  1 s,1r (p,q),p,(d−1,1)(Rd, (1, vμ); Y ).

4.3. Traces of anisotropic mixed-norm spaces

The goal of this subsection is to prove the trace result Theorem4.6, which is a weighted vector-valued version of [36, Theorem 2.2].

In contrast to Theorem4.6, the trace result [36, Theorem 2.2] is formulated for the distributional trace operator; see Remark4.8for more information. However, all estimates in the proof of that result are carried out for the “working definition of the trace.” The proof of Theorem4.6presented below basically consists of modifications of these estimates to our setting. As this can get quite technical at some points, we have decided to give the proof in full detail.

4.3.1. The working definition of the trace

Letϕ ∈ d,a(Rd) with associated family of convolution operators (Sn)n∈N ⊂ L(S_(Rd_{; X)) be fixed. In order to motivate the definition to be given in a moment, let}

us first recall that f =∞_n=0Snf inS(Rd; X) (respectively, in S(Rd; X)) whenever f ∈ S(Rd; X) (respectively, f ∈ S(Rd; X)), from which it is easy to see that

f_|{0}×Rd−1 = ∞

n=0

(Snf)_|{0}×Rd−1 in S(Rd−1; X), f ∈ S(Rd; X).

Furthermore, given a general tempered distribution f ∈ S(Rd; X), recall that Snf ∈ OM(Rd; X); in particular, each Snf has a well-defined classical trace with respect

to {0} × Rd−1. This suggests to define the trace operator τ = τϕ : D(γϕ) ⊂ S_(Rd_{; X) −→ S}_(Rd−1_{; X) by} τϕf := ∞ n₌₀ (Snf)_|{0}×Rd−1 (25)

on the domainD(τϕ) consisting of all f ∈ S(Rd; X) for which this defining series converges inS(Rd−1; X). Note that F−1E(Rd; X) is a subspace of D(τϕ) on which τϕ_{coincides with the classical trace of continuous functions with respect to{0}×R}d−1_;

of course, for an f belonging toF−1E(Rd; X) there are only finitely many Snf

nonzero.

4.3.2. The distributional trace operator

Let us now introduce the concept of distributional trace operator. The reason for us to introduce it is the right inverse from Lemma4.5.

The distributional trace operator r (with respect to the hyperplane{0} × Rd−1) is defined as follows. Viewing C(R; D(Rd−1; X)) as subspace of D(Rd; X) = D(R× Rd−1_{; X) via the canonical identification D}_{(R; D}_(Rd−1_{; X)) = D}_{(R × R}d−1_{; X)}

(arising from the Schwartz kernel theorem),

we define r∈ L(C(R; D(Rd−1; X)), D(Rd−1; X)) as the ’evaluation in 0 map’ r : C(R; D(Rd−1; X)) −→ D(Rd−1; X), f → ev0f.

Then, in view of

C(Rd; X) = C(R × Rd−1; X) = C(R; C(Rd−1; X)) → C(R; D(Rd−1; X)), we have that the distributional trace operator r coincides on C(Rd; X) with the classical trace operator with respect to the hyperplane{0} × Rd−1, i.e.,

r: C(Rd; X) −→ C(Rd−1; X), f → f_|{0}×Rd−1. The following lemma can be established as in [36, Section 4.2.1].

Lemma 4.5. Let ρ ∈ S(R) such that ρ(0) = 1 and supp ˆρ ⊂ [1, 2], a1 ∈ R,

˜d ∈ (Z>0)l−1withd = (1, ˜d ), ˜a ∈ (0, ∞)l−1, and(φn)n∈N ∈ ˜d ,˜a(Rd−1). Then, for each g∈ S(Rd−1; X), ext g:= ∞ n=0 ρ(2na1· ) ⊗ [φ n∗ g] (26)

defines a convergent series inS(Rd; X) with suppF [ρ ⊗ [φ0∗ g]] ⊂ {ξ | |ξ|d,a ≤ c}

suppF [ρ(2na1· ) ⊗ [φ

n∗ g]] ⊂ {ξ | c−12n≤ |ξ|d,a ≤ c2n} , n ≥ 1,

(27) for some constant c> 0 independent of g. Moreover, the operator ext defined via this formula is a linear operator

ext: S(Rd−1; X) −→ Cb(R; S(Rd−1; X))

which acts as a right inverse of r: C(R; S(Rd−1; X)) −→ S(Rd−1; X). 4.3.3. Trace spaces of Triebel–Lizorkin, Sobolev and Bessel potential spaces

Theorem 4.6. Let X be a Banach space,d1 = 1, a ∈ (0, ∞)l, p ∈ [1, ∞)l, q ∈

[1, ∞], γ ∈ (−1, ∞) and s > a1 p1(1 + γ ). Let w ∈ l j=1A∞(Rdj) be such that w1(x1) = wγ(x1) = |x1|γandw∈ l j=2Apj/rj(Rdj) for some r= (r2, . . . , rl) ∈ (0, 1)l−1_{satisfying s−}a1 p1(1 + γ ) > l j=2ajdj(_r1 j − 1).

