Complex interpolation with Dirichlet boundary conditions on the half line
Lindemulder, Nick; Meyries, Martin; Veraar, Mark DOI
10.1002/mana.201700204 Publication date
2018
Document Version Final published version Published in
Mathematische Nachrichten
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Lindemulder, N., Meyries, M., & Veraar, M. (2018). Complex interpolation with Dirichlet boundary conditions on the half line. Mathematische Nachrichten, 291(16), 2435-2456. https://doi.org/10.1002/mana.201700204
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DOI: 10.1002/mana.201700204
O R I G I N A L PA P E R
Complex interpolation with Dirichlet boundary conditions on the
half line
Nick Lindemulder
1Martin Meyries
2Mark Veraar
11Delft Institute of Applied Mathematics,
Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands
2Institut für Mathematik,
Martin-Luther-Universität Halle-Wittenberg, 06099, Halle (Saale), Germany
Correspondence
Mark Veraar, Delft Institute of Applied Math-ematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands. Email: m.c.veraar@tudelft.nl
Funding information
Netherlands Organisation for Scientific Research (NWO), Grant/Award Number: 639.032.427
Abstract
We prove results on complex interpolation of vector-valued Sobolev spaces over the half-line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space-valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.
K E Y W O R D S
𝐴𝑝-weights, Bessel potential spaces, complex interpolation with boundary conditions,𝐻∞-calculus,
point-wise multipliers, Sobolev spaces, UMD
M S C ( 2 0 1 0 )
Primary: 46E35; Secondary: 42B25, 46B70, 46E40, 47A60
1
INTRODUCTION
The main result of the present paper is the following. Let𝑊01,𝑝(ℝ+;𝑋) be the first order Sobolev space over the half line with values in a UMD Banach space𝑋 vanishing at 𝑡 = 0, where 𝑝 ∈ (1, ∞). Then for complex interpolation we have
[ 𝐿𝑝(ℝ +;𝑋), 𝑊 1,𝑝 0 (ℝ+;𝑋) ] 𝜃 =𝐻 𝜃,𝑝 0 (ℝ+;𝑋), 𝜃 ∈ (0, 1), 𝜃 ≠ 1∕𝑝,
see Theorems 6.7 and (6.6). Here𝐻0𝜃,𝑝denotes the fractional order Bessel potential space with vanishing trace for𝜃 > 1∕𝑝, and𝐻0𝜃,𝑝=𝐻𝜃,𝑝for𝜃 < 1∕𝑝. In more generality, we consider spaces with Muckenhoupt power weights 𝑤𝛾(𝑡) = 𝑡𝛾, where the critical value 1∕𝑝 is shifted accordingly.
In the scalar-valued case𝑋 = ℂ, the result is well-known and due to Seeley [43]. The vector-valued result was already used several times in the literature without proof. Seeley also considers the case𝜃 = 1∕𝑝, which we ignore throughout for simplicity, and the case of domains Ω⊆ ℝ𝑑. The corresponding result for real interpolation is due to Grisvard [17] and more elementary to prove.
At the heart of complex interpolation theory with boundary conditions is the pointwise multiplier property of the characteristic function of the half-space𝟏ℝ
+ on𝐻
𝜃,𝑝(ℝ; 𝑋) for 0 < 𝜃 < 1∕𝑝. It is due to Shamir [44] and Strichartz [45] in the scalar-valued
case. In [36] by the second and third author, a general theory of pointwise multiplication of weighted vector-valued functions was developed. As a main application the multiplier result was extended to the vector-valued and weighted setting. An alternative approach to this was found by the first author in [27] and is based on a new equivalent norm for vector-valued Bessel potential spaces. In Section 4 we present a new and simpler proof of the multiplier property of𝟏ℝ
+, which is based on the representation of fractional powers of the negative Laplacian as a singular integral and the Hardy–Hilbert inequality.
For future reference and as it is only a minimal extra effort, we will formulate and prove some elementary assertions for the half spaceℝ𝑑+for𝑑 ≥ 1 or even domains, and general 𝐴𝑝 weights𝑤. In order to make the presentation as self-contained as possible, we further fully avoid the use of Triebel–Lizorkin spaces and Besov spaces, but we point out where they could be used. We will only use the UMD property of𝑋 through standard applications of the Mihlin multiplier theorem. Several results will be presented in such a way that the UMD property is not used. A detailed explanation of the theory of UMD spaces and their connection to harmonic analysis can be found in the monograph [20]. In their reflexive range, all standard function spaces are UMD spaces.
The complex interpolation result has applications in the theory of evolution equations, as it yields a characterization of the fractional power domains of the time derivative D((𝑑∕𝑑𝑡)𝜃)and D((−𝑑∕𝑑𝑡)𝜃)onℝ+. Here the half line usually stands for the time variable and𝑋 is a suitable function space for the space variable. For instance such spaces can be used in the theory of Volterra equations (see [38,48,49]), in evolution equations with form methods (see [9,12]), in stochastic evolution equations (see [37]).
In order to deal with rough initial values it is useful to consider a power weights𝑤𝛾(𝑡) = 𝑡𝛾in the time variable. Examples of papers in evolution equation where such weights are used include [3,8,23,28,31,32,35,39,41]. The monographs [2,29,40] are an excellent source for applications of weighted spaces to evolution equations. In order to make our results available to this part of the literature as well, we present our interpolation results for weighted spaces. For the application to evolution equations it suffices to consider interpolation of vector-valued Sobolev spaces overℝ+with Dirichlet boundary conditions and therefore we focus on this particular case. In a future paper we extend the results of [17] and [43] to weighted function spaces on more general domains Ω⊆ ℝ𝑑, in the scalar valued situation, where one of the advantages is that Bessel potential spaces have a simple square function characterization.
Notation. ℝ𝑑+= (0, ∞) × ℝ𝑑−1denotes the half space. We write𝑥 = (𝑥1, ̃𝑥) ∈ ℝ𝑑 with𝑥1∈ℝ and ̃𝑥 ∈ ℝ𝑑−1and define the weight𝑤𝛾by𝑤𝛾(𝑥1, ̃𝑥) = |𝑥1|𝛾. Sometimes it will be convenient to also write (𝑡, 𝑥) ∈ ℝ𝑑with𝑡 ∈ ℝ and 𝑥 ∈ ℝ𝑑−1. The operator denotes the Fourier transform. We write 𝐴 ≲𝑝𝐵 whenever 𝐴 ≤ 𝐶𝑝𝐵 where 𝐶𝑝is a constant which depends on the parameter 𝑝. Similarly, we write 𝐴 ≂𝑝𝐵 if 𝐴 ≲𝑝𝐵 and 𝐵 ≲𝑝𝐴.
2
PRELIMINARIES
2.1
Weights
A locally integrable function𝑤 ∶ ℝ𝑑 → (0, ∞) will be called a weight function. Given a weight function 𝑤 and a Banach space 𝑋 we define 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)as the space of all strongly measurable𝑓 ∶ ℝ𝑑→ 𝑋 for which
‖𝑓‖𝐿𝑝(ℝ𝑑,𝑤;𝑋) ∶= (
∫ ‖𝑓(𝑥)‖𝑝𝑤(𝑥) 𝑑𝑥 )1
𝑝
is finite. Here we identify functions which are a.e. equal.
Although we will be mainly interested in a special class of weights, it will be natural to formulate some of the result for the class of Muckenhoupt𝐴𝑝-weights. For𝑝 ∈ (1, ∞), we say that 𝑤 ∈ 𝐴𝑝if
[𝑤]𝐴𝑝 = sup 𝑄 1 |𝑄| ∫𝑄𝑤(𝑥) 𝑑𝑥 ⋅ ( 1 |𝑄| ∫𝑄𝑤(𝑥) − 1 𝑝−1𝑑𝑥 )𝑝−1 < ∞.
Here the supremum is taken over all cubes𝑄 ⊆ ℝ𝑑with sides parallel to the coordinate axes. For𝑝 ∈ (1, ∞) and a weight 𝑤 ∶ ℝ𝑑→ (0, ∞) one has 𝑤 ∈ 𝐴
𝑝if and only the Hardy–Littlewood maximal function is bounded on𝐿𝑝(ℝ𝑑, 𝑤). We refer the reader
to [16, Chapter 9] for standard properties of𝐴𝑝-weights. For a fixed𝑝 and a weight 𝑤 ∈ 𝐴𝑝, the weight𝑤′=𝑤−1∕(𝑝−1)∈𝐴𝑝′ is the𝑝-dual weight. By Hölder's inequality one checks that
∫ |𝑓(𝑥)||𝑔(𝑥)| 𝑑𝑥 ≤ ‖𝑓‖𝐿𝑝(ℝ𝑑,𝑤)‖𝑔‖𝐿𝑝′(ℝ𝑑,𝑤′) (2.1)
for𝑓 ∈ 𝐿𝑝(ℝ𝑑, 𝑤)and𝑔 ∈ 𝐿𝑝′(ℝ𝑑, 𝑤′). Using this, for each𝑤 ∈ 𝐴
𝑝one can check that𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)⊆ 𝐿1loc (
ℝ𝑑;𝑋).
Example 2.1. Let
𝑤𝛾(𝑥1, ̃𝑥) = |𝑥1|𝛾, 𝑥1∈ℝ, ̃𝑥 ∈ ℝ𝑑−1. As in [16, Example 9.1.7]) one sees that𝑤𝛾 ∈𝐴𝑝if and only if𝛾 ∈ (−1, 𝑝 − 1).
