Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 964794, 15 pages http://dx.doi.org/10.1155/2014/964794

Research Article

Anisotropic, Mixed-Norm Lizorkin-Triebel Spaces and Diffeomorphic Maps

J. Johnsen,1 S. Munch Hansen,1 and W. Sickel2

1 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg 0st, Denmark

2 Institute of Mathematics, Ernst-Abbe-Platz 2, 07740 Jena, Germany

Correspondence should be addressed to J. Johnsen; jjohnsen@math.aau.dk Received 22 May 2013; Accepted 28 August 2013; Published 2 March 2014 Academic Editor: Gen-QiXu

Copyright © 2014 J. Johnsen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper gives general results on invariance of anisotropic Lizorkin-Triebel spaces with mixed norms under coordinate transformations on Euclidean space, open sets, and cylindrical domains.

1. Introduction

This paper continues a study of anisotropic Lizorkin-Triebel spaces F!'^(R") with mixed norms, which was begun in [1, 2] and followed up in our joint work [3].

First Sobolev embeddings and completeness of the scale F!'^(R") were established in [1], using the Nikol'skil-Planche-rel-Polya inequality for sequences of functions in the mixed-norm space Lp(R"), which was obtained straightforwardly in [1]. Then a detailed trace theory for hyperplanes in R" was worked out in [2], for example, with the novelty that the well-known borderline s = 1/p has to be shifted upwards in some cases, because of the mixed norms.

Secondly, our joint paper [3] presented some general characterisations of F!'^(R"), which may be specialised to kernels of local means, in Triebel's sense [4]. One interest of this is that local means have recently been useful for obtaining wavelet bases of Sobolev spaces and especially of their generalisations to the Besov and Lizorkin-Triebel scales (cf. works of Vybiral [5, Theorem 2.12], Triebel [6, Theorem 1.20], Hansen [7, Theorem 4.3.1]).

In the present paper, we treat the invariance of F!'1^ under coordinate changes. During the discussions below, the results in [3] are crucial for the entire strategy.

Indeed, we address the main technical challenge to obtain invariance of F!'^(R") under the map

f^foa, (1)

when a is a bounded diffeomorphism on R". (Cf. Theorems 20 and 21) Not surprisingly, this will require the condition on a that only affects blocks of variables Xj in which the corresponding integral exponents pj are equal, and similarly for the anisotropic weights aj. Moreover, when estimating the operator norm of f ^ f ° a, that is, obtaining the inequality

l<cll/

the Fourier analytic definition of the spaces seems difficult to manage directly, so as done by Triebel [4] we have chosen to characterise F!'a (R") in terms oflocal means as developed in [3].

However, the diffeomorphism invariance relies not just on the local means, but first of all also on techniques underlying them. In particular, we use the following inequality for the maximal function y* f(x) of Peetre-Fefferman-Stein type, which was established in [3, Theorem 2] for mixed norms and with uniformity with respect to a general parameter 0:

2S; suP Vej

Hereby the "cut-off" functions fj, fj should fulfill a set of Tauberian and moment conditions; cf. Theorem 14 for the full statement. In the isotropic case this inequality originated in a well-known article of Rychkov [8], which contains a serious

flaw (as pointed out in [7]); this and other inaccuracies were corrected in [3].

A second adaptation of Triebel's approach is caused by the anisotropy a we treat here. In fact, our proof only extends to, for example, s < 0 by means of the unconventional lift operator

Ar = OP (Ar), Ar = + 0

Moreover, to cover all a = (a1,...,an), especially to allow irrational ratios aj/ak, we found it useful to invoke the corresponding pseudodifferential operators (1 - 92)^ = OP((1 + that for ^ e R are shown here to be bounded

FS/(R") ^ F!_2fl'M(R") for all s.

Local versions of our result, in which <r is only defined on subsets of R", are also treated below. In short form we have, for example, the following result (cf. Theorem 22).

Theorem 1. Let U,Vc R" be open and let <r : U ^ V be a Cm-bijection on the form a(x) = (a'(x1,..., x„_1), xn). When f e has compact support and all pj are equal for j < n,

and similarly for the a^, then f ° a e F!'^(U") and

11/ o a

^ (U)||<c(supp (V)|

This is useful for introduction of Lizorkin-Triebel spaces on cylindrical manifolds. However, this subject is postponed to ourforthcomingpaper [9]. (Already this part of the mixed-norm theory has seemingly not been elucidated before.) Moreover, in [9] we also carry over trace results from [2] to spaces over a smooth cylindrical domain in Euclidean space, for example, by analysing boundedness and ranges for traces on the flat and curved parts of its boundary

To elucidate the importance of the results here and in [9], we recall that the F!'a are relevant for parabolic differential equations with initial and boundary value conditions: when solutions are sought in a mixed-norm Lebesgue space Lp (in order to allow different properties in the space and time directions), then Ff'^-spaces are in general inevitable for a correct description of nontrivial data on the curved boundary.

This conclusion was obtained in works of Weidemaier [10-12], who treated several special cases; one may also consult the introduction of [2] for details.

Contents. Section 2 contains a review of our notation, and the definition of anisotropic Lizorkin-Triebel spaces with mixed norms is recalled, together with some needed properties, a discussion of different lift operators and a pointwise multiplier assertion.

In Section 3 results from [3] on characterisation of F!'a-L J M

spaces by local means are recalled and used to prove an

important lemma for compactly supported elements in F!'1^. Sufficient conditions for / ^ /°<r to leave the spaces F!'^(R") invariant for all s e R are deduced in Section 4, when

0 is a bounded diffeomorphism. Local versions for spaces on domains are derived in Section 5 together with isotropic results.

2. Preliminaries

2.1. Notation. The Schwartz space S(R") contains all rapidly decreasing Cm-functions. It is equipped with the family of seminorms, using Da := (-¿9^)"' • • • (-i9x )"" for each multiindex a = (a1,..., an) with e N0 := N U {0}, and (x)2 :=

1 +w2,

pM (<p) := sup |Da<p (%)| | % e R", |a| < m| ,

or with

in,a '

(y):= [ Ne N0, ae N". (7)

The Fourier transformation = #(£) = JR„ e

for g e S(R") extends by duality to the dual space S'(R") of

temperate distributions.

Inequalities for vectors p = (p1,...,p„) are understood componentwise; as are functions, for example, p ! = p1! • • • p„!. Moreover, t+ := max(0, i) for ( e R.