1_{Then, the trace operator}

τ = τϕ _{(25) is well defined on F}s,a p,q,d(R

d_{, (w}

γ, w); X), where it is independent of ϕ, and restricts to a retraction

τ : Fs,a p,q,d(R d_{, (w} γ, w); X) −→ F s−a1_p1(1+γ ),a p,p1,d (R d−1_{, w}_{; X) (28)}

for which the extension operator ext from Lemma4.5(with ˜d = d and ˜a = a) restricts to a corresponding coretraction.

1_{This technical condition onw}_{is in particular satisfied when p}_{∈ (1, ∞)}l−1_andw_∈l

j=2Apj(R

Remark 4.7. In the situation of Theorem4.6, suppose that q< ∞. Then, S(Rd; X) is a dense linear subspace of Fs_p,a_,q,d(Rd, (w_γ, w); X) and τ is just the unique extension of the classical trace operator

S(Rd_{; X) −→ S(R}d−1_{; X), f → f}

|{0}×Rd−1,

to a bounded linear operator (28).

Remark 4.8. In contrary to the unweighted case considered in [36], one cannot use translation arguments to show that

Fs_p,a_,q,d(Rd, (w_γ, w); X) → C(R; D(Rd−1; X)) for s>a1 p1(1+γ ). However, for s > a1 p1(1+γ+), p ∈ (1, ∞) l_andw_∈l j=2Apj(Rdj), the inclusion Fs_p,a_,q,d(Rd, (w_γ, w); X) → C(R; S(Rd−1; X)) can be obtained as follows: picking˜s with s > ˜s > a1

p1(1 + γ+), there holds the chain of inclusions Fs_p,a_,q,d(Rd, (w_γ, w); X) → B˜s,a_p,1,d(Rd, (w_γ, w); X) (30) → Cb(R, ρp1,γ; B ˜s−a1 p1(1+γ+),a  p,1,d (R d−1_{, w}_{; X))} → C(R; S_(Rd−1_{; X)).}

Here, the restriction s> a1

p1(1 + γ+) when γ < 0 is natural in view of the necessity of s> a1

p1 in the unweighted case with p1> 1 (cf. [36, Theorem 2.1]).

Note that the trace space of the weighted anisotropic Triebel–Lizorkin space is independent of the microscopic parameter q ∈ [1, ∞]. As a consequence, if E is a normed space with

F_ps,a_,1,d(Rd, (w_γ, w); X) → E → Fs_p,a_,∞,d(Rd, (w_γ, w); X),

then the trace result of Theorem4.6also holds forE in place of Fs_p,a_,q,d(Rd,(w_γ, w);X). In particular, we have:

Corollary 4.9. Let X be a Banach space,d1 = 1, a ∈ (0, ∞)l, p ∈ (1, ∞)l,γ ∈ (−1, p1− 1) and s > a_p1

1(1 + γ ). Let w ∈ l

j₌₁Apj(Rdj) be such that w1(x1) =

wγ(x1) = |x1|γ. Suppose that either

• E = Wn p_,d(Rd, (wγ, w); X), n ∈ (Z≥1)l, n= sa−1; or • E = Hs,a p,d(R d_{, (w} γ, w); X); or • E = Hs p,d(R d_{, (w} γ, w); X), s ∈ (0, ∞)l, s= sa−1.

Then, the trace operatorτ = τϕ(25) is well defined onE, where it is independent of ϕ, and restricts to a retraction

τ : E −→ Fs−a1p1(1+γ ),a p_,p1,d (R

d−1_{, w}_{; X)}

for which the extension operator ext from Lemma4.5(with ˜d = d and ˜a = a) restricts to a corresponding coretraction.