Lemma 2.2. Let𝑝 ∈ (1, ∞) and let 𝑤 ∈ 𝐴𝑝. Assume𝜙 ∈ 𝐿1(ℝ𝑑)and∫ 𝜙 𝑑𝑥 = 1. Let 𝜙𝑛(𝑥) = 𝑛𝑑𝜙(𝑛𝑥). Assume that 𝜙 satisfies any of the following conditions:
1. 𝜙 is bounded and compactly supported
2. There exists a radially decreasing function𝜓 ∈ 𝐿1(ℝ𝑑)such that|𝜙| ≤ 𝜓 a.e.
Then for all𝑓 ∈ 𝐿𝑝(ℝ𝑑;𝑋),𝜙𝑛∗𝑓 → 𝑓 in 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)as𝑛 → ∞. Moreover, there is a constant 𝐶 only depending on 𝜙 such that ‖𝜙𝑛∗𝑓‖ ≤ 𝐶𝑀𝑓 almost everywhere.
Proof. For convenience of the reader we include a short proof. By [20, Theorem 2.40 and Corollary 2.41]𝜙𝑛∗𝑓 → 𝑓 almost everywhere and‖𝜙𝑛∗𝑓‖ ≤ ‖𝜓‖𝐿1(ℝ𝑑)𝑀𝑓 almost everywhere, where 𝑀 denotes the Hardy–Littlewood maximal function.
Therefore, the result follows from the dominated convergence theorem. □
2.2
Fourier multipliers and UMD spaces
Let(ℝ𝑑;𝑋)be the space of𝑋-valued Schwartz functions and let ′(ℝ𝑑;𝑋)=((ℝ𝑑), 𝑋)be the space of𝑋-valued tempered distributions. For𝑚 ∈ 𝐿∞(ℝ𝑑)let𝑇𝑚∶(ℝ𝑑;𝑋)→ ′(ℝ𝑑;𝑋)be the Fourier multiplier operator defined by
𝑇𝑚𝑓 = −1(𝑚 ̂𝑓).
There are many known conditions under which𝑇𝑚 is a bounded linear operator on𝐿𝑝(ℝ𝑑;𝑋). In the scalar-valued the set of all Fourier multiplier symbols on𝐿2(ℝ𝑑)for instance coincides with𝐿∞(ℝ𝑑). In the case𝑝 ∈ (1, ∞) ⧵ {2} a large set of multipliers for which𝑇𝑚 is bounded is given by Mihlin's multiplier theorem. In the vector-valued case difficulties arise and geometric conditions on𝑋 are needed already if 𝑑 = 1 and 𝑚(𝜉) = sign(𝜉); in fact, in [5,6] it was shown that in this specific case the boundedness of𝑇𝑚 on𝐿𝑝(ℝ; 𝑋) characterizes the UMD property of 𝑋. Since the work of [5,6,30] it is well-known that the right class of Banach spaces for vector-valued harmonic analysis is the class of UMD Banach spaces, as many of the classical results in harmonic analysis, such as the classical Mihlin multiplier theorem, have been extended to this setting. We refer to [7,20,42] for details on UMD spaces and Fourier multiplier theorems.
All UMD spaces are reflexive. Conversely, all spaces in the reflexive range of the classical function spaces have UMD: e.g.:𝐿𝑝, Bessel potential spaces, Besov spaces, Triebel–Lizorkin spaces, Orlicz spaces.
The following result is a weighted version of the Mihlin multiplier theorem which can be found in [36, Proposition 3.1] and is a simple consequence of [19].
Proposition 2.3. Let𝑋 be a UMD space, let 𝑝 ∈ (1, ∞) and let 𝑤 ∈ 𝐴𝑝. Assume that𝑚 ∈ 𝐶𝑑+2(ℝ𝑑⧵ {0})satisfies
𝐶𝑚∶= sup
|𝛼|≤𝑑+2sup𝜉≠0|𝜉|
|𝛼||𝜕𝛼𝑚(𝜉)| < ∞.
Then𝑇𝑚is bounded on𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)and has an operator norm that only depends𝐶𝑚, 𝑑, 𝑝, 𝑋, [𝑤]𝐴 𝑝.
3
WEIGHTED FUNCTION SPACES
In this section we present several results on weighted function spaces, which do not require the UMD property of the underlying Banach space (except in Proposition 3.2).
3.1
Definitions and basic properties
For an open set Ω⊆ ℝ𝑑let(Ω) denote the space compactly supported smooth functions on Ω equipped with its usual inductive limit topology. For a Banach space𝑋, let ′(Ω;𝑋) = ((Ω), 𝑋) be the space of 𝑋-valued distributions. For a distribution 𝑢 ∈ ′(Ω;𝑋) and an open subset Ω
0⊆ Ω, we define the restriction 𝑢|Ω0 ∈′ ( Ω0;𝑋 ) as𝑢|Ω0(𝑓) = 𝑢(𝑓) for 𝑓 ∈ ( Ω0 ) .
For 𝑝 ∈ (1, ∞) and 𝑤 ∈ 𝐴𝑝 let 𝑊𝑘,𝑝(Ω, 𝑤; 𝑋) ⊆ ′(Ω;𝑋) be the Sobolev space of all 𝑓 ∈ 𝐿𝑝(Ω, 𝑤; 𝑋) with 𝜕𝛼𝑓 ∈ 𝐿𝑝(Ω, 𝑤; 𝑋) for all |𝛼| ≤ 𝑘 and set
‖𝑓‖𝑊𝑘,𝑝(Ω,𝑤;𝑋)= ∑ |𝛼|≤𝑘 ‖𝜕𝛼𝑓‖𝐿𝑝(Ω,𝑤;𝑋), [𝑓]𝑊𝑘,𝑝(Ω,𝑤;𝑋)= ∑ |𝛼|=𝑘 ‖𝜕𝛼𝑓‖𝐿𝑝(Ω,𝑤;𝑋). Here for𝛼 ∈ ℕ𝑑,𝜕𝛼=𝜕𝛼1 1 …𝜕 𝛼𝑑 𝑑 .
Let𝑠denote the Bessel potential operator of order𝑠 ∈ ℝ defined by
𝑠𝑓 = (1 − Δ)𝑠∕2𝑓 ∶= −1(1 +| ⋅ |2)𝑠∕2𝑓,̂
where ̂𝑓 denotes the Fourier transform of 𝑓 and Δ =∑𝑑𝑗=1𝜕2𝑗. For 𝑝 ∈ (1, ∞), 𝑠 ∈ ℝ and 𝑤 ∈ 𝐴𝑝 let 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)⊆ ′(ℝ𝑑;𝑋)denote the Bessel potential space of all𝑓 ∈ ′(ℝ𝑑;𝑋)for which
𝑠𝑓 ∈ 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)and set
‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤;𝑋) = ‖𝑠𝑓‖𝐿𝑝(ℝ𝑑,𝑤;𝑋). In the following lemma we collect some properties of the operators𝑠.
Lemma 3.1. Fix𝑠 > 0. There exists a function 𝐺𝑠∶ℝ𝑑 → [0, ∞) such that 𝐺𝑠∈𝐿1(ℝ𝑑)and −𝑠𝑓 = 𝐺𝑠∗𝑓 for all 𝑓 ∈ ′(ℝ𝑑;𝑋). Moreover,𝐺
𝑠has the following properties:
1. For all|𝑦| ≥ 2, 𝐺𝑠(𝑦) ≲𝑠,𝑑 𝑒−|𝑦|2 . 2. For|𝑥| ≤ 2, 𝐺𝑠(𝑥) ≲𝑠,𝑑 ⎧ ⎪ ⎨ ⎪ ⎩ |𝑥|𝑠−𝑑, 𝑠 ∈ (0, 𝑑), 1 + log ( 2 |𝑥| ) , 𝑠 = 𝑑, 1, 𝑠 > 𝑑.
3. For all𝑠 > 𝑘 ≥ 0 and all |𝛼| ≤ 𝑘, there exists a radially decreasing function 𝜙 ∈ 𝐿1(ℝ𝑑)such that|𝜕𝛼𝐺𝑠| ≤ 𝜙 pointwise. In particular, if𝑑 = 1, 𝑝 ∈ (1, ∞), 𝛾 ∈ (−1, 𝑝 − 1) and 𝑠 > 1+𝛾 𝑝 , then𝐺𝑠∈𝐿𝑝 ′( ℝ, 𝑤′ 𝛾 ) .
Proof. The fact that the positive function𝐺𝑠∈𝐿1(ℝ𝑑)exists, together with (1) and (2), follows from [16, Section 6.1.b]. To prove (3), we use the following representation of𝐺𝑠(see [16, Section 6.1.b]):
𝐺𝑠(𝑥) = 𝐶𝑠,𝑑∫ ∞ 0 𝑒−𝑡𝑒−|𝑥|2 4𝑡 𝑡 𝑠−𝑑 2 𝑑𝑡 𝑡 .
By induction one sees that𝜕𝛼𝐺𝑠(𝑥) is a linear combination of functions of the form 𝐺𝑠−2𝑗(𝑥)|𝑥|𝛽with|𝛽| ≤ 𝑗 ≤ 𝑘. Therefore, by (2) for|𝑥| ≤ 2, |𝜕𝛼𝐺𝑠(𝑥)| ≲𝑠,𝑑,𝛼|𝑥|𝜀−𝑑for some𝜀 ∈ (0, 𝑑). On the other hand for |𝑥| ≥ 2, |𝜕𝛼𝐺𝑠(𝑥)| ≲𝑠,𝑑,𝛼|𝑥|𝛽𝑒−|𝑥|2 ≲𝑑,𝑠,𝑘 𝑒−|𝑥|4
. Now the function𝜙(𝑥) = 𝐶1|𝑥|𝜀−𝑑for|𝑥| ≤ 2 and 𝜙(𝑥) = 𝐶2𝑒−
|𝑥|
4 for certain constants𝐶
1, 𝐶2> 0. satisfies the required conditions.
