In Section 4.3 we noticed the importance of compatibility embeddings as in Proposition 4.17 and Propo-sition 4.18. Thus, it is worthwhile to consider such compatibility embeddings for Triebel-Lizorkin spaces.
As in Section 4.3 we are faced to the interpolation of intersections of spaces and therefore it is helpful to have a characterization of these intersections as anisotropic Triebel-Lizorkin spaces. Hence, we provide such a characterization as well as the behavior under interpolation in this section.
Lebesgue, Bessel-potential, and Besov spaces with mixed norms are well-known in the literature, see for example in [BP61], [Bug71], [BIN78], and [BIN79]. However, there is not much literature concerning
5.2: Anisotropic Triebel-Lizorkin spaces and representation by intersections 107 Triebel-Lizorkin spaces with mixed norms. In [JS08] one can find general results on the trace problem of scalar-valued anisotropic Triebel-Lizorkin spaces with mixed norms. There, the authors show that the trace spaces are also of this type but there is no representation of the anisotropic Triebel-Lizorkin spaces with mixed norms by an intersection of spaces as in [Ber87a], [Wei05], and [DHP07]. In the following we want to show that the anisotropic Triebel-Lizorkin spaces with mixed norms used in [JS08]
can be represented as an intersection of two spaces in some special cases. First, we state some results of M.Z. Berkolaiko which are of interest for our purpose.
For anisotropic Besov and Bessel-potential spaces a representation by intersections can be found in [Ama09, Theorem 3.6.3, Theorem 3.7.2] for the non-mixed norm case. These proofs, however, cannot be carried over directly to the mixed norm case because we cannot change the order of variables.
We want to state the definitions of the anisotropic Triebel-Lizorkin spaces with mixed norms given in [Ber85] and [JS08].
Definition 5.15(Anisotropic Triebel-Lizorkin space with mixed norms). (i) LetX be an ar-bitrary Banach space and ~p∈(1,∞)n. Then we define Lp~(Rn, X) as the space of all measurable all measurable functionsf:Rn→Csuch that
kfkL~r
Here the anisotropy is given by~r.
(iii) ([JS08]): For~a∈(0,∞)n andξ∈Rn we define the anisotropic distance function|ξ|~a as the unique
Here the anisotropy is given by~a.
Remark 5.16. To avoid confusion we want to mention that in the works of M.Z. Berkolaiko, J. Johnsen, W. Sickel the last variablexn is associated with the time variable and(x1, . . . , xn−1)are associated with the space variables. The definitions ofL~r~p,q(Rn)andF~p,qs,~a(Rn)differ in their decomposition and partition of unity, respectively. In Definition 5.15 (ii) the author uses a partition based on a cubic structure whereas in Definition 5.15 (iii) the partition is based on an ellipsoid structure. It is conjectured that both definitions are equivalent but this is not necessary for our purposes. We only need the relation given in Remark 5.19 below.
Definition 5.17 (Anisotropic Bessel-potential space with mixed norms). For ~p ∈ (1,∞)n,
~a∈(0,∞)n, ands≥0 we define Hp~s,~a(Rn)as the set of allu∈L~p(Rn)such that F−1(1 +|ξ|2~a)s/2Fu∈L~p(Rn).
The norm is then given by kukHs,~a
~
p (Rn):=kF−1(1 +|ξ|2~a)s/2FukLp~(Rn), u∈H~ps,~a(Rn).
Remark 5.18. Let~p∈(1,∞)n,~a∈(0,∞)n, ands≥0. Ifm~ := (m1, . . . , mn) := (s/a1, . . . , s/an)∈Nn0, then we have
H~ps,~a(Rn) =W~pm~(Rn)
where the anisotropic Sobolev space W~pm~(Rn)consists of allu∈Lp~(Rn)such that
∂mku∈Lp~(Rn), k= 1, . . . , n, cf. [JS08, Proposition 2.10 (ii)]. The canonical norm is given by
kukWm~
~
p(Rn):=kukL~p(Rn)+
n
X
k=1
k∂kmkukL~p(Rn).
Remark 5.19 ([Ber87a, Theorem 4 B.], [JS08, Proposition 2.10 (i)]). For all ~p∈(1,∞)n,~r∈(0,∞)n, ands≥0 we have
H~ps,~a(Rn) =L~r~p,2(Rn) =F~p,2s,~a(Rn), ~a:= (s/r1, . . . , s/rn)
whereH~ps,~a(Rn)denotes the anisotropic Bessel-potential spaces with mixed norms. In Soviet mathematical literature the Bessel-potential spaces are usually called Liouville spaces and are denoted by ’Lsp’.