4.3.4. Traces by duality for Besov spaces

Let i∈ {1, . . . , l}. For b ∈ Rdi_{, we define the hyperplane}

[d ;i],b:= Rd1 × Rdi−1× {b} × Rdi+1× Rdl

and we simply put_{[d ;i]} := _{[d ;i],0}. Furthermore, given sets S1, . . . , Sl and x = (x1, . . . , xl) ∈

l

j=1Sj, we write x[i]= (x1, . . . , xi−1, xi+1, . . . , xl).

Proposition 4.10. Let X be a Banach space, i∈ {1, . . . , l}, a ∈ (0, ∞)l, p∈ (1, ∞)l, q∈ [1, ∞), γ ∈ (−di, ∞) and s > a_pi

i(di+ γ ). Let w ∈

l

j₌₁A∞(Rdj) be such that wi(xi) = wγ(xi) = |xi|γ andwj ∈ Apj for each j= i. Then, the trace operator

tr_{[d ;i],b}: S(Rd; X) −→ S(Rd−di; X), f → f |[d ;i], extends to a retraction tr_{[d ;i],b}: Bs_p,a_,q,d(Rd, w; X) −→ Bs− ai pi(di+γ ),a[i] p[i],q,d[i] (R d−di, w[i]; X) ₍₂₉₎

for which the extension operator ext from Lemma4.5(with ˜d = d[i]and ˜a = a[i], modified in the obvious way to the i th multidimensional coordinate) restricts to a corresponding coretraction. Furthermore, if s> ai

pi(di+ γ+), then Bs_p,a_,q,d(Rd, w; X) → Cb(Rdi, ρpi,γ; B s−ai_pi(di+γ+),a[i] p[i]_,q,d[i] (R d−1_{, w}[i]_{; X))} → C(Rdi; S(Rd−di; X)), ₍₃₀₎ whereρpi,γ := max{| · |, 1} −γ−_pi.

Corollary 4.11. Let X be a Banach space, a∈ (0, ∞)l, p ∈ (1, ∞)l, q ∈ [1, ∞),

γ ∈l j=1(−dj, ∞) and s > l j=1 aj pj(dj+ γj,+). Let w ∈ l j=1A∞(Rdj) be such thatwj(xj) = wγ(xj) = |xj|γ for each j∈ {1, . . . , l}. Then,

Bs_p,a_,q,d(Rd, w; X) → Cb(Rd1, ρpl,γl; . . . Cb(Rdl, ρp1,γ1; X) . . .).

Proof. Thanks to the Sobolev embedding of Proposition5.1, it is enough to treat the casew ∈l_j=1Apj(Rdj), which can be obtained by l iterations of Proposition4.10.

Remark 4.12. The above proposition and its corollary remain valid for q = ∞. In this case the norm estimate corresponding to (29) can be obtained in a similar way, from which the unique extendability to a bounded linear operator (29) can be derived via the Fatou property, (10) and the case q = 1. The remaining statements can be established in the same way as for the case q< ∞.

Remark 4.13. Note that ifγ ∈ [0, ∞)l in the situation of the above corollary, then Bs_p,a_,q,d(Rd, w; X) → BUC(Rd; X)

by density of the Schwartz spaceS(Rd; X) ⊂ BUC(Rd; X) in Bs_p,a_,q,d(Rd, w; X). This could also be established in the standard way by the Sobolev embedding Propo-sition5.1, see for instance [49, Proposition 7.4].

Let X be a Banach space. Then, [S_(Rd_{; X)]= S(R}d_{; X∗)}

and [S(Rd; X)]= S(Rd; X∗) via the pairings induced by

 f ⊗ x∗, g ⊗ x =  f, x∗, g, x; see [4, Corollary 1.4.10].

Let i ∈ {1, . . . , l} and b ∈ Rdi_{. Let tr[d ;i],b} ∈ L(S(Rd; X), S(Rd−1; X)) be

given by tr_{[d ;i],b} f := f|[d ;i],b. Then, the adjoint operator T[d ;i],b := [tr[d ;i],b] ∈

L(S_(Rd−1_{; X}∗_{), S}_(Rd_{; X}∗_{)) is given by T}

[d ;i],bf = δb⊗[d ;i] f , which can be seen

by testing on the dense subspaceS(Rdi)⊗_{[d ;i]}S(Rd−di) of S(Rd). Now suppose that E

is a locally convex space withS(Rd; X)→ E and that F is a complete locally convexd space withS(Rd−di; X) → F. Then, Ed  → S(Rd; X∗) and F → S(Rd−di; X∗)

under the natural identifications, and tr_{[d ;i],b}extends to a continuous linear operator tr_E→FfromE to F if and only if T_{[d ;i],b}restricts to a continuous linear operator T_F_→E fromFtoE, in which case[tr_E→F]= T_F_→E.