To prove the final assertion for𝑑 = 1, note that the blow-up behaviour near 0 gets worse as 𝑠 decreases. Therefore, without loss of generality we may assume that𝑠 ∈(1+𝑝𝛾, 1), in which case (2) yields
|𝐺𝑠(𝑥)|𝑝′ 𝑤′ 𝛾(𝑥) ≲𝑠,𝑝,𝛾|𝑥| (𝑠−1)𝑝−𝛾 𝑝−1 =|𝑥|−1+𝑝−1𝑝 (𝑠− 1+𝛾 𝑝 ) for|𝑥| ≤ 2,
which is integrable. Integrability, for|𝑥| > 2, is clear from (1). □
The following result is proved in [36, Proposition 3.2 and 3.7] by a direct application of Proposition 2.3.
Proposition 3.2. Let𝑋 be a UMD space, let 𝑝 ∈ (1, ∞), 𝑘 ∈ ℕ0, and let𝑤 ∈ 𝐴𝑝. Then𝐻𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)=𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋) with norm equivalence only depending on𝑑, 𝑋, 𝑝, 𝑘 and [𝑤]𝐴
The UMD property is necessary in Proposition 3.2 (see [20, Theorem 5.6.12]). Sometimes it can be avoided by instead using the following simple embedding result which holds for any Banach space. The sharper version 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)→ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)if𝑠 < 𝑘 and 𝑘 ∈ ℕ
0. can be obtained from [33, Propositions 3.11 and 3.12] but is more complicated.
Lemma 3.3. Let𝑋 be a Banach space, let 𝑝 ∈ (1, ∞), 𝑘 ∈ ℕ0,𝑠 ∈ (𝑘, ∞) and let 𝑤 ∈ 𝐴𝑝. Then the following continuous embeddings hold
𝑊2𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)→ 𝐻2𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋), 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)→ 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋), with embedding constants which only depend on𝑑, 𝑠, 𝑘 and [𝑤]𝐴
𝑝.
Proof. The first embedding is immediate from 𝐽2𝑘𝑓 = (1 − Δ)𝑘𝑓 and Leibniz' rule. For the second embedding let 𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)and write𝑓
𝑠=𝐽𝑠𝑓 ∈ 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋). By Lemma 3.1 (3) and Lemma 2.2, for all|𝛼| ≤ 𝑘,
‖𝜕𝛼𝑓‖𝑋=‖𝜕𝛼𝐺𝑠∗𝑓𝑠‖𝑋≤ 𝜙 ∗ ‖𝑓𝑠‖𝑋≤ 𝐶𝜙𝑀(‖𝑓𝑠‖𝑋),
where𝜙 ∈ 𝐿1(ℝ𝑑)is a radially decreasing function depending on𝛼, 𝑘 and 𝑠. Therefore, by the boundedness of the Hardy– Littlewood maximal function, we have𝜕𝛼𝑓 ∈ 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)with
‖𝜕𝛼𝑓‖𝐿𝑝(ℝ𝑑,𝑤;𝑋) ≲𝑝,[𝑤]
𝐴𝑝 ‖𝑓𝑠‖𝐿𝑝(ℝ𝑑,𝑤;𝑋) = ‖𝑓 ‖𝐻𝑠,𝑝(ℝ𝑑,𝑤;𝑋).
Now the result follows by summation over all𝛼. □
We proceed with two density results.
Lemma 3.4. Let 𝑋 be a Banach space, let 𝑝 ∈ (1, ∞), 𝑠 ∈ ℝ and let 𝑤 ∈ 𝐴𝑝. Then (ℝ𝑑;𝑋)→ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)→ ′(ℝ𝑑;𝑋). Moreover,𝐶∞
𝑐
(
ℝ𝑑)⊗ 𝑋 is dense in 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋).
Proof. First we prove that(ℝ𝑑;𝑋)→ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋). It suffices to prove this in the case𝑠 = 0 by continuity of 𝑠= (1 − Δ)𝑠∕2on(ℝ𝑑;𝑋). In the case𝑠 = 0, the continuity of the embedding follows from
‖𝑓‖𝐿𝑝(ℝ𝑑,𝑤;𝑋) ≤‖‖‖(1 +|𝑥|2)−𝑛‖‖‖ 𝐿𝑝(ℝ𝑑,𝑤)‖‖‖ ( 1 +|𝑥|2)𝑛𝑓‖‖‖ 𝐿∞(ℝ𝑑;𝑋)≲𝑑,𝑛,𝑝,𝑤 ∑ |𝛼|≤2𝑛 sup 𝑥∈ℝ𝑑‖𝑥 𝛼𝑓(𝑥)‖
for𝑛 ∈ ℕ with 𝑛 ≥ 𝑑𝑝 (see [33, Lemma 4.5]).
To prove the density assertion note that𝐿𝑝(ℝ𝑑, 𝑤)⊗ 𝑋 is dense in 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)and(ℝ𝑑)is dense in𝐿𝑝(ℝ𝑑, 𝑤)(see [16, Exercise 9.4.1]) it follows that(ℝ𝑑)⊗ 𝑋 is dense in 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋). Since𝐽−𝑠leaves(ℝ𝑑)invariant, also(ℝ𝑑)⊗ 𝑋 is dense in𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋). Combining this with(ℝ𝑑;𝑋)→ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)and the fact that𝐶𝑐∞(ℝ𝑑)is dense in(ℝ𝑑) (see [10, Lemma 14.7]) we obtain the desired density assertion.
To prove the embedding𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)→ ′(ℝ𝑑;𝑋) it suffices again to consider 𝑠 = 0. In this case from (2.1) and (ℝ𝑑)→𝐿𝑝′(
ℝ𝑑, 𝑤′)densely, we deduce
𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)→ (𝐿𝑝′(
ℝ𝑑, 𝑤′), 𝑋)→ ((ℝ𝑑), 𝑋)=′(ℝ𝑑;𝑋).
□
Lemma 3.5. Let 𝑋 be a Banach space, let 𝑝 ∈ (1, ∞), 𝑘 ∈ ℕ and let 𝑤 ∈ 𝐴𝑝. Then (ℝ𝑑;𝑋)→ 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)→ ′(ℝ𝑑;𝑋). Moreover,𝐶∞
𝑐
(
ℝ𝑑)⊗ 𝑋 is dense in 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋).
Proof. The case𝑘 = 0 follows from Lemma 3.4 and the case 𝑘 ≥ 1 follow by differentiation. Let 𝜙 ∈ 𝐶∞
𝑐
(
ℝ𝑑) be such that ∫
ℝ𝑑𝜙 𝑑𝑥 = 1 and define 𝜙𝑛∶=𝑛𝑑𝜙(𝑛 ⋅ ) for every 𝑛 ∈ ℕ. Then, by Lemma 2.2 and standard properties of convolutions, 𝑓𝑛∶=𝜙𝑛∗𝑓 → 𝑓 in 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋) as 𝑛 → ∞ with 𝜙𝑛∗𝑓 ∈ 𝑊∞,𝑝(ℝ𝑑, 𝑤; 𝑋)= ⋂
𝑙∈ℕ𝑊𝑙,𝑝
(
ℝ𝑑, 𝑤; 𝑋). In particular, 𝑊2𝑘+2,𝑝(ℝ𝑑, 𝑤; 𝑋) is dense in 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋). This yields 𝐻𝑘+1,𝑝(ℝ𝑑, 𝑤; 𝑋)→𝑑 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)by Lemma 3.3. The density of𝐶∞
𝑐
(
ℝ𝑑)⊗ 𝑋 in 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)now follows from Lemma 3.4.
Lemma 3.6. Let𝑋 be a Banach space, let 𝑝 ∈ (1, ∞), 𝑠 ∈ ℝ and let 𝑤 ∈ 𝐴𝑝. Assume𝜙 ∈ 𝐶𝑐∞(ℝ) with ∫ 𝜙 𝑑𝑥 = 1. Let 𝜙𝑛(𝑥) = 𝑛𝑑𝜙(𝑛𝑥). Then, for all 𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋),
‖𝜙𝑛∗𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤;𝑋) ≲𝑠,𝑝,[𝑤],𝑑 ‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤;𝑋)
Proof. The first part of the statement follows from Lemma 2.2 and𝑠(𝜙𝑛∗𝑓) = 𝜙𝑛∗𝑠𝑓. For the last part, note that 𝜙𝑛∗𝑓 = −𝑠
[
𝜙𝑛∗𝑠𝑓]∈𝐻∞,𝑝(ℝ𝑑, 𝑤; 𝑋)by basic properties of convolutions in combination with Lemma 3.3. □ The following version of the Hardy inequality will be needed (see [33, Corolllary 1.4] for a related result). The result can be deduced from [34, Theorem 1.3 and Proposition 4.3] but for convenience we include an elementary proof.
Lemma 3.7(Hardy inequality with power weights). Let 𝛾 ∈ (−1, 𝑝 − 1) and 𝑠 ∈ (0, 1). Let 𝑤𝛾(𝑡, 𝑥) = |𝑡|𝛾 for 𝑡 ∈ ℝ and 𝑥 ∈ ℝ𝑑−1. Then𝐻𝑠,𝑝(ℝ𝑑, 𝑤
𝛾;𝑋)→ 𝐿𝑝(ℝ𝑑, 𝑤𝛾−𝑠𝑝;𝑋).