For 0 < p < to the space L^(R") consists of all Lebesgue measurable functions such that

II« | Lz

\pi/pi

i/p„

with the modification of using the essential supremum over Xj in case = to. Equipped with this quasinorm, L^(R") is a quasi-Banach space (normed if > l for all j).

Furthermore, for 0 < q < to we will use the notation L^(l?)(R") for the space of all sequences {wfc}£=0 of Lebesgue measurable functions wfc : R" ^ C such that

|«fc}~ |L i(«,)(R"

Xk (of) n,

with supremum over k in case q = to. This quasi-norm is often abbreviated to ||wfc | Land when p = (p,..., p) we simplify L^ to LIf max(p1,..., pn, < to sequences of C™-functions are dense in L

Generic constants will primarily be denoted by c or C and when relevant, their dependence on certain parameters will be explicitly stated. B(0, r) stands for the ball in R" centered at 0 with radius r > 0, and U denotes the closure of a set U c R".

2.2. Anisotropic Lizorkin-Triebel Spaces with Mixed Norms. The scales of mixed-norm Lizorkin-Triebel spaces refine the scales of mixed-norm Sobolev spaces (cf. [2, Proposition 2.10]), and hence the history of these spaces goes far back in time; the reader is referred to [3, Remark 2.3] and [1, Remark 10] for a brief historical overview, which also list some of the ways to define Lizorkin-Triebel spaces.

Our exposition uses the Fourier-analytic definition, but first we recall the definition of the anisotropic distance function | • \s, where a = (a1,...,an) e [1,on R" and some of its properties. Using the quasihomogeneous dilation tax := (taix1 ,...,ta"xn) for t > 0, \x\a is for % e R" \ {0} defined as the unique t > 0 such that T3 % e S"-1 (|0|3 := 0), that is,

t2a„

By the Implicit Function Theorem, | • \s is Cm on R" \ {0}. We also recall the quasi-homogeneity |tax|3 = ^x^ together with (cf. [1, Section 3])

\X + y\a^Mä+\

max ( \x

ii/«i

■ .,\xn

l/a„

< \X\a < R|

|1/«i

+ ••• + \X,

|1/«n

The definition of Fsf( P'i

1) uses a Littlewood-Paley decomposition, that is, 1 = ®^(O, which (for convenience) is based on a fixed f e C™ such that 0 < y(%) < 1 for all fc = 1 if Ma < 1 and ?(!■) = 0 if > 3/2; setting O = f- f(2s•), we define

Oo (i;) := f ($) , Oj (O := O (2-j3^) , j= 1,2,.

Definition 2. The Lizorkin-Triebel space Fs^(Rn) with s e R,

0 < p < <m and 0 < q < <m consists of all u e S'(R") such that

Definition 3. The Besov space Bs:a^(Rn) consists of all u e S'(R") such that

\\u\Bf (R")l| :=(T2iSqh \Lp (Rn) \}=0

<OT. (16)

In [1, 2] many results on these classes are elaborated, and hence we just recall a few facts. They are quasi-Banach spaces (Banach spaces if min(p1,..., pn, q) > 1) and the quasinorm is subadditive, when raised to the power d := min(1,pl,...,pn,q),

\\u + v \Fsfq (R")

^ i rs,a /rm n

< \\U \ FJ

II 1 M

)\\ +\\v\FS&

u, v e Fi'" ( p,q

Also the spaces do not depend on the chosen anisotropic decomposition of unity (up to equivalent quasinorms) and there are continuous embeddings

ps,a ( p,q

where S is dense in Fsf for q < >x>.

Since for A > 0, the space Fs/ coincides with FXJM, cf. [2, Lemma 3.24], most results obtained for the scales when a > 1 can be extended to the case 0 < a < 1 (for details we refer to [3, Remark 2.6]). The subspace L1

Joc(R") c D'(Rn) of locally integrable functions is equipped with the Frechet space topology defined from the seminorms u ^ \\x\<j \u(x)\dx, j e N. By

Cb(R") we denote the Banach space of bounded, continuous functions, endowed with the supremum norm.

Lemma 4. Let s e R and a e N" be arbitrary.

ps,a ( p,q

'n \1/q

Y^F-1 (0;- (?) Fu(i,))(^ ) \Lp (R»)

< >x>.

The number q is called the sum exponent and the entries in p are integral exponents, while s is a smoothness index. Usually the statements are valid for the full ranges 0 < p < >x>, 0 < q < >x>, so we refrain from repeating these. Instead we focus on whether s e R is allowed or not. In the isotropic case, that is, a = (1,..., 1), the parameter a is omitted.

We will also consider the closely related Besov spaces, recalled using the abbreviation

Uj (x) := F-1 (Oj (O Fu ($)) (x), xe R", j e N0

(i) The differential operator Da is bounded FS^(R") ^

^s-a-U'd /rrr¡n

(ii) For s > Y!e=1((ae/pe) - ae)+ there is an embedding

F£(R«) Uoc(Rn).

(iii) The embedding F^ ^ Cb(R") holds true whenever s > (a-i/pi) + + (an/pn).

Proof. For part (i) the reader is referred to [2, Lemma 3.22], where a proof using standard techniques for is indicated (though the cross-reference in that proof in [2] should have been to Proposition 3.13 instead of 3.14).

Part (ii) is obtained from the Nikol'skij inequality (cf. [1, Corollary 3.8]), which allows a reduction to the case in which pj > 1 for j = 1,...,n, while s > 0; then the claim follows

from the embedding F^ L 1>loc. Part (iii) follows at once from [2, (3.20)]. ' □

A local maximisation over a ball can be estimated in Lat least for functions in certain subspaces of Cfc(R") (cf. Lemma 4(iii)).

Lemma 5 (see [3]). When s > «;/ min(p1;..., p), then for each C > 0

sup |wM||L , (R"

|x-y|<C

< c II« | Fs

Next we extend a well-known embedding to the mixed-norm setting. Let C^ (R") denote the Holder class of order p > 0, which by definition consists of all u e Cfc(R") satisfying ||w||p := j sup|D"w(x)|

jajsfc5

+ X sup | D"m(x)-D"w(y) | | x-y |

j«j=fcx-yeR"\{0}

whereby fc is the integer satisfying k < ^ < fc + 1.