Remark 5.20 ([JS08, Proposition 3.15]). Let~a∈ (0,∞)n,p~∈ (1,∞)n,q ∈(1,∞), and s, r ≥0 with s−r≥0. Then we have the isomorphism
Λr,~a:=F−1(1 +|ξ|2~a)r/2F ∈LIsom
F~p,qs,~a(Rn), F~p,qs−r,~a(Rn) . Remark 5.21 (Trace result of M.Z. Berkolaiko).
(i) In [Ber85, Theorem 2], respectively [Ber87b, Corollary 1], M.Z. Berkolaiko proved the following trace result for the semi-isotropic case: If κ:= 1−(pνrν)−1>0,ν∈ {1, . . . , n}, and if there exists r >0 such that rν+1=. . .=rn=r, then
γxν=0 L~r~p,2(Rn)
=L~pz(Rn−ν, Bρ~p~w
w,pν(Rν−1))∩Lρ~p
z,pν(Rn−ν, L~pw(Rν−1))
where ~ρw := (κr1, . . . , κrν−1), ρ := κr, ~pw := (p1, . . . , pν−1), and p~z := (pν+1, . . . , pn) (i.e. the restriction operator is continuous and there exists a continuous extension operator). For the defi-nition of the anisotropic Besov space and the vector-valued Triebel-Lizorkin space by differences we refer to [Ber87b]. In the following we only need the isotropic and non-mixed norm versions of these spaces.
(ii) Note that for the trace in the ’space’ variable xν=n−1 = 0 (see Remark 5.16) the assumption rν+1=. . .=rn=ris always fulfilled.
(iii) Let ν = n−1, ~r := (l, . . . , l, t) ∈ (0,∞)n, and ~p := (q, . . . , q, p) ∈ (1,∞)n such that we have κ= 1−(lq)−1>0. Then we get
~
pw= (q, . . . , q)∈(1,∞)n−2, pν=n−1=q, ~pz=p, r=t, ~ρw:= (κl, . . . , κl), ρ=κt.
Hence (i) reads as follows
γxn−1=0 L~r~p,2(Rn)
=Lp(R, Bκlq,q(Rn−2))∩Lκtp,q(R, Lq(Rn−2)). (5.13) At this point we want to emphasize that on the right-hand side of (5.13) only isotropic vector-valued spaces appear. In this semi-isotropic and semi-mixed norm situation the variablesx1 andxn−1can be interchanged without consequences. Hence, we also get
γx1=0 L~r~p,2(Rn)
=Lp(R, Bq,qκl(Rn−2))∩Lκtp,q(R, Lq(Rn−2)). (5.14)
5.2: Anisotropic Triebel-Lizorkin spaces and representation by intersections 109 (iv) In the works of M.Z. Berkolaiko the vector-valued Triebel-Lizorkin spaces on the right-hand side of (5.13) and (5.14) are defined by differences. Due to Remark 5.2 this yields the same space as in Definition 5.1. Thus, we finally obtain
γx1=0 L~rp,2~ (Rn)
=Lp(R, Bqκl(Rn−2))∩Fp,qκt(R, Lq(Rn−2)).
Remark 5.22 (Trace result of J. Johnsen and W. Sickel). In [JS08, Theorem 2.2] one can find the following results:
(i) Let~p∈(1,∞)n,~a∈(0,∞)n, ands > a1/p1. Then we have γx1=0
F~p,2s,~a(Rn)
=F~ps−a00,p11/p1,~a00(Rn−1)
where ~a00 := (a2, . . . , an) and ~p00 := (p2, . . . , pn) (i.e. the restriction operator is continuous and there exists a continuous extension operator).
(ii) Let ~p= (q, . . . , q, p)∈(1,∞)n and~a= (l−1, . . . , l−1, t−1)∈(0,∞)n such that s >1/(lq)then (i) reads as follows
γx1=0
F~p,2s,~a(Rn)
=Fs−1/(lq),~a
00
~
p00,q (Rn−1) where~a00= (l−1, . . . , l−1, t−1)and~p00= (q, . . . , q, p).
Proposition 5.23 (Representation of anisotropic Triebel-Lizorkin spaces by intersections).