Proof. It suffices to prove‖𝐺𝑠∗𝑓‖𝐿𝑝(𝑤𝛾−𝑠𝑝;𝑋)≲𝑝,𝑠,𝑑,𝛾 ‖𝑓‖𝐿𝑝(𝑤𝛾;𝑋), where𝐺𝑠is as in Lemma 3.1 and𝑓 ∈ 𝐿𝑝(𝑤𝛾;𝑋). Since 𝐺𝑠≥ 0, by the triangle inequality it suffices to consider the case of scalar functions 𝑓 with 𝑓 ≥ 0.
To prove the result we first apply Minkowski's and Young's inequality inℝ𝑑−1: ‖𝐺𝑠∗𝑓(𝑡, ⋅)‖𝐿𝑝(ℝ𝑑−1)≤ ∫
ℝ‖𝐺𝑠(𝑡 − 𝜏, ⋅)‖𝐿
1(ℝ𝑑−1)‖𝑓(𝜏, ⋅)‖𝐿𝑝(ℝ𝑑−1)𝑑𝜏 = 𝑔𝑠∗𝜙(𝜏). Here𝑔𝑠(𝑡) = ‖𝐺𝑠(𝑡, ⋅)‖𝐿1(ℝ𝑑−1)and𝜙(𝜏) = ‖𝑓(𝜏, ⋅)‖𝐿𝑝(ℝ𝑑−1). Then for|𝑡| ≤ 2, by Lemma 3.1 (1) and (2),
𝑔𝑠(𝑡) ≲𝑠,𝑑 ∫
ℝ𝑑−1(|𝑡| + |𝑥|)
𝑠−𝑑𝑑𝑥 = |𝑡|𝑠−1
∫ℝ𝑑−1(1 +|𝑥|)
𝑠−𝑑𝑑𝑥 = 𝐶|𝑡|𝑠−1,
where we used𝑠 < 1. For |𝑡| > 2, by Lemma 3.1 (2) and |(𝑡, 𝑥)| ≂ |𝑡| + |𝑥|, we find 𝑔𝑠(𝑡) ≲𝑠,𝑑 𝑒−|𝑡| 2 ∫ℝ𝑑𝑒 −|𝑥| 2 𝑑𝑥 ≂𝑑 𝑒− |𝑡| 2.
Finally by the weighted version of Young's inequality (see and [22, Theorem 3.4(3.7)]) in dimension one, we find that ‖𝐺𝑠∗𝑓‖𝐿𝑝(ℝ𝑑,𝑤𝛾−𝑠𝑝)≤ ‖𝑔𝑠∗𝜙‖𝐿𝑝(ℝ,𝑤𝛾−𝑠𝑝)≤ 𝐶‖𝜙‖𝐿𝑝(ℝ,𝑤𝛾)=𝐶‖𝑓‖𝐿𝑝(ℝ𝑑,𝑤𝛾),
where𝐶 = sup𝑡∈ℝ|𝑡|1−𝑠𝑔𝑠(𝑡) < ∞. □
We end this section with a weighted version of the classical Hardy–Hilbert inequality.
Lemma 3.8(Hardy–Hilbert inequality with power weights). Let𝑝 ∈ (1, ∞) and 𝛾 ∈ (−1, 𝑝 − 1). Let 𝑤𝛾(𝑥1, ̃𝑥) = |𝑥1|𝛾and
𝑘(𝑥, 𝑦) = 1
((|𝑥1|+|𝑦1|)2+| ̃𝑥− ̃𝑦|2)𝑑∕2
, where𝑥 = (𝑥1, ̃𝑥) and 𝑦 = (𝑦1, ̃𝑦). Then the formula
𝐼𝑘ℎ(𝑥) ∶= ∫
ℝ𝑑𝑘(𝑥, 𝑦)ℎ(𝑦) 𝑑𝑦 yields a well-defined bounded linear operator𝐼𝑘on𝐿𝑝(ℝ𝑑, 𝑤𝛾).
Proof. It suffices to considerℎ ≥ 0. Moreover, by symmetry it is enough to consider 𝑥1, 𝑦1> 0. Thus we need to show that ‖‖ ‖‖ ‖𝑥 → ∫ℝ𝑑 + 𝑘(𝑥, 𝑦)ℎ(𝑦) 𝑑𝑦‖‖‖‖ ‖𝐿𝑝(ℝ𝑑 +,𝑤𝛾 )≲𝑝,𝑑,𝛾‖ℎ‖𝐿𝑝(ℝ𝑑+,𝑤𝛾 ), ℎ ∈ 𝐿𝑝(ℝ𝑑 +, 𝑤𝛾 ) , ℎ ≥ 0. Step I. The case𝑑 = 1. Replacing 𝑘 by
𝑘𝛽(𝑥, 𝑦) = 𝑤𝛾(𝑥) 1∕𝑝𝑤 𝛾(𝑦)−1∕𝑝 (|𝑥| + |𝑦|) = |𝑥|𝛽|𝑦|−𝛽 |𝑥| + |𝑦|, with𝛽 = 𝛾∕𝑝, it suffices to consider the unweighted case.
To prove the required result we apply Schur's test in the same way as in [14, Theorem 5.10.1]. Let𝑠(𝑥) = 𝑡(𝑥) = 𝑥− 1 𝑝𝑝′. Then since −1< 𝛽 −𝑝1′ < 0 ∫ ∞ 0 𝑠(𝑥)𝑝𝑘𝛽(𝑥, 𝑦) 𝑑𝑥 = ∫ ∞ 0 𝑥𝛽−𝑝′1𝑦−𝛽 𝑥 + 𝑦 𝑑𝑥 = 𝑡(𝑦)𝑝∫ ∞ 0 𝑧𝛽−1𝑝 𝑧 + 1𝑑𝑧 = 𝐶𝑝,𝛽𝑡(𝑦)𝑝.
Similarly, since −1< −𝛽 −1𝑝 < 0 ∫ ∞ 0 𝑡(𝑦)𝑝′ 𝑘𝛽(𝑥, 𝑦) 𝑑𝑦 = ∫ ∞ 0 𝑥𝛽𝑦−𝛽−1𝑝 𝑥 + 𝑦 𝑑𝑦 = 𝑠(𝑥)𝑝 ′ ∫ ∞ 0 𝑧−𝛽−1 𝑝 1 +𝑧 𝑑𝑧 = 𝐶𝑝,𝛽𝑠(𝑥) 𝑝′ . Step II. The general case. By Minkowski's inequality we find
‖𝐼𝑘𝑓(𝑥1, ⋅)‖𝐿𝑝(ℝ𝑑−1)≤ ∫ ∞ 0 ( ∫ℝ𝑑−1 ( ∫ℝ𝑑−1 𝑓(𝑦1, ̃𝑦) ((𝑥1+𝑦1)2+| ̃𝑥 − ̃𝑦|2)𝑑∕2 𝑑 ̃𝑦 )𝑝 𝑑 ̃𝑥 )1∕𝑝 𝑑𝑦1. Fix𝑦1> 0 and let 𝑔𝑟(̃𝑦) = 𝑓(𝑦1, 𝑟 ̃𝑦). Setting 𝑟 = 𝑥1+𝑦1and substituting𝑢 ∶= ̃𝑥∕𝑟 and 𝑣 ∶= ̃𝑦∕𝑟 we can write
∫ℝ𝑑−1 ( ∫ℝ𝑑−1 𝑓(𝑦1, ̃𝑦) (|𝑥1+𝑦1|2+| ̃𝑥 − ̃𝑦|2)𝑑∕2 𝑑 ̃𝑦 )𝑝 𝑑 ̃𝑥 =𝑟−𝑝+𝑑−1 ∫ℝ𝑑−1 ( ∫ℝ𝑑−1 𝑔𝑟(𝑣) (1 +|𝑢 − 𝑣|2)𝑑∕2𝑑𝑣 )𝑝 𝑑𝑢 ≤ 𝑟−𝑝+𝑑−1‖𝑔 𝑟‖𝑝𝐿𝑝(ℝ𝑑−1)‖‖‖ ( 1 +| ⋅ |2)−𝑑∕2‖‖‖𝑝 𝐿1(ℝ𝑑−1)=𝐶𝑑,𝑝𝑟 −𝑝‖𝑔 1‖𝑝𝐿𝑝(ℝ𝑑−1), where we applied Young's inequality for convolutions. Therefore,
‖𝐼𝑘𝑓(𝑥1, ⋅)‖𝐿𝑝(ℝ𝑑−1)≤ 𝐶𝑑,𝑝∫ ∞ 0 ‖𝑓(𝑦1, ⋅)‖𝐿𝑝(ℝ𝑑−1) 𝑥1+𝑦1 𝑑𝑦1.
Taking𝐿𝑝((0, ∞), 𝑤𝛾)-norms in𝑥1and applying Step I yields the required result. □
Remark 3.9. Actually, the kernel𝑘 of Lemma 3.8 is a standard Calderón–Zygmund kernel, because 𝑘 is a.e. differentiable and |∇𝑥𝑘(𝑥, 𝑦)| + |∇𝑦𝑘(𝑥, 𝑦)| ≤ |𝑥 − 𝑦|−𝑑−1, 𝑥 ≠ 𝑦.
Although we will not need it below let us note that [19, Corollary 2.10] implies that𝐼𝑘is bounded on𝐿𝑝(ℝ𝑑, 𝑤)for any𝑤 ∈ 𝐴𝑝
4
POINTWISE MULTIPLICATION WITH 𝟏
ℝ𝒅 +In this section we prove the pointwise multiplier result, which is central in the characterization of the complex interpolation spaces of Sobolev spaces with boundary conditions in Section 6. Let𝑤𝛾(𝑥1, ̃𝑥) = |𝑥1|𝛾, where𝑥1∈ℝ and ̃𝑥 ∈ ℝ𝑑−1.
Theorem 4.1. Let𝑋 be a UMD space, let 𝑝 ∈ (1, ∞), 𝛾 ∈ (−1, 𝑝 − 1), 𝛾′= −𝛾∕(𝑝 − 1), and assume −𝛾′+1
𝑝′ < 𝑠 <
𝛾+1 𝑝 . Then
for all𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤𝛾;𝑋)∩𝐿𝑝(ℝ𝑑, 𝑤𝛾;𝑋), we have𝟏ℝ𝑑 +𝑓 ∈ 𝐻 𝑠,𝑝(ℝ𝑑, 𝑤 𝛾;𝑋 ) and ‖‖ ‖𝟏ℝ𝑑 +𝑓‖‖‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋)≲𝑋,𝑝,𝛾,𝑠‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋), and therefore, pointwise multiplication by𝟏ℝ𝑑
+extends to a bounded linear operator on𝐻
𝑠,𝑝(ℝ𝑑, 𝑤𝛾;𝑋).
To prove this the UMD property will only be used through the norm equivalence of Lemma 4.2 below.
Lemma 4.2. Let𝑋 be a UMD space, let 𝑝 ∈ (1, ∞), 𝑠 ∈ ℝ, 𝜎 ≥ 0, and let 𝑤 ∈ 𝐴𝑝. Then (−Δ)𝜎∕2∶(ℝ𝑑;𝑋)←→ ′(ℝ𝑑;𝑋), 𝑓 → ℱ−1
[
(𝜉 → |𝜉|𝜎) ̂𝑓 ]
defines (by extension by density) a bounded linear operator from 𝐻𝑟+𝜎,𝑝(ℝ𝑑, 𝑤; 𝑋) to 𝐻𝑟,𝑝(ℝ𝑑, 𝑤; 𝑋), independent of 𝑟 ∈ ℝ and 𝑤 (in the sense of compatibility), which we still denote by (−Δ)𝜎∕2. Moreover,𝑓 ∈ 𝐻𝑠+𝜎,𝑝(ℝ𝑑, 𝑤; 𝑋)if and only if
𝑓, (−Δ)𝜎∕2𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋), in which case
Proof. All assertions follow from the fact that the symbols 𝜉 → |𝜉|𝜎 (1 +|𝜉|2)2∕𝜎, 𝜉 → 1 (1 +|𝜉|2)2∕𝜎, 𝜉 → (1 +|𝜉|2)2∕𝜎 1 +|𝜉|𝜎
satisfy the conditions of Proposition 2.3. □
In the proof of Theorem 4.1 we will use the norm equivalence of the above lemma via (a variant of) a well known representation for (−Δ)𝜎∕2as a singular integral. For𝑓 ∈ 𝐻𝜎,𝑝(ℝ𝑑)this representation reads as follows:
(−Δ)𝜎∕2𝑓 = lim
𝑟→0+𝐶𝑑,𝜎∫
ℝ𝑑⧵𝐵(0,𝑟)
𝑇ℎ𝑓 − 𝑓
ℎ 𝑑ℎ,
with limit in𝐿𝑝(ℝ𝑑)(see [26, Theorem 1.1(e)]); here𝑇ℎdenotes the left translation and𝐶𝑑,𝜎 is a constant only depending on 𝑑 and 𝜎.
In the proof we want to use a formula as above for𝑓 replaced by 𝟏ℝ𝑑
+𝑓, which in general is an irregular function even if 𝑓 is smooth; in particular, a priori it is not clear that𝟏ℝ𝑑
+𝑓 ∈ 𝐻
𝜎,𝑝(ℝ𝑑). We overcome this technical obstacle by Proposition 4.4
below, which provides a (non sharp) representation formula for (−Δ)𝜎∕2in spaces of distributions. For the proof of Proposition 4.4 we need the following simple identity.
Lemma 4.3. For each𝜎 ∈ (0, 1) there exists a constant 𝑐𝑑,𝜎 ∈ (−∞, 0) such that |𝜉|𝜎 =𝑐
𝑑,𝜎∫
ℝ𝑑
𝑒𝚤ℎ⋅𝜉− 1
|ℎ|𝑑+𝜎 𝑑ℎ, 𝜉 ∈ ℝ𝑑.
Moreover, for all𝜙 ∈ (ℝ𝑑)
[𝜉 → |𝜉|𝜎]( 𝜙) ∶= ∫ℝ𝑑|𝜉| 𝜎𝜙(𝜉) 𝑑𝜉 = 𝑐 𝑑,𝜎∫ ℝ𝑑∫ℝ 𝑒𝚤ℎ𝜉− 1 |ℎ|𝑑+𝜎 𝜙(𝜉) 𝑑𝜉 𝑑ℎ =∶ 𝑐𝑑,𝜎∫ℝ𝑑 [ 𝜉 → 𝑒𝚤ℎ𝜉− 1 |ℎ|𝑑+𝜎 ] (𝜙) 𝑑ℎ. (4.1)
Proof. Let𝜉 ∈ ℝ𝑑⧵ {0} and choose 𝑅 ∈ O(𝑛) with 𝑅𝜉 = |𝜉|𝑒1. Thenℎ ⋅ 𝜉 = 𝑅ℎ ⋅ 𝑅𝜉 = |𝜉|𝑅ℎ ⋅ 𝑒1and the substitution𝑦 = |𝜉|𝑅ℎ yields ∫ℝ𝑑 𝑒𝚤ℎ⋅𝜉− 1 |ℎ|𝑑+𝜎 =|𝜉|𝜎∫ℝ𝑑 𝑒𝚤𝑦1− 1 |𝑦|𝑑+𝜎 𝑑𝑦.
Observing that the integral on the right is a number in (−∞, 0), the first identity follows. Next we show (4.1). Given𝜙 ∈ (ℝ𝑑), the first identity gives
[𝜉 → |𝜉|𝜎]( 𝜙) = ∫ℝ𝑑|𝜉| 𝜎𝜙(𝜉) 𝑑𝜉 = 𝑐 𝑑,𝜎∫ ℝ𝑑∫ℝ𝑑 𝑒𝚤ℎ𝜉− 1 |ℎ|𝑑+𝜎 𝑑ℎ 𝜙(𝜉) 𝑑𝜉. Since𝜙 ∈ (ℝ𝑑)and |𝑒𝚤ℎ𝜉− 1| |ℎ|𝑑+𝜎 ≤ 1|ℎ|≤1ℎ−(𝑑−1+𝜎)|𝜉| + 2 ⋅ 1|ℎ|>1|ℎ|−(𝑑+𝜎),
we may invoke Fubini's theorem in order to get [𝜉 → |𝜉|𝜎](𝜙) = 𝑐 𝑑,𝜎∫ ℝ𝑑∫ℝ 𝑒𝚤ℎ𝜉− 1 |ℎ|𝑑+𝜎 𝜙(𝜉) 𝑑𝜉 𝑑ℎ = 𝑐𝑑,𝜎∫ℝ𝑑 [ 𝜉 → 𝑒𝚤ℎ𝜉− 1 |ℎ|𝑑+𝜎 ] (𝜙) 𝑑ℎ, as desired. □
For𝑓 ∈ ′(ℝ𝑑;𝑋)let𝛿ℎ𝑓 = 𝑇ℎ𝑓 − 𝑓, where 𝑇ℎdenotes the left translation byℎ. For 0 < 𝑟 < 𝑅 let 𝐴(𝑟, 𝑅) ∶={𝑥 ∈ ℝ𝑑 ∶ 𝑟 < |𝑥| < 𝑅}be an annulus.
Proposition 4.4 (Representation of (−Δ)𝜎2). Let 𝑝 ∈ (1, ∞) and 𝜎 ∈ (0, 1). For all 𝑠 ≥ 0 and 𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑)⊗ 𝑋 ⊂ 𝐿𝑝(ℝ𝑑;𝑋)we have (−Δ)𝜎2𝑓 = 1 𝑐𝑑,𝜎 𝑟↘0,𝑅↗∞lim [ 𝑥 → ∫𝐴(𝑟,𝑅)𝛿|ℎ|ℎ𝑓(𝑥)𝑑+𝜎 𝑑ℎ ] in 𝐻𝑠−2,𝑝(ℝ𝑑;𝑋),
where𝑐𝑑,𝜎 is the constant of Lemma 4.3.
The weights are left out on purpose, because translations are not well-behaved on weighted𝐿𝑝-spaces. Moreover, no UMD is required in the result above.
Proof. We prove this proposition by proving the following three statements:
1. The linear operator
𝑓 → [
ℎ → 𝛿ℎ𝑓 |ℎ|𝑑+𝜎
]
is bounded from𝐻𝑠,𝑝(ℝ𝑑;𝑋)to𝐿1(ℝ𝑑;𝐻𝑠−2,𝑝(ℝ𝑑;𝑋))for all𝑠 ∈ ℝ and thus gives rise to a bounded linear operator
𝜎∶𝐻𝑠,𝑝(ℝ𝑑;𝑋)←→ 𝐻𝑠−2,𝑝(ℝ𝑑;𝑋), 𝑓 → ∫ ℝ𝑑
𝛿ℎ𝑓 |ℎ|𝑑+𝜎 𝑑ℎ.