Lemma 6. For p > 0 and s e R with s < p there is an embeddingC£(R") £°f»(R").

Proof. The claim follows by adapting the proof of [14, Proposition 8.6.1] to the anisotropic case, that is,

||M I 5°°U|| = sup2sj sup |F-1 (o;Fm) (x)| < cp

The expressions in the Besov norm are for j > 1 estimated using that F-10 has vanishing moments of arbitrary order,

F-1 (0;-Fw) (%)

j F-10(y)

jajsfc

Using a Taylor expansion of order fc - 1 with fc e N chosen such that fc < ^ < fc+1 (or directly if fc = 0), we get an estimate of the parenthesis by

jaj=fc

j1 (i - 0)fc-1 (a"M (* - 2-j30y) - a"« (x)) c

< X -|2-V||M||J2-'Vffc j1 (1-0)fc-1d0

| ^ | | Jo

„ /U-ia |P|| II

< cp|2 7 ||w||

Now we obtain, since a > 1,

sup | F-1 (o.-Fw) (%) | < c'2-jp||w|L [ | F-10 (y) 11 y | p dy

xeR" | | P J | |

< cp2-^||M||p.

This bound can also be used for j = 0, if cp is large enough, so (21) holds for p > s. □

As a tool we also need to know the mapping properties of certain Fourier multipliers A(D)w := F- (A(£)w(£)). For generality's sake, we give the following.

Proposition 7. When A e C° seminorms of the form

*) for some r e R has finite

Ca (A):= sup |D"A(2^)| | j 6 No, I < |*|a < 4},

a 6 N0, (25)

then A(D) is continuous on moreover bounded

F!'3(R") ^ F!-r,3(R") for a// s e R, with operator norm

|WD)|| < c^ Ca(A).

Proof. The quasi-homogeneity of | • |5 yields that |DaA(£)| < cCa(A)(1 + |^|a)r-aa, and hence every derivative is of polynomial growth (cf. (12)), so A(D) is a well-defined continuous map on S'. Boundedness follows as in the proof of [2, Proposition 3.15], mutatis mutandis. In fact, only the last step there needs an adaptation to the symbol A(£), but this is trivial because finitely many of the constants Ca(A) can enter the estimates. □

2.3. Lift Operators. The invariance under coordinate transformations will be established below using a somewhat unconventional lift operator A r, re R,

Ar« = OP (Ar ($)« = F-1 (Ar (?)«(?)),

a, (i) = jr (1+ii)"'2-(26)

To apply Proposition 7, we derive an estimate uniformly in j e N0 and over the set 1/4 < |£|3 < 4: while the mixed derivatives vanish, the explicit higher order chain rule in the Appendix yields

(2-j% (2^))| < £Cfc(2-2ja- ^^^V^'-2^ ' fc=1

x X (2(2^^i))"'2"2 < to.

fc=«1 +«2

Indeed, the precise summation range gives = n1 + 2(k-n1), so the harmless power 2"1+"2 results. (Note that this means that \DaXr(2j5^)\ < Ca2i(r-S'a).)

Now Xr(¡-) has no zeros, and for Xr(^)-1 it is analogous to obtain such estimates uniformly with respect to j of Da (2jrXr(2jS 0-1), using the Appendix and the aforementioned. So Proposition 7 gives both that Ar is a homeomor-phism on S' (although A-!1 =A-r) and the proof of our lemma.

Lemma 8. The map A r is a linear homeomorphism Fs/(Rn) ^ Fi-r'S (Rn) for s e R.

In a similar way one also finds the next auxiliary result.

Lemma 9. For any ^ e R, k e {1,...,n} the operator (1 - dXkfu = OP((1 + £,kT)u is a linear homeomorphism

Fs/(Rn) ^ f:-2^'3 (Rn) for all s e R.

p,q p,q J

A standard choice of an anisotropic lift operator is obtained by associating each £ e R" with (1,£) e R +",which is given the weights (1,a), and by setting

<03 = |U|(1'3). (28)

This is in C™, as \ ■ \(13) is so outside the origin. (Note the analogy to {£,) = ^1 + |£|2.) Moreover, da{%)i is for each t e R estimated by powers of \£\,(cf. [15, Lemma 1.4]). Therefore, there is a linear homeomorphism Et : S' ^ S' given by

Eiu:= OP(<Ol)u = F-1 (<01^(0), te R. (29)

In our mixed-norm setup, it is a small exercise to show that it restricts to a homeomorphism

El : Fs* (R") Fs—'d (R"), V e R. (30)

Indeed, invoking Proposition 7, the taskis as in (27) to show a uniform bound, and using the elementary properties of (cf. [15, Lemma 1.4]) one finds for t - a ■ a > 0,

¡Da (2-jt(2j3^)ts)\ = 2^-t) iD^X-vH|

< c2j(d-a-t) (2jdS)t-a'a < c{i)t-i'a.

When t - a ■ a < 0, then \^\tcfaa is the outcome on the right-hand side. But the uniformity results in both cases, since the estimates pertain to (1/4) < < 4.

We digress to recall that the classical fractional Sobolev space Hspa(Rn), for s e R and 1 < p < <x>, consists of the u e

S' for which Etu e L^(R"), with \\u \ Hs/\\ := \\Eiu \ Lp\\.

If mk := s/ak e N0 for all k, then Hsf coincides (as shown by

Lizorkin [16]) with the space )(R") of u e Lp having

dT" u in L g for all k.

This characterisation is valid for FSJ*. with 1 < p < >x> in

view of the identification

u e Hsf (R") ^ u e Fsf2 (R"), (32)

which by use of Es reduces to the case

Littlewood-Paley inequality that maybe proved with general

methods of harmonic analysis (cf. [2, Remark 3.16]).

A general reference on mixed-norm Sobolev spaces is the

classical book of Besov et al. [13,17]. Schmeisser and Triebel

[18] treated Fs/ for n = 2. L J M

Remark 10. Traces on hyperplanes were considered for Hs/(Rn) by Lizorkin [16] and for wf(Rn) by Bugrov [19], who raised the problem of traces at {xj = 0} for j < n. This was solved by Berkolalko, who treated traces in the Fspa^(Rn)-

scales for 1 < p < >x> in, for example, [20]. The range 0 < p < >x> was covered on R" for j = 1 and j = n in [2], and in our forthcoming paper [9], we carry over the trace results to F!'^-spaces over a smooth cylindrical domain Qx]0, T[.

Remark 11. We take the opportunity to correct a minor inaccuracy in [2], where a lift operator (also) called Ar unfortunately was defined to have symbol (1 + \Z,\\)r/2. However, it is not in C™(R") for a = (1,..., 1); this can be seen from the example for n = 2 with a = (2,1) where [15, Example 1.1] gives the explicit formula

\t\a=2-1/2(e2+4e1)m)m. (33)

Here an easy calculation shows that D^\%\2i is discontinuous along the line (?1,0), which is inherited by the symbol, for example, for r = 2. The resulting operator is therefore not defined on all of S'. However, this is straightforward to avoid by replacing the lift operator in [2] by the better choice Er given in (30). This gives the space H*:a(Rn) in (32).

2.4. Paramultiplication. This section contains a pointwise

multiplier assertion for the Fi'a-scales. We consider the

densely defined product on S x S , introduced in [21, Definition 3.1] and in an isotropic setup in [22, Chapter 4],

u ■ v := lim F-1 (f (2-jS$) Fu ($))

■ F-1 (y(2-idZ) Fv (V),

which is considered for those pairs (u, v) in S x which the limit on the right-hand side exists in D and is independent of Here y e C™ is the function used in the construction of the Littlewood-Paley decomposition (in principle the independence should be verified for all y e C™ equalling 1 near the origin, but this is not a problem here).

To illustrate how this product extends the usual one and to prepare for an application below, the following is recalled.

Lemma 12 (see [21]). When f e C™(R") has derivatives of any order of polynomial growth, and when g e S'(R") is arbitrary, then the limit in (34) exists and equals the usual product f ■ g,as defined on C™ x D.

Using this extended product, we introduce the usual space of multipliers

M (f!4) := {m e S | w • v e F!'3, Vv e F5/} V m/ l 1 M MJ

equipped with the induced operator quasinorm

)||:= sup {||w V |F^||||v i^1}. (36)

As Lemma 6 at once yields C£° c ns>0£^TO (a well-known result in the isotropic case) for C£° := e Cm | Va : D"^ e LTO|, the next result is in particular valid for «eC™ .

Lemma 13. Let s e R and tafce S, >5 suc^ tfeat also

si > I

1 V min (l><?> Fi>--->Fi)

-a» I - s.

To estimate n3(M, v) we first consider the case s > 0 and pick t e ]s, Sj [. The dyadic corona criterion together with the formula vJ = v0 + • • • + and a summation lemma, which exploits that t - s, <0 (cf. [15, Lemma 3.8]), gives

Ps (m, v) | fH < csup2fcSl ||wfc | Lc

|| fceN„

x ||2(t-Sl)V-2 | L¿(l?)||

2(t-Sl)jI |vfc|H*(*,)

< c ||w | £s"a

< C ||M | ^co^||||V ^

ri-Sj ||

Since t - s1 < 0 < s implies F:

Fl-5"3 and also FÎ'3

F!'a holds, the above yields

Then eacfr m e BCCC

defines a multiplier of F!'^ and

||w | M ( F!'a

< c ||w | fis"a ||

— || 1 C'C||

Proof. The proof will be brief as it is based on standard arguments from paramultiplication, (cf. [21] and [22, Chapter 4] for details). In particular we will use the decomposition

m • v = nl (m, v) + n2 (m, v) + n3 (m, v) .

The exact form of this can also be recalled from the below formulae. In terms of the Littlewood-Paley partition 1 = £°=0 0;-(£) from Definition 2, we set = O0 + • • • + for j > 1 and ¥0 = 00. These are used in Fourier multipliers, now

written with upper indices as mj =

Note first that s, > 0, whence ^ L„, which is

useful since the dyadic corona criterion for F!'^ (cf. [2, Lemma 3.20]), implies the well-known simple estimate

||ni ("> v)|^||<c||«|Lc|| ||v | F-||

Furthermore, since

:= s, + s >

l min (l.^.pl,...

-|fl|,

using the dyadic ball criterion for F!'1^ (cf. [2, Lemma 3.19]), we find that

n2 («. v)|^;;:?3||<c||2jS2 M;.v;. |l

<Csup2fcSl ||Mfc | Lc|| ||2* |v;-| | Lp (l?)||

fceN„

^H^H ||v |^||

||ns (w, v) | F;

< c ||m

C'C ||

||v | F!'

(38) For s < 0 the procedure is analogous, except that (43) is derived for t e ]0, s, + s[ , which is nonempty by assumption (37) on s; then standard embeddings again give (44).

In closing, we remark that as required the product u • v is independent of the test function f appearing in the definition. Indeed for q < to this follows from Lemma 12,

(39) which gives the coincidence between this product on S' x S and the usual one, hence by density of S (cf. (18)) and the above estimates, the map v ^ u • v extends uniquely by continuity to all a e F!'a. For a = to the embedding F!'a

for e > 0 yields the independence using the previous case. □

3. Characterisation by Local Means

Characterisation of Lizorkin-Triebel spaces by local means is due to Triebel, [4, 2.4.6], and it was from the outset an important tool in proving invariance of the scale under dif-feomorphisms. An extensive treatment of characterisations of mixed-norm spaces F!'^ in terms of quasinorms based on convolutions, in particular the case of local means, was given in [3], which to a large extent is based on extensions to mixed norms of inequalities in [8]. For the reader's convenience we recall the needed results.

Throughout this section we consider a fixed anisotropy a > 1 with a := min(a1; ...,«„) and functions y e S(R") that fulfil Tauberian conditions in terms of some e > 0 and/or a moment condition of order > -1 (My = -1 means that the condition is void),

Fo (£)| > 0 on {Ç||Ç|3<2e}, (45)

|Fy(Ç)|>0 on {Ç| e/2< | Ç| 3<2e}, (46) (42) D" (Fy) (0) = 0 for |a| < My. (47)

Note by (12) that in case (45) is fulfilled for the Euclidean distance, it holds true also in the anisotropic case, perhaps with a different e.

We henceforth change notation, from (15), to

<Pj (x) = 2M<p (2fax), <pe S, j e N,

which gives rise to the sequence (fj)^ . The nonlinear Peetre-Fefferman-Stein maximal operators induced by (Vj)jeN0 are for an arbitrary vector r = (r1,...,rn) > 0 and any f e S'(R") given by (dependence on a and r is omitted)

Wi *f(y)\

r> f(x)="£ m, a+2* k -y,\Y'

x e R", j e N0.

Later we will also refer to the trivial estimate

\fj *f(x)\ < f-f(x).

Finally for an index set 0, we consider f8 0, fe e S(R"), d e 0, where the yg satisfy (47) for some MVe independent of d e 0, and also f0, (p e S(R") that fulfil (45)-(46) in terms of an £ > 0. Setting f9tj(x) = 2j]i]fe(2jdx) for j e N, we can state the first result relating different quasinorms.

Theorem 14 (see [3]). Let 0 < p < >x>, 0 < q < >x>, and ->x> < s < (Mye + 1)a. For a given r in (49) and an integer M > -1 chosen so large that (M + 1)a- 2a ■ r + s > 0, we assume that

A := sup max |\DaFye | L^J < >x>,

B := sup max 11(1 + \£\)M+1 DYFye (Ç) \L J < œ,

C := sup max |\DaFfd 0 \ Lœ|| < œ,

D := sup max I I (1 + \Ç\) DYFyefi (Ç) \L Jl < œ,

where the maxima are over a such that la\ < + 1 or a < \r + 2], respectively, over y with yj < rj + 2. Then there exists a constant c > 0 such that for f e S' (R"),

2$1 sup VeJ

<c(A + B + C + D)

^f^olt P (0

It is also possible to estimate the maximal function in terms of the convolution appearing in its numerator.

Theorem 15 (see [3]). Let e S(R") satisfy the Tauberian conditions (45)-(46). When 0 < p < m, 0 < q < m, -m < s < m, and

1< min(q,pi,...,pn), l=l,...,n, (53) ri

there exists a constant c > 0 such that for f e S'(R"),

^ r,f\l0\L P (eq) <c ||{2SJ Y, *f}lo\L P (eq)\\.

As a consequence of Theorems 14 and 15 (the first applied for a trivial index set like 0 = {1}), we obtain the characterisation of Ff'^-spaces by local means.

Theorem 16 (see [3]). Let k0,k° e S such that J k0(x)dx = 0= Jk0(x)dx andset k(x) = ANk0(x) forsome N e N. When 0 < p < >x>, 0 < q < >x>, and ->x> < s < 2Na, then a distribution f e S'(R") belongs to Fs/(Rn) if and only if(cf. (48) for the kj)

f\>?J := "0 *f\L I

+ }i{2'i kJ *f\h\L P < œ.

Furthermore, \\f | Fp^W is an equivalent quasinorm on Fs/(Rn).

Application of Theorem 16 yields a useful result regarding Lizorkin-Triebel spaces on open subsets, when these are defined by restriction:

Definition 17. Let U c R" be open. The space F^(U) is

defined as the set of all u e D'(U) such that there exists a distribution f e Fs:^(Rn) satisfying

f(f) = u(f) VcpeC™ (U). (56)

We equip Fs-f^U) with the quotient quasinorm given by \\u | F^(U)W = inf ruf=u\\f I F^(Rn)\\; it is normed if p,q> 1.

In (56) it is tacitly understood that on the left-hand side f is extended by 0 outside U. For this we henceforth use the operator notation %<p. Likewise % denotes restriction to U, whereby u = rjjf in (56).

The Besov spaces Bsf^(U) on U can be defined analogously. The quotient norms have the well-known advantage that embeddings and completeness can be transferred directly from the spaces on R". However, the spaces are probably of little interest, if dU does not satisfy some regularity conditions because we then expect (as in the isotropic case) that they do not coincide with those defined intrinsically.

Lemma 18. Let U c R" be open and r > 0. When FS* (U) has the infimum quasinorm given by derived from the local means in Theorem 16 fulfilling supp k0, supp k c B(0, r), and

dist (supp f, R" \U)> 2r (57)

holds for some f e F^U) with compact support, then

\\f \ FSl(U)\\ = \\euf \ FSsl(Rn)\\. (58)

In other words, the infimum is attained at euf for such f.

Proof. For any other extension f e S'(Rn) the difference g = f - euf is nonzero in S'(Rn) and supp eu/ n supp g = 0.So by the properties of r,

supp (kj * evf) n supp (kj * g) — 0, je Nq

Since g= 0, there is some j such that supp(fc;- * g) = 0, and hence kj * g(x) = 0 on an open set disjoint from supp(fc;- * %/). This term therefore effectively contributes to the L

norm in (55) and thus \\f \ Ff\\ = \\euf + g \ FfJ > \\euf \

F!'a II, which shows (58).

M11' v '

4. Invariance under Diffeomorphisms

The aim of this section is to show that Fsf (R") is invariant under suitable diffeomorphisms o : R" ^ R" and from this deduce similar results in a variety of setups.

4.1. Bounded Diffeomorphisms. A one-to-one mapping y = a(x) of R" onto R" is here called a diffeomorphism if the components Oj : R" ^ R have classical derivatives DaOj for all a e N". We set r(y) = a-1(y).

For convenience a is called a bounded diffeomorphism when a and t furthermore satisfy

Ca,a '■= max,llDai \Lœ\\ < je{1,...,K} 11 11

C ■= max , ||D"Tj \LJ\ < rn.

je{1,...,n} 11 "

(60) (61)

In this case there are obviously positive constants (when Ja denotes the Jacobian matrix)

cv := inf \det Ja (x) \ > 0, cr := inf |det Jr (y)\ > 0.

XGR ^GR

For example, by the Leibniz formula for determinants, ca >

i/(»inia|=iCa>T) > 0.

Conversely, whenever a Cm-map a : R" ^ R" fulfils (60) and that ca > 0, then t is Cm (as Jr(y) = 1/ (det Ja(r(y)))Adj Jo(r(y)), if Adj denotes the adjugate, each TjTk is in Cm if t is so) and using, for example, the Appendix it is seen by induction over \a\ that also (61) is fulfilled. Hence such a a is a bounded diffeomorphism.

Recall that for a bounded diffeomorphism a and a temperate distribution f, the composition f ° a denotes the temperate distribution given by

(f °a,y) = (f,y°r\det Jt\) for ye S. (63)

It is continuous S' ^ S' as the adjoint of the continuous map y ^ y ° t\ det Jt\ on S: since \ det Jt\ is in C™ , continuity on S can be shown using the higher-order chain rule to estimate each seminorm qNa (y °r),cf. (7), by Z|^|<|«| (Y) (changing variables, {&(■)) can be estimated using the Mean Value Theorem on each a)

We need a few further conditions, due to the anisotropic situation: one can neither expect f ° a to have the same regularity as f, for example, if a is a rotation, nor that f °a e Ls when f e L s. On these grounds we first restrict to the situation in which

Un ■— Ui — ^ — ••• — Ut

•n-1>

Po ■= Pi — Pn-1'

a(x) = (a' (x1,...,xn_1),xn), Vx e R". (65)

To prepare for Theorem 20, which gives sufficient conditions for the invariance of Ff'a under bounded diffeomor-phisms of the type (65), we first show that it suffices to have invariance for sufficiently large s.

Proposition 19. Let a be a bounded diffeomorphism on R" on the form in (65). When (64) holds and there exists s1 e R with the property that f ^ f ° a is a linear homeomorphism of F!'a(R") onto itself for every s > s1, then this holds truefor all s e R.

Proof. It suffices to prove for s < s1 that ||/°a|F',a1|<c||/|F'

ik 1 p,qll ik 1 i

with some constant independent of f, as the reverse inequality then follows from the fact that the inverse of a is also a bounded diffeomorphism with the structure in (65).

First r > s1 - s + 2an is chosen such that d0 := (r/2a0) is a natural number. Setting dn = (r/2an) and taking ^ e [0,1 [ such that dn - p e N,we have that r^ := r - 2^an > s1 - s.

Now Lemma 8 yields the existence of he ps_+r'a such that f = A rh, that is,

f==(l-< Th + t(l-*lk )d" h (67)

Setting g1 = ((1 -d2x )^h) ° a and g0 = h ° a, we may apply the higher-order chain rule to for example h = g0 ° t (using denseness of S in S' and the S'-continuity of composition in (63), the Appendix extends to S'). Taking into account that t(x) = (t'(x'), xn), and letting prime indicate summation over multi-indices with pn = 0,

f=^nn,i ^01 Z °r, (68)

1=0 fc=1 |£|<2d0

where ^ := (-1)' ( ) and the are functions containing derivatives at least of order 1 of r, and these can be estimated, say by cni<m<2d0 (9™r)2d°. Composingwith a and applying Lemma 4(i) gives for d := min(1, pn), when

denotes the F!'a-norm,

d IK2I

-iZi ^ 011

+ I I' •"|M(i$|| ||

fc=1 |ß|<2d0

< c|L | F^l

HMd I I' •*|M(i$|.

fc=1 |£|<2d0

According to the remark preceding Lemma 13, the last sum is finite because ^ ^ e . Finally, since s + r^ > s1 and s + r > s1, the stated assumption means that h ^ and h ^ are bounded, which in view of r^ + = r and Lemmas 8 and 9 yields

d . II, , ^+r„+2p«„;ä||d + IL | Fs+r>3||d < Jlf |

IM A? II " IK A?

11/nr Hh^;;

proving the boundedness of / R.

f.ff in F!;a for all s e

In addition to the reduction in Proposition 19, we adopt in Theorem 20 the strategy for the isotropic, unmixed case developed by Triebel [4, 4.3.2], who used Taylor expansions for the inner and outer functions for large s.

While his explanation was rather sketchy, our task is to account for the fact that the strategy extends to anisotropies and to mixed norms. Hence we give full details. This will also allow us to give brief proofs of additional results in Sections 4.2 and 5.

To control the Taylor expansions, it will be crucial for us to exploit both the local means recalled in Theorem 16 and the parameter-dependent setup in Theorem 14. This is prepared for with the following discussion.

The functions fc0 and k in Theorem 16 are for the proof of Theorem 20 chosen (as we may) so that N in the definition of k fulfils s < and so that both are even functions and

supp fc0, supp k cjie R" | |x| < 1}.

The set 0 in Theorem 14 is chosen to be the set of (n-1)x(n-1) matrices B = (fei>fc) that, in terms of the constants cCT, Ca>0. in (62) and (60), respectively, satisfy

|det B| > cff,

maxM < max=:

>,fc |a|=1

Splitting z = (z',zn), we set ^(z) = zy fc(z) for some y' e (chosen later) and define

Ve M = (74)

where 0 is identified with A-1 := /a'(x'), which obviously belongs to 0 (for each x').

To verify that the above functions 0 6 0, satisfy the moment condition (47) for an M^ such that the assumption s < (M^ + 1)« in Theorem 14 is fulfilled, note that

fo (£) = |det V^U,,)- (75)

Hence D^fo vanishes at £ = 0 when D0^ = D"(-D?,)r

does so. As = -|^i2Nfc°(^) and fc°(0) = 0, we have

D"ij(0) = 0 for a satisfying |a| + |y'| < 2N - 1. In the course of the proof below (cf. Step 3), we obtain a 0-independent estimate of |y'|, hence of M^.

Moreover, the constant A in Theorem 14 is finite: basic properties of the Fourier transform give the following estimate, where the constant is independent of A-1 6 0:

= |detA

<c(a,Cff)[ |fc(z)|dz.

J|z|<1

To estimate B we exploit that F : B"/2(R") ^ L 1(R") is bounded according to Szasz's inequality (cf. [18, Proposition 1.7.5]) and obtain

||(1 + |.|)M+1 DyFfd | Li|| < C||/4A/,y„) | Bfi+1+("/2)|| <c(y,Cff,Cr) ||fc|C-||

when me N is chosen so large that m > M + 1 + n/2.In fact, the last inequality is obtained using the embeddings C™ Hm ^ B^+1+"/2 and the estimate

>0 | co II = sup

(f(Ay')yk(Ay',yn)

<c(y,cff,cT) ||fc|c;

This relies on the higher-order chain rule (cf. the Appendix and the support of fc): it suffices to use the supremum over |a| < m and R" | |A/|2 + y2 < 1}, and for a point in this set |y'| < ||A-1|||Ay| < c(CCT), so we need only estimate an A-independent cylinder.

Replacing fc by fc0 in the definition of g and setting := ^(Ay',y„),thefinitenessof C and D follows analogously. The Tauberian properties follow from J fc0

Hence all assumptions in Theorem 14 are satisfied, and we are thus ready to prove our main result.

Theorem 20. If a is a bounded diffeomorphism on Rn on the form in (65), then f ^ f ° a is a linear homeomorphism FS/(Rn) ^ Ff(Rn) for all s e R when (64) holds.

Proof. According to Proposition 19, it suffices to consider s > Sj, say for

h ■= K0a0 + (n-l)-0 + . " ,, (79) Po min [Po'PJ

where by K0 is the smallest integer satisfying

K0a0 > (n-1) — + —:

Po min (Po'PnY

We now let sejjj.ml be given and take some K > K0, that is, K solving (80), such that

Kao + (n-1) — + —r-r < s < 2Kao. (81)

Po min [Po'Pn)

(The interval thus defined is nonempty by (80), and the left end point is at least Sj.)

Note that (81) yields that every f e F^* is continuous,

(cf. Lemma 4(iii)), so are even the derivatives D'6 f for p= (p1,...,pn_1,0),\p\ < K,since s-p-a = s-\p\a0 > a-1/p.

Step 1. For the norms \\f°a \ F^W and \\f \ F^W in inequality (66), which also here suffices, we use Theorem 16 with 2N > (K-1)(2K- 1)+s/a.

By the symmetry of k0 and k in (71), we will estimate

kj *(foa)(x)=\ k(z) f(a(x + 2-jaz))dz, je N, J|z|<1

together with the corresponding expression for k0,where k is replaced by k0.

First we make a Taylor expansion of the entries in o'(x') := (a1(x')t... ,an_1(x1)) to the order 2K - 1. So for l = 1,.. ,,n- 1 there exists we e ]0,1[ such that

i > >\ v (x') " ae (x +z)= X —¿j-z

la'l<2K

la'l=2K

da ae (x + wez')

For convenience, we let denote summation over multi-indices a e N having an = 0 and define the vector of Taylor polynomials, respectively, entries of a remainder R,

P2K-1 (*') = I

, daa' (x')

lal=2K

lal<2K-1

I daae (x' + <Mez')

Applying the Mean Value Theorem to f (cf. (81)), now yields an ¿5 e ]0,1| so that

\k: *(fo0) (%)|

^ k (z) f (P2K-1 (2-ja' z) , Xn + 2~'a" Zn) dz

\k(z)dXd f(y',Xn + 2-Ja Zn) d=1Jlzli1 \

xRd (2-ja' z' )\dz,

when y' := P2K_1(2_ja z') + ti(Ri (2_,a z'),..., Rn-1(2_}a z')). Using (60) and (83), it is obvious that this y fulfils

\ a (x) - (y, xn + 2_ia"zn)\ < \a' (x') -y'\ + \2_ja"zn\ < C

for each z e supp k and some constant C depending only on

n and Ca a with \<x\ < 2K.

Step 2. Concerning the remainder terms in (85) we exploit (86) to get

^ ^ \k (z) dXd f (y', xn + 2_iaZn) Rd (2_ia'z')\dz

-2jKaQ

la' l=2K

№ °d \L Jl

x I \k(z)\dz sup \dXdf(y)\.

\o(x)-y\<C

The exponent in 2-2'Ka" is a result of (64) and the chosen Taylor expansion of a(x + 2-}az), and since s - 2Ka0 < 0 the norm of £„ is trivial to calculate, whence

2s I \k(z)d f(y',Xn + 2~*>Zn)Rd

J|z|<1 1

x(2-ja' z')\dz\L p(eq)

sup ^ f(y)\\L p(RX)

\a(x)-y\<C

Now we use that p1 = ••• = pn-1 to change variables in the resulting integral over t' denoting (a') 1. Since

Lemma 5 in view of (81) applies to dXd f, d = 1,... ,n - 1, the right-hand side of the last inequality can be estimated, using also Lemma 4(i), by

c[ sup) \detJr' (y)\ ) \\dxdf \ F^'a\ < c\\f \ F^

Step 3. To treat the first term in (85), we Taylor expand

f(-,xn), which is in CK(Rn—1). Setting P(z') = P2K—1(z') -

f —' ' ! Px(z ), expansion at the vector P-i(2 —a z ) gives

f(p2K_1 (l-*' z'),xn + 2-ja" zn)

^ DPf(p1 (2->a' z'),xn + 2~'a" Zn)

0<\ß\<K-1

xp(l-ja' z')

Dßf(y',xn + 2-aZn) n(^ ,

P(2-}a z ) ,

where y is a vector analogous to that in (85) and satisfies (86), perhaps with another C.

To deal with the remainder in (90), note that the order was chosen to ensure that, in the powers P(2-ja z')^,the I'th factor is the ^'th power of a sum of terms each containing a factor 2-Jao\«'\ with \a'\> 2. Hence each = K in total contributes by 0(2-2jKa°). More precisely as in Step 2 we obtain

)\z\<1

, Dßf(y',Xn + 2-ja"Zn)

P(2-ia z

-j2Ka0

f \k(z)\dzi £ C«,c)

s1 V2<\a\<2K-1 /

x£ sup \Dßf(y)\.

\ß\=K\o(x)-y\<C

In view of (81), Lemma 5 barely also applies to Dßf for \ß\ = K, so the above gives

f k(z)X

, Dßf(y',xn + 2 ja"Zn)

xp(2-ja' z')ßdz\L p(eq)

<c( sup \det Jr' (y)\) X' l^H^r

Now it remains to estimate the other terms resulting from (90), that is,

;\ß\<K-1 J\Z\S1

Dßf(P1

2-ja z'),xn + 2-ja" zr

0<\ß\<K-1 JIZIS1 ß!

x p(2-ja'z')ßdz.

Using the multinomial formula on the entries in P(z') = Yl2<\y\<2K-\ zydya (x')lyl and the g and yd discussed in (74), the above task is finally reduced to controlling terms like

IJ,P,Y (a' (x')'xn)

.= 2-2№a [ g(z)DP

J\z\<1

xf (a' (x')+2-ja° Jo' (x') z, xn+2-ja"zn) dz

= 2-2j\ß\at

'» \det A\ffe (y)Dß

x f (a' (x')+2-ja°y', xn+2-ja"yn) dy.

Note that in g, fe we have 2 < \y\ < \p\(2K - 1) and \p\<K-1, ^ = 0 = Yn.

Step 4. Before we estimate (94), it is first observed that all previous steps apply in a similar way to the convolution k0 * (f°o)—except in this case there is no dilation, so the l^-norm is omitted and the function yg is replaced by f8 0.

So, when collecting the terms of the form (94)withfinitely many p, y in both cases (omitting remainders from Steps 23), we obtain with two changes of variables and (50),

Xl0,ß,y (x')>xn) \ Lp

2,sX'hß, (-' (x')>xn)\L p(eq)

/ \ 1/p0 < ¿X ( sup \det Jr (y)| )

ßrXy^R"-1 )

\\n,o (y)Dßf(x-y)dy\L

ao) \fe (y)Dßf(x-2-jay)dy\Lp(eq)

j(.s-2\ß\ao'.

2j(s-2\ßMsup fljDßf\ \Lp(eq)

Here we apply Theorem 14 to the family of functions yg o, fe with the fj chosen as the Fourier transformed of the system

in the Littlewood-Paley decomposition, (cf. (13)). Estimating |y|, the yg satisfy the moment condition (47) with MVe := 2N- 1-(K- 1)(2K- 1), which fulfils s < (M^g + 1)a, because of the choice of N in Step 1. So, by applying Theorem 15 and Lemma 4(i), usings-2lfila0 < s-fi-a, theaboveisestimated thus

<c(A + B + C + D)

L рЫ

H {2j(s-rna„ Dßfro\Lp(eq)\\

ß.y j

< ¿Y^ \\Dßf | ps-2ßa<"ä ß.y \\

0'u II ^ jr I rs,a

\\ < с \\t I F: \\ pq \\ Г p'4\\

This proves the necessary estimate for the given s > Sj. □

4.2. Groups of Bounded Diffeomorphisms. It is not difficult to see that the proofs in Section 4.1 did not really use that xn is a single variable. It could just as well have been replaced by a whole group of variables x", corresponding to a splitting x = (x1, x"), provided that a acts as the identity on x".

Moreover, x' could equally well have been "embedded" into x" ,that is, x" could contain variables xk both with k < j0 and with k > j1 when x = (xj , •••,Xj1) (but no interlacing); in particular the changes of variables yielding (89) would carry over to this situation when pjo = ••• = p^. It is also not difficult to see that Proposition 19 extends to this situation when a^ = ••• = aj (perhaps with several g1 -terms, each having a value of

Thus we may generalise Theorem 20 to situations with a splitting into m >2 groups, that is, R" = RNl x - - - x RN™ where N1 +----+ Nm = n, namely, when

5. Derived results

5.1. Diffeomorphisms on Domains. The strategies of Proposition 19 and Theorem 20 also give the following local version.

For example, for the paraboloid U = [x\xn > x\ +----+ x„-i}

we may take a to consist in a rotation around the xn-axis (cf. (65)).

Theorem 22. Let U,V c R" be open and a : U ^ V a Cm-bijection as in (65). If (64) is fulfilled and f e Fi'a(V) has

compact support, then f ° a e and

f°°II% (U)l<c\\fIFp: (V)

holds for a constant c depending only on a and the set supp f.

Proof. Step 1. Let us consider s > s1 (cf. (79)), and adapt the proof of Theorem 20 to the local set-up. We will prove the statement for the f e Fi'a (V) satisfying supp f c K c V for some arbitrary compact set K. First we fix r e ]0,1[ so small that

6r < min (dist (K, R" \V), dist (a-1 (K), R" \ u)) . (102)

Then, by Lemma 18, we have \\f ° a | FS/(U)\\ = \\eu(f ° a) l Fpaq\\ when Theorem 16 is utilised for k0,k e S, say, so that supp k0, supp k c B(0,r) (cf. also (71)). Extension by 0 outside U of f °o is redundant, for it suffices to integrate over x e W := supp(f o a) + B(0, r). However, to apply the Mean Value Theorem (cf. (85)), we extend f by 0 instead; that is, we consider (82) with integration over lzl < r and with f replaced by evf.

Since evf inherits the regularity of f (cf. Lemma 18) and daa can be estimated on the compact set W, the proof of Theorem 20 carries over straightforwardly. For example, one obtains a variant of (89) where l det Jr'(x')l1/p0 is estimated over {x' l 3xn : (x , xn) e a(W)}, and the integration is then extended to R", which by Lemma 18 yields

p = I pi, . . . , pvp2, . . . , p2,...,pm, . . . , pm

a={ai,...,ai,a2,...,a2,...,am,...,am)

a(x) = [o[ (x(1)),...,a!n (x(m)))

\x-y\<C

\dXd evf{y)\

<c\^evfIF^l(Rn)

Fsf (V)\\

with arbitrary bounded diffeomorphisms a. on RN' and

x(j) e Rn> .

Indeed, viewing a as a composition of a1 := o[ ® idRn-w1, and so forth on R", the above gives

\\f о о I Fsf \ \ < с \ \f о am о... о a2 | Fi'" \\<---<c\\f | Fsf \\ 1 p.q\\ m 21 p.q\\ P 1 p.q\\

To estimate the first term in (85) in this local version, the argumentation there is modified as above and the set 0 is chosen to be the set of all (n - 1) x (n - 1) matrices satisfying (72) with infimum over x e W and (73) with Ca := max1ijinM=1supxeWlDaajMl

Before applying Theorem 14 to the new estimate (95), the integration is extended to R" (using evf). Then application of Theorems 14 and 15 together with Lemma 18 finishes the proof for s > s1.

Theorem 21. f ^ f о a is a linear homeomorphism on Fp? when (97), (98), and (99) hold.

Step 2. For s < s1 we use Lemma 8 to write evf = A rh for some h e Ft+™(Rn); hence the identity (67) holds in

D'(R") for ev/ and h. Applying rv to both sides and using that it commutes with differentiation on C™, hence on D',we obtain (68) as an identity in D'(V) for the new := (rvh)»a and gx := (rv(1-92n )'h).ff.

Composing with o we obtain an identity in o a is treated using cut-off functions. For example, we can

take x>X\ e C™(U) with ^ = 1 onsupp(/oa)+ £(0,r) =: Wr and supp ^ c W2r, while ^ = 1 on W3r and supp ^ c W4r. This entails

As a preparation for our coming work [9], we include a natural extension to the case of an infinite cylinder, where supp / is only required to be compact on cross-sections.

Theorem 24. Let o : U x R ^ R be a C™-bijection on the form in (65), and U,Vc R"-1 open, f (64) holds and f e x R) has supp f