Let s > 0, ~a = (l−1, . . . , l−1, t−1) ∈ (0,∞)n, p, q ∈ (1,∞), and ~p := (q, . . . , q, p) ∈ (1,∞)n. Then we obtain the following representation of anisotropic spaces with mixed norms
F~p,qs,~a(Rn) = Fp,qst(R, Lq(Rn−1))∩Lp(R, Bqsl(Rn−1)). (5.15) Proof. Lets >0,~a= (l−1, . . . , l−1, t−1)∈(0,∞)n, p, q∈(1,∞), and~p:= (q, . . . , q, p)∈(1,∞)n. Then we defines0:=s+ 1/(lq)>1/(lq)and get
Fp,q~s,~a(Rn) =Fs
0−1/(lq),~a
~
p,q (Rn).
Due to Remark 5.22 (ii) we have γx1=0
Fs
0,~a+
~
p+,2 (Rn+1)
=F~p,qs0−1/(lq),~a(Rn)
where ~a+ := (l−1, . . . , l−1, t−1) ∈ (0,∞)n+1 and ~p+ := (q, . . . , q, p) ∈ (1,∞)n+1. According to Re-mark 5.19 we have F~ps0,~a+
+,2 (Rn+1) = L~r~p+
+,2(Rn+1) where ~r+ := (s0l, . . . , s0l, s0t) ∈ (0,∞)n+1. In this situation Remark 5.21 (iv) states
γx1=0
L~rp~+
+,2(Rn+1)
= Lp(R, Bqκs0l(Rn−1))∩Fp,qκs0t(R, Lq(Rn−1))
= Lp(R, Bqs0l−1/q(Rn−1))∩Fp,qs0t−t/(lq)(R, Lq(Rn−1))
= Lp(R, Bqsl(Rn−1))∩Fp,qst(R, Lq(Rn−1)) withκ= 1−1/(s0lq). Next, we want to apply Lemma 2.8 with
X:=Hs
0,~a+
~
p+ (Rn+1), Y0:=Fp,qst(R, Lq(Rn−1))∩Lp(R, Bqsl(Rn−1)), Y1:=F~p,qs,~a(Rn).
Remarks 5.21 and 5.22 then yield that there exist trace operators
R0:=γxB1=0∈L(X, Y0), R1:=γxJS1=0∈L(X, Y1)
with corresponding extension operators E0:=extBx
1=0∈L(Y0, X), E1:=extJSx
1=0∈L(Y1, X).
We haveD :=S(Rn+1),→d X and triviallyR0f =R1f for all f ∈D. Lemma 2.8 then yields Y0=Y1
with equivalence of norms.
The next aim is to show that anisotropic Triebel-Lizorkin spaces with mixed norms are an interpolation scale. This can be proved by a common retraction argument for which we need some results on Fourier-multipliers on mixedLp-spaces.
Remark 5.24(Fourier multipliers on spaces with mixed norms (cf. [Hyt05, Theorem 2.2])).
Let X be a Banach space of classHT with property(α)and~p∈(1,∞)n. If m∈Cn(Rn\ {0}, L(X))
is given such that{ξαDαm(ξ) :ξ∈Rn\ {0}, α∈ {0,1}n} ⊆L(X)isR-bounded, thenmis anL~p-Fourier multiplier, i.e. we have
Tm∈L(L~p(Rn, X)).
Lemma 5.25. Using the functions (Φk)k∈N0 introduced in Definition 5.15 (iii) we define Φe0:= Φ0+ Φ1
andΦek:= Φk−1+ Φk+ Φk+1 fork∈N.
(i) For all α∈Nn0 there existsC(α, ~a)>0 such that
kξαDαΦkk∞ ≤ C(α, ~a), k∈N0, (5.16) kξαDαΦekk∞ ≤ 3C(α, ~a), k∈N0. (5.17) (ii) Letp~∈(1,∞)n,q∈(1,∞), and g= (gk)k∈N0 ∈L~p(Rn, `q). Then the series P∞
k=0F−1ΦekFgk is convergent in S0(Rn).
Proof. (i) For~a∈(0,∞)n we introduce the notation
t~ax:= (ta1x1, . . . tanxn), ts~ax:= (ts)~ax
where t≥0, x∈Rn, ands∈R. Due totη(ξ) =η(t~aξ)(t >0) forη(ξ) :=|ξ|~a we derive (∂jη)(ξ) = (∂jη)(t~aξ)·taj−1, t >0.