Estimates in the classical Besov and Triebel–Lizorkin spaces for the tensor product with the one-dimensional delta-distributionδ0can be found in [34, Proposition 2.6],

where a different proof is given than the one below.

Lemma 4.14. Let X be a Banach space, i ∈ {1, . . . , l}, a ∈ (0, ∞)l, p ∈ [1, ∞)l, q∈ [1, ∞], γ ∈ (−di, ∞). Let w ∈

l

j=1A∞(Rdj) be such that wi(xi) = wγ(xi) =

|xi|γ. For each b∈ Rdi consider the linear operator T_{[d ;i],b}: S(Rd−di; X) −→ S(Rd; X), f → δ b⊗[d ;i] f. (i) If s ∈ (−∞, ai  di+γ pi − di 

), then T[d ;i],0is bounded from Bs+ai  di−di +γ_pi ,a[i] p[i],q,d (Rd−di, w[i]; X) to Bs,a p,q,d(R d_{, w; X).}

(ii) If s∈ (−∞, ai

di+γ−

pi − di

), then T[d ;i],bis bounded from B s+ai  di−di +γ_pi− ,a[i] p[i]_,q (Rd−di, w[i]; X) to Bs,a p,q,d(R

d_{, w; X) with norm estimate}

||T[d ;i],b|| B(Bs+ai  di −di +γ_pi  ,a[i]

p[i ],q (Rd−di,w[i];X),Bsp,q(Rd,wγ))

 max{|b|, 1}γ+p .

In order to perform all the estimates in Lemma4.14, we need the following two lemmas.

Lemma 4.15. Letψ : Rd −→ C be a rapidly decreasing measurable function and

putψR := Rdψ(R · ) for each R > 0. Let p ∈ [1, ∞) and γ ∈ (−1, ∞). For every R> 0 and a ∈ Rd, the following estimate holds true:

||ψR( · − a)||Lp_(Rd_{,| · |}γ₎ Rd− d+γ

p (|a|R + 1)γ+/p

Proof. By [11, Condition Bp] (see [49, Lemma 4.5] for a proof), ifw is an Aq-weight

onRd_{with q∈ (1, ∞), then} ˆ Rd(1 + |x − y|) −dq_{d y} [w]Aq,q ˆ B(x,1) w(y) dy. (31)

So let us pick q∈ (1, ∞) so that | · |γ ∈ Aq. Then, asψ is rapidly decreasing, there

exists C > 0 such that |ψ(x)| ≤ C(1 + |x|)−q/p for every x ∈ Rd. We can thus estimate ||ψR( · − a)||Lp_(Rd_{,| · |}γ₎= Rd− d+γ p ||ψ( · − Ra)|| Lp_(Rd_{,| · |}γ₎ ≤ C Rd₋d+γ_p ||t → (1 + |t − Ra|)−q/p||Lp_(Rd_{,| · |}γ₎ (31)  Rd−d+γ_p ˆ B(|a|R,1) |y|γd y 1/p  Rd₋d+γ_p (|a|R + 1)γ+/p_.  Lemma 4.16. For every r ∈ [1, ∞] and t > 0, there exists a constant C > 0 such

that, for all sequences(bk)k∈N ∈ CN, the following two inequalities hold true:

!!2t k∞_n=k+1|bn|  k_∈N!! r_(N)≤ C||(2 t k_b k)k∈N|| r_(N), !! !2−tkk_n=0|bn| k∈N !! ! r_(N)≤ C||(2 −tk_b k)k∈N|| r_(N).

Proof. See [36, Lemma 4.2] (and the references given there). 

Proof of Lemma4.14. Takeϕ = (ϕn)n∈N ∈ (d ,a)(Rd) with ϕ0 = φ0⊗_{[d ;i]}ψ0,

where φ = (φn)n∈N ∈ ai(Rdi) and ψ = (ψn)n ∈ (d

[i]_,a[i]₎

(Rd−di). For f ∈

S_(Rd−di; X), we then have

S₀ϕ(δb⊗_{[d ;i]} f) = S₀φδb⊗_{[d ;i]}S₀ψf = [φ0∗ δb] ⊗[d ;i][S0ψf] = φ0( · − b) ⊗[d ;i]S ψ 0 f