2. For all𝑠 ≥ 0 we have
𝜎𝑓 = lim 𝑟↘0,𝑅↗∞ [ 𝑥 → ∫𝐴(𝑟,𝑅)𝛿|ℎ|ℎ𝑓(𝑥)𝑑+𝜎 𝑑ℎ ] in 𝐻𝑠−2,𝑝(ℝ𝑑;𝑋) for every𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑;𝑋)⊂ 𝐿𝑝(ℝ𝑑;𝑋). 3. For all𝑓 ∈ 𝐻−∞,𝑝(ℝ𝑑)⊗ 𝑋, 𝜎𝑓 = 𝑐𝑑,𝜎(−Δ)𝜎2𝑓 in ′(ℝ𝑑;𝑋), (4.2)
where𝑐𝑑,𝜎is the constant of Lemma 4.3. Here𝐻−∞,𝑝(ℝ𝑑)=⋃𝑠∈ℝ𝐻𝑠,𝑝(ℝ𝑑).
(1): To prove this it is enough to establish the boundedness from𝐻𝑠,𝑝(ℝ𝑑;𝑋)to𝐿1(ℝ𝑑;𝐻𝑠−2,𝑝(ℝ𝑑;𝑋)). As the Bessel potential operator𝑠 commutes with𝛿ℎ, we may restrict ourselves to the case𝑠 = 2. Since by Lemma 3.3 𝐻2,𝑝(ℝ𝑑;𝑋)→ 𝑊1,𝑝(ℝ𝑑;𝑋), we only need to estimate
∫ℝ𝑑
‖𝛿ℎ𝑓‖𝐿𝑝(ℝ𝑑;𝑋)
|ℎ|𝑑+𝜎 𝑑ℎ ≲𝑑,𝜎,𝑝‖𝑓‖𝑊1,𝑝(ℝ𝑑;𝑋), 𝑓 ∈ 𝑊
1,𝑝(ℝ𝑑
;𝑋). (4.3)
To this end, let𝑓 ∈ 𝑊1,𝑝(ℝ𝑑;𝑋). Then 𝛿ℎ𝑓 ℎ1+𝜎 = 1|ℎ|≤1|ℎ| −(𝑑−1+𝜎) ∫ 1 0 𝑇𝑡ℎ [ ∇𝑓 ⋅ ℎ|ℎ| ] 𝑑𝑡 + 1|ℎ|>1|ℎ|−(𝑑+𝜎)(𝑇 ℎ𝑓 − 𝑓),
where the integral is an𝐿𝑝(ℝ𝑑;𝑋)-valued Bochner integral. It follows that ‖𝛿ℎ𝑓‖𝐿𝑝(ℝ𝑑;𝑋) |ℎ|𝑑+𝜎 ≤ 1|ℎ|≤1|ℎ|−(𝑑−1+𝜎)∫ 1 0 ‖𝑇𝑡ℎ‖∇𝑓‖𝑋𝑑‖𝐿𝑝(ℝ𝑑) 𝑑𝑡 + 1|ℎ|>1|ℎ|−(𝑑+𝜎)(‖𝑇ℎ𝑓‖𝐿𝑝(ℝ;𝑋)+‖𝑓‖𝐿𝑝(ℝ;𝑋) ) = 1|ℎ|≤1ℎ−(𝑑−1+𝜎)‖∇𝑓‖𝐿𝑝(ℝ;𝑋𝑑)+ 2 ⋅ 1|ℎ|>1|ℎ|−(𝑑+𝜎)‖𝑓‖𝐿𝑝(ℝ;𝑋).
(2): Let𝑠 ≥ 0 and 𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑;𝑋)⊂ 𝐿𝑝(ℝ𝑑;𝑋). By the first assertion and the Lebesgue dominated convergence theorem, 𝜎𝑓 = lim 𝑟↘0,𝑅↗∞∫𝐴(𝑟,𝑅) 𝛿ℎ𝑓 |ℎ|𝑑+𝜎 𝑑ℎ in 𝐻𝑠−2,𝑝 ( ℝ𝑑;𝑋), (4.4)
where the integrals ∫𝐴(𝑟,𝑅)|ℎ|𝛿ℎ𝑑+𝜎𝑓 𝑑ℎ are Bochner integrals in 𝐻𝑠−2,𝑝(ℝ𝑑;𝑋). As 𝑓 ∈ 𝐿𝑝(ℝ𝑑;𝑋), ℎ → |ℎ|𝛿ℎ𝑑+𝜎𝑓 is in 𝐿1(𝐴(𝑟, 𝑅); 𝐿𝑝(ℝ𝑑;𝑋))for every 0< 𝑟 < 𝑅 < ∞. Since 𝐿𝑝(ℝ𝑑;𝑋), 𝐻𝑠−2,𝑝(ℝ𝑑;𝑋)→ ′(ℝ𝑑;𝑋), it follows that the inte-grals∫𝐴(𝑟,𝑅)|ℎ|𝛿ℎ𝑑+𝜎𝑓 𝑑ℎ in (4.4) can also be considered as Bochner integrals in 𝐿𝑝(ℝ𝑑;𝑋), implying that∫𝐴(𝑟,𝑅)|ℎ|𝛿ℎ𝑑+𝜎𝑓 𝑑ℎ = [
𝑥 → ∫𝐴(𝑟,𝑅)𝛿ℎ𝑓(𝑥)
|ℎ|𝑑+𝜎 𝑑ℎ ]
(see [20, Proposition 1.2.25]).
(3) By linearity it suffices to consider the scalar case𝑓 ∈ 𝐻𝑠,𝑝(ℝ𝑑)for some𝑠 ∈ ℝ. By the density of (ℝ𝑑)⊆ 𝐻𝑠,𝑝(ℝ𝑑) (see Lemma 3.4) it suffices to consider𝑓 ∈ (ℝ𝑑). Indeed, this follows from the boundedness of𝜎and (−Δ)𝜎∕2(see (1). Now (4.2) follows from well-known results (see [26, Theorem 1.1(e)]). For convenience we include a direct proof. Using Lemma 4.3, for each𝑓 ∈ (ℝ𝑑;𝑋)we find (−Δ)𝜎∕2𝑓 = ℱ−1 [ (𝜉 → |𝜉|𝜎) ̂𝑓]=ℱ−1 [ 𝑐𝑑,𝜎∫ ℝ𝑑 [ 𝜉 → 𝑒𝚤ℎ𝜉− 1 |ℎ|𝑑+𝜎 𝑓(𝜉)̂ ] 𝑑ℎ ] =𝑐𝑑,𝜎 ∫ℝ𝑑ℱ −1 [ 𝜉 → 𝑒𝚤ℎ𝜉− 1 |ℎ|𝑑+𝜎 𝑓(𝜉)̂ ] 𝑑ℎ = 𝑐𝑑,𝜎∫ ℝ𝑑 𝛿ℎ𝑓 |ℎ|𝑑+𝜎 𝑑ℎ,
where all integrals are in′(ℝ𝑑;𝑋). By (1), for every𝑓 ∈ (ℝ𝑑;𝑋)⊂ 𝐻0,𝑝(ℝ𝑑;𝑋)we have𝜎𝑓 = ∫ℝ𝑑 |ℎ|𝛿ℎ𝑑+𝜎𝑓 𝑑ℎ, where the integral is taken in𝐻−1,𝑝(ℝ𝑑;𝑋)→ ′(ℝ𝑑;𝑋). This proves (4.2), as desired. □
Finally we are in position to prove the pointwise multiplier result.
Proof of Theorem 4.1. We only consider𝑠 ≥ 0. The case 𝑠 < 0 follows from a duality argument using [36, Proposition 3.5]. By Lemma 3.4 it is enough to prove ‖‖‖𝟏ℝ𝑑
+𝑓‖‖‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋)≲𝑠,𝑝,𝑑,𝛾,𝑋‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋) for an arbitrary𝑓 ∈ ( ℝ𝑑)⊗ 𝑋. Let 𝑔 ∶= 𝟏ℝ𝑑 +𝑓 ∈ 𝐿 𝑝(ℝ𝑑)⊗ 𝑋. By Lemma 4.2, we have ‖𝑔‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋)≲𝑠,𝑝,𝑑,𝛾,𝑋 ‖𝑔‖𝐿𝑝(ℝ𝑑,𝑤𝛾;𝑋)+ ‖‖‖(−Δ)𝑠∕2𝑔‖‖‖𝐿𝑝 (ℝ𝑑,𝑤𝛾;𝑋). Clearly,‖𝑔‖𝐿𝑝(ℝ𝑑,𝑤𝛾;𝑋)≤ ‖𝑓‖𝐿𝑝(ℝ𝑑,𝑤𝛾;𝑋)from which we see that it suffices to show
‖‖ ‖(−Δ)𝑠∕2𝑔‖‖‖𝐿𝑝(ℝ𝑑,𝑤𝛾;𝑋)≲𝑠,𝑝,𝑑,𝛾 ‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋). (4.5) By Proposition 4.4, 𝑠,𝑗𝑔 ∶= [ 𝑥 → ∫𝐴(1 𝑗,𝑗) 𝛿ℎ𝑔(𝑥) |ℎ|𝑑+𝑠 𝑑ℎ ] 𝑗→∞ ←→ (−Δ)𝑠∕2𝑔 in 𝐻𝑠−2,𝑝(ℝ𝑑;𝑋)→ ′(ℝ𝑑;𝑋).
In order to finish the proof, it is thus enough to show that𝑠,𝑗𝑔 converges in 𝐿𝑝(ℝ𝑑, 𝑤𝛾;𝑋)+𝐿𝑝(ℝ𝑑;𝑋)→ ′(ℝ𝑑;𝑋)to some𝐺 satisfying
‖𝐺‖𝐿𝑝(ℝ𝑑,𝑤𝛾;𝑋)≲𝑠,𝑝,𝑑,𝛾,𝑋‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋). (4.6)
Indeed, then (−Δ)𝑠∕2𝑔 = 𝐺 and (4.5) holds. Defining
we have 𝑠,𝑗𝑔 = 𝐺1,𝑗+𝐺2,𝑗∶=𝟏ℝ𝑑 +𝑠,𝑗𝑓 + [ 𝑥 → −sgn(𝑥1)∫ 𝐴(1 𝑗,𝑗) 𝟏𝑆(𝑥1, ℎ1) 𝑓(𝑥 + ℎ) |ℎ|𝑑+𝑠 𝑑ℎ ] , (4.7)
where𝑠,𝑗𝑓 is defined analogously to 𝑠,𝑗𝑔:
𝑠,𝑗𝑓 ∶= [ 𝑥 → ∫𝐴(1 𝑗,𝑗) 𝛿ℎ𝑓(𝑥) |ℎ|𝑑+𝑠 𝑑ℎ ] . We first consider {𝐺1,𝑗 } 𝑗∈ℕ. Since 𝑠,𝑗𝑓 𝑗→∞
←→ (−Δ)𝑠∕2𝑓 in 𝐿𝑝(ℝ𝑑;𝑋) by Proposition 4.4, it follows that 𝐺
1∶= 𝟏ℝ𝑑 +(−Δ) 𝑠∕2𝑓 = lim 𝑗→∞𝐺1,𝑗in𝐿𝑝 ( ℝ𝑑;𝑋). By Proposition Lemma 4.2, ‖𝐺1‖𝐿𝑝(ℝ𝑑,𝑤𝛾;𝑋)≤ ‖‖‖(−Δ)𝑠∕2𝑓‖‖‖𝐿𝑝 (ℝ𝑑,𝑤𝛾;𝑋)≲𝑠,𝑝,𝑑,𝛾,𝑋‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋). We next consider {𝐺2,𝑗}𝑗∈ℕ. Observing that
|ℎ| =(|ℎ1|2+|̃ℎ|2 )1∕2 =((|𝑡| + |ℎ1+𝑡|)2+|̃ℎ|2 )1∕2 for allℎ =(ℎ1, ̃ℎ ) ∈ℝ𝑑and𝑡 ∈ ℝ with (𝑡, ℎ1) ∈𝑆, we find ∫𝐴(1 𝑗,𝑗) 𝟏𝑆(𝑥1, ℎ1) ‖𝑓(𝑥 + ℎ)‖𝑋 |ℎ|𝑑+𝑠 𝑑ℎ ≤ ∫ℝ𝑑 ‖𝑓(𝑥 + ℎ)‖𝑋 ( (|𝑥1| + |ℎ1+𝑥1|)2+|̃ℎ|2 )𝑑+𝑠 2 𝑑ℎ = ∫ℝ𝑑 ‖𝑓(𝑦)‖𝑋 ( (|𝑥1| + |𝑦1|)2+| ̃𝑦 − ̃𝑥|2)𝑑+𝑠2 𝑑𝑦 ≤ ∫ℝ𝑑𝑘(𝑥, 𝑦)|𝑦1| −𝑠‖𝑓(𝑦)‖ 𝑋𝑑𝑦,
where𝑘(𝑥, 𝑦) =((|𝑥1| + |𝑦1|)2+| ̃𝑦 − ̃𝑥|2)𝑑2. Applying Lemma 3.8 to the function𝜙(𝑦) = |𝑦
1|−𝑠‖𝑓(𝑦)‖𝑋we thus obtain ‖‖ ‖‖ ‖‖𝑥 → ∫𝐴(1 𝑗,𝑗 ) 𝟏𝑆(𝑥1, ℎ1) ‖𝑓(𝑥 + ℎ)‖𝑋 |ℎ|𝑑+𝑠 𝑑ℎ ‖‖ ‖‖ ‖‖𝐿𝑝(ℝ𝑑,𝑤𝛾) ≤ ‖𝐼𝑘𝜙‖𝐿𝑝(ℝ𝑑,𝑤𝛾) ≲𝑝,𝑑,𝛾 ‖𝜙‖𝐿𝑝(ℝ𝑑,𝑤𝛾) = ‖𝑓‖𝐿𝑝(ℝ𝑑,𝑤𝛾−𝑠𝑝;𝑋). ≲𝑝,𝑑,𝛾 ‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋), where in the last step we applied Lemma 3.7. It follows that the limit𝐺2∶= lim𝑗→∞𝐺2,𝑗exists in𝐿𝑝
( ℝ𝑑, 𝑤 𝛾;𝑋 ) and, moreover, ‖𝐺2‖𝐿𝑝(ℝ𝑑,𝑤𝛾;𝑋)≲𝑝,𝑑,𝛾 ‖𝑓‖𝐻𝑠,𝑝(ℝ𝑑,𝑤𝛾;𝑋).
Finally, combining the just obtained results for {𝐺1,𝑗}𝑗∈ℕ and {𝐺2,𝑗}𝑗∈ℕ, we see that𝐺 ∶= 𝐺1+𝐺2= lim𝑗→∞𝑠,𝑗𝑔 in 𝐿𝑝(ℝ𝑑, 𝑤
𝛾;𝑋)+𝐿𝑝(ℝ𝑑;𝑋)→ ′(ℝ; 𝑋) and (4.6) holds as desired. □
5
INTERPOLATION THEORY WITHOUT BOUNDARY CONDITIONS
For details on interpolation theory we refer the reader to [4,46]. In this section we present some weighted and vector-valued versions of known results.
Lemma 5.1(Extension operator). Let𝑋 be a Banach space. Let 𝑝 ∈ (1, ∞), and 𝑚 ∈ ℕ0. Let𝑤 ∈ 𝐴𝑝be such that𝑤(−𝑥1, ̃𝑥) = 𝑤(𝑥1, ̃𝑥) for 𝑥1∈ℝ and ̃𝑥 ∈ ℝ𝑑−1. Then there exists an operator+𝑚∶𝐿𝑝
( ℝ𝑑 +, 𝑤; 𝑋 ) → 𝐿𝑝(ℝ𝑑, 𝑤; 𝑋)such that 1. For all𝑓 ∈ 𝐿𝑝(ℝ𝑑+, 𝑤; 𝑋),(+𝑚𝑓)||| ℝ𝑑 + =𝑓;
2. for all𝑘 ∈ {0, … , , 𝑚}, +𝑚∶𝑊𝑘,𝑝(ℝ+𝑑, 𝑤; 𝑋)→ 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)is bounded.
Moreover, if𝑓 ∈ 𝐿𝑝(ℝ𝑑+, 𝑤; 𝑋)∩𝐶𝑚(ℝ𝑑+;𝑋), then+𝑚𝑓 is 𝑚-times continuous differentiable on ℝ𝑑. By a reflection argument the same holds forℝ𝑑−. The corresponding operator will be denoted by−𝑚.
Proof. The result is a simple extension of the classical construction given in [1, Theorem 5.19] to the weighted setting. The final
assertion is clear from the construction of+𝑚. □
To define Bessel potential spaces on domains, we proceed in an abstract way using factor spaces.
Definition 5.2. Let𝔽 → ′(ℝ𝑑;𝑋)be a Banach space. Define the restricted space/factor space to an open set Ω⊆ ℝ𝑑as 𝔽 (Ω) ∶={𝑓 ∈ ′(ℝ𝑑;𝑋)∶ ∃𝑔 ∈ 𝔽 , 𝑓 = 𝑔| Ω } and let ‖𝑓‖𝔽 (Ω)= inf{‖𝑔‖𝔽 ∶𝑔|Ω=𝑓 } . We say that is an extension operator for 𝔽 (Ω) if
1. for all𝑓 ∈ 𝔽 (Ω), (𝑓)|Ω=𝑓;
2. ∶ 𝔽 (Ω) → 𝔽 is bounded.
For𝑝 ∈ (1, ∞), 𝑤 ∈ 𝐴𝑝and an open set Ω⊂ ℝ𝑑, we define the Bessel potential space𝐻𝑠,𝑝(Ω, 𝑤; 𝑋) as the factor space 𝐻𝑠,𝑝(Ω, 𝑤; 𝑋) ∶=[𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)](Ω).
By Lemma 5.1 and for𝑤 as stated there, we find that 𝑊𝑘,𝑝(ℝ𝑑+, 𝑤; 𝑋)can be identified (up to an equivalent norm) with the factor space[𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)](ℝ𝑑+), where an extension operator can also be found. Indeed, let 𝑊factor𝑘,𝑝 (ℝ𝑑+, 𝑤; 𝑋)= [
𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)](ℝ𝑑
+ )
denote the factor space. For𝑓 ∈ 𝑊factor𝑘,𝑝 (ℝ𝑑, 𝑤; 𝑋)let𝑔 ∈ 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)be such that𝑔|ℝ𝑑 +=𝑓. Then
‖𝑓‖𝑊𝑘,𝑝(ℝ𝑑+,𝑤;𝑋)≤ ‖𝑔‖𝑊𝑘,𝑝(ℝ𝑑,𝑤;𝑋). Taking the infimum over all of the above𝑔, we find
‖𝑓‖𝑊𝑘,𝑝(ℝ𝑑+,𝑤;𝑋)≤ ‖𝑓‖𝑊𝑘,𝑝
factor(ℝ𝑑+,𝑤;𝑋). Next let𝑓 ∈ 𝑊𝑘,𝑝(ℝ𝑑+, 𝑤; 𝑋). Then+𝑓 ∈ 𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)and
‖𝑓‖𝑊𝑘,𝑝
factor(ℝ𝑑+,𝑤;𝑋)≤ ‖+𝑓‖𝑊𝑘,𝑝 (
ℝ𝑑,𝑤;𝑋) ≤ 𝐶‖𝑓‖𝑊𝑘,𝑝(ℝ𝑑+,𝑤;𝑋).
Next we present two abstract lemmas to identify factor spaces in the complex interpolation scale. The result is a straightforward consequence of [46, Theorem 1.2.4]. We include the short in order to be able to track the constants. For details on complex interpolation theory we refer to [46, Section 1.9.3].
Lemma 5.3. Let (𝑋0, 𝑋1) and (𝑌0, 𝑌1) be interpolation couples and let 𝑋𝜃 = [𝑋0, 𝑋1]𝜃 and 𝑌𝜃= [𝑌0, 𝑌1]𝜃 for a given 𝜃 ∈ (0, 1). Assume that 𝑅 ∶ 𝑋0+𝑋1→ 𝑌0+𝑌1and𝑆 ∶ 𝑌0+𝑌1→ 𝑋0+𝑋1are linear operators such that𝑆 ∈
( 𝑌𝑗, 𝑋𝑗), 𝑅 ∈ (𝑋𝑗, 𝑌𝑗)and𝑅𝑆 is the identity operator on 𝑌𝑗 for𝑗 ∈ {0, 1}. Then 𝑆𝑅 defines a projection on 𝑋𝜃 and𝑅 is an isomor-phism from𝑆𝑅(𝑋𝜃)onto𝑌𝜃with inverse𝑆. Moreover, the following estimates hold:
𝐶−1 𝑆 ‖𝑆𝑦‖𝑋𝜃 ≤ ‖𝑦‖𝑌𝜃 ≤ 𝐶𝑅‖𝑆𝑦‖𝑋𝜃, 𝑦 ∈ 𝑌𝜃, ‖𝑅𝑥‖𝑌𝜃 ≤ 𝐶𝑅‖𝑥‖𝑋𝜃, 𝑥 ∈ 𝑋𝜃, ‖𝑥‖𝑋𝜃 ≤ 𝐶𝑆‖𝑅𝑥‖𝑌𝜃, 𝑥 ∈ 𝑆𝑅(𝑋𝜃), where𝐶𝑅 = max𝑗∈{0,1}‖𝑅‖(𝑋 𝑗,𝑌𝑗)and𝐶𝑆 = max𝑗∈{0,1}‖𝑆‖(𝑋𝑗,𝑌𝑗).
Proof. By complex interpolation we know
‖𝑆‖(𝑌𝜃,𝑋𝜃)≤ 𝐶𝑆, and ‖𝑅‖(𝑋𝜃,𝑌𝜃)≤ 𝐶𝑅
and𝑅𝑆 is the identity operator on 𝑌𝜃. This proves the upper estimates for𝑆 and 𝑅. To see that 𝑆𝑅 is a projection note that (𝑆𝑅)(𝑆𝑅) = 𝑆𝑅. The lower estimate for 𝑆 follows from
‖𝑦‖𝑌𝜃 =‖𝑅𝑆𝑦‖𝑌
𝜃 ≤ 𝐶𝑅‖𝑆𝑦‖𝑋𝜃, 𝑦 ∈ 𝑌𝜃. To prove the lower estimate for𝑅 note that for 𝑥 ∶= 𝑆𝑅𝑢 ∈ 𝑆𝑅(𝑋𝜃)
‖𝑥‖𝑋𝜃 =‖𝑆𝑅𝑆𝑅𝑢‖𝑋
𝜃 ≤ 𝐶𝑆‖𝑅𝑆𝑅𝑢‖𝑌𝜃 =𝐶𝑆‖𝑅𝑥‖𝑋𝜃. □
Lemma 5.4. Let𝔽0, 𝔽1→ ′(ℝ𝑑;𝑋)be two Banach spaces. For𝜃 ∈ (0, 1), let 𝔽𝜃 =[𝔽0, 𝔽1]
𝜃.
Let Ω⊆ ℝ𝑑 be an open set, and define𝔽𝜃(Ω) as in Definition 5.2, and assume there is an extension operator for 𝔽𝑠(Ω) for 𝑠 ∈ {0, 1}. Then[𝔽0(Ω), 𝔽1(Ω)]
𝜃=𝔽𝜃(Ω) and
𝐶−1‖𝑓‖
𝔽𝜃(Ω)≤ ‖𝑓‖[𝔽0(Ω),𝔽1(Ω)]
𝜃 ≤ ‖𝑓‖𝔽𝜃(Ω)
where𝐶 only depends on the norms of the extension operator. Moreover, is an extension operator for 𝔽𝜃(Ω).
Proof. Define𝑅 ∶ 𝔽𝑗 → 𝔽𝑗(Ω) by𝑅𝑓 = 𝑓|Ω and𝑆 ∶ 𝔽𝑗(Ω)→ 𝔽𝑗 as 𝑆 = . Then ‖𝑅‖ ≤ 1, ‖𝑆‖ ≤ 𝐶 and 𝑅𝑆 = 𝐼. From Lemma 5.3 we conclude that for all𝑓 ∈[𝔽0(Ω), 𝔽1(Ω)]𝜃
𝐶−1‖𝑓‖
𝔽𝜃(Ω)≤ 𝐶−1‖𝑓‖𝔽𝜃 ≤ ‖𝑓‖[𝔽0(Ω),𝔽1(Ω)] 𝜃.
Conversely, let𝑓 ∈ 𝔽𝜃(Ω). Choose,𝑔 ∈ 𝔽𝜃such that𝑅𝑔 = 𝑔|Ω=𝑓. Since ‖𝑅‖ ≤ 1, by complex interpolation we find ‖𝑓‖[𝔽0(Ω),𝔽1(Ω)]
𝜃 ≤ ‖𝑔‖[𝔽0,𝔽1]
𝜃 =‖𝑔‖𝔽𝜃. Taking the infimum over all𝑔 as above, the result follows.
To show the final assertion, note that ∈ (𝔽𝜃(Ω), 𝔽𝜃)by the above. Moreover, for𝑓 ∈ 𝔽0(Ω) ∩𝔽1(Ω), (𝑓)|Ω=𝑓. By
density (see [46, Theorem 1.9.3]) this extends to all𝑓 ∈ 𝔽𝜃(Ω). □
Proposition 5.5. Let𝑋 be a UMD space, let 𝑝 ∈ (1, ∞), 𝑘 ∈ ℕ0and assume that𝑤 ∈ 𝐴𝑝 is such that𝑤(𝑥1, ̃𝑥) = 𝑤(−𝑥1, ̃𝑥) for𝑥1∈ℝ and ̃𝑥 ∈ ℝ𝑑−1. Then𝐻𝑘,𝑝(ℝ+𝑑, 𝑤; 𝑋)=𝑊𝑘,𝑝(ℝ𝑑+, 𝑤; 𝑋).
Proof. This is immediate from Proposition 3.2 and the fact that 𝑊𝑘,𝑝(ℝ𝑑+, 𝑤; 𝑋) coincides with the factor space [
𝑊𝑘,𝑝(ℝ𝑑, 𝑤; 𝑋)](ℝ𝑑
+ )
. □
Next we identify the complex interpolation spaces of𝐻𝑠,𝑝(Ω, 𝑤; 𝑋). Here the UMD property is needed to obtain bounded imaginary powers of −Δ.
Proposition 5.6. Let𝑋 be a UMD space and let 𝑝 ∈ (1, ∞). Let 𝑤 ∈ 𝐴𝑝be such that𝑤(−𝑥1, ̃𝑥) = 𝑤(𝑥1, ̃𝑥) for all 𝑥1∈ℝ and ̃𝑥 ∈ ℝ𝑑−1.
(1) Let𝜃 ∈ [0, 1] and let 𝑠0, 𝑠1, 𝑠 ∈ ℝ be such that 𝑠 = 𝑠0(1 −𝜃) + 𝑠1𝜃. Then for Ω = ℝ𝑑or Ω =ℝ𝑑+one has [
𝐻𝑠0,𝑝(Ω, 𝑤; 𝑋), 𝐻𝑠1,𝑝(Ω, 𝑤; 𝑋)]
𝜃 =𝐻𝑠,𝑝(Ω, 𝑤; 𝑋).
(2) For each𝑚 ∈ ℕ0there exists an+𝑚∈(𝐻−𝑚,𝑝(ℝ𝑑+, 𝑤; 𝑋), 𝐻−𝑚,𝑝(ℝ𝑑, 𝑤; 𝑋))such that • for all|𝑠| ≤ 𝑚, +∈(𝐻𝑠,𝑝(ℝ+𝑑, 𝑤; 𝑋), 𝐻𝑠,𝑝(ℝ𝑑, 𝑤; 𝑋)),
• for all|𝑠| ≤ 𝑚, 𝑓 → (+𝑓)|ℝ𝑑
+equals the identity operator on𝐻
𝑠,𝑝(ℝ +, 𝑤; 𝑋). Moreover, if𝑓 ∈ 𝐿𝑝(ℝ𝑑+, 𝑤; 𝑋)∩𝐶𝑚 ( ℝ𝑑 +;𝑋 ) , then+𝑚𝑓 ∈ 𝐶𝑚(ℝ𝑑;𝑋).