Setting t:=|ξ|~−1a we get
|∂jη(ξ)|=|∂jη(|ξ|~−~aaξ)| · |ξ|~1−aa j ≤C(~a)|ξ|1−a~a j, ξ∈Rn\ {0}
due to the compactness of {ζ∈Rn\ {0}:|ζ|~a= 1} and||ξ|−~~aaξ|~a =|ξ|~−1a |ξ|~a= 1. With this we get fork≥1
|ξj||∂jΦk(ξ)| = |ξj||2−k(∂jψ)(2−k|ξ|~a)−2−(k−1)(∂jψ)(2−(k−1)|ξ|~a)||∂jη(ξ)|
≤ 3·2−kk∂jψk∞·χsupp Φk(ξ)|ξj||∂jη(ξ)|
≤ 3·2−kC(~a)k∂jψk∞·χsupp Φk(ξ)|ξj||ξ|~1−aa j. Forξ∈supp Φk we have |ξj| ≤11/10·2kaj and|ξ|~a ≤11/10·2k, which yield
|ξj||∂jΦk(ξ)| ≤ Ck∂jψk∞, ξ∈Rn.
Iterating these arguments we obtain (5.16). The second assertion (5.17) then follows from (5.16) and the definition of Φek.
5.2: Anisotropic Triebel-Lizorkin spaces and representation by intersections 111 (ii) The Fourier transform is continuous inS0(Rn)and so it suffices to show that the scalar sequence
PN one dimensional Hölder inequality, see for example in [BIN78, Ch. 1, 2.4] or [AF03, 2.49]. For all x∈Rn andα, γ∈Nn0 we get With the same arguments as in the proof of (i) we can show that there existsC(α, γ)>0such that kDα−βΦekk∞≤C(α, γ)for allk∈N0. It is obvious that we have#{k∈N0: x∈suppΦek} ≤4 so
Proof. We use the standard method of retraction and coretraction together with Lemma 2.5. First, we
p,q(Rn)trivially yields thatS is an isometry and hence S∈L
F~p,q0,~a(Rn), L~p(Rn, `q) . To obtain a corresponding retraction we define
R: L~p(Rn, `q) → S0(Rn),
where the series converges inS0(Rn)according to Lemma 5.25 (ii). Considering the supports ofΦj and Φej we get
In contrast to other proofs on this topic we use operator-valued Fourier multipliers associated with R-boundedness. In the following we want to apply Remark 5.24, where the necessary R-boundedness can be proved by the square function estimate (Remark B.15). Note that `q is of class HT and has property(α)due to Remarks B.18 and B.21. We define the symbol
m:Rn\ {0} → L(`q(N0)),
According to Lemma 5.25 we can show that there existsC>0such that
|ξαDαmjk(ξ)| ≤ C, ξ∈Rn\ {0}, α∈ {0,1}n, j, k∈N0.
5.3: Auxiliary results on Bessel-valued Triebel-Lizorkin spaces 113
With the triangle inequality for the2-norm andk · k`q we then derive by Remark 5.24 and Remark B.15.
Forf ∈F~p,q0,~a(Rn)we deduce with corresponding coretractionS. This implies
h
by Lemma 2.5, Theorem A.11, and Theorem A.12. The general case then follows by Remark 5.20 and the compatibility of interpolation and isomorphisms stated in Lemma 2.6.
5.3 Auxiliary results on Bessel-valued Triebel-Lizorkin spaces
In this section we want to lift the definitions, shifts, and other analog results of Chapter 2 and Chapter 4 to the setting of Bessel-valued Triebel-Lizorkin spaces.
Similar to Definition 2.25 we define an abbreviation for Bessel-valued Triebel-Lizorkin spaces.
Definition 5.27 (Spaces of mixed scales). For%≥0,1 < p <∞, and q ∈(2p/(1 +p),2p),s >0, r∈R, and
F=Fpq, K=Hq
we define
WF,K,%s,r (Rn+1+ ) :=0Fpq,%s (R+, Hqr(Rn)).
Remark 5.28. Note that the restriction on q always comes from the fact that we obtain Bessel-valued Triebel-Lizorkin spaces by interpolation of a pure Bessel-potential scale and the Bessel-valued Besov scale, cf. Corollary 5.11. The Besov spaces can also be obtained by interpolation of Bessel spaces of course. In this sense our approach to Triebel-Lizorkin spaces is purely Bessel oriented.
It might be possible to drop the restriction on q by an approach which follows the intrinsic structure of the Bessel-valued Triebel-Lizorkin spaces. The results of Chapters 2 and 4 mainly depend on Fourier multiplier theorems. Versions of Fourier multiplier theorems on Triebel-Lizorkin spaces can be found in [BK05] and [BK09], for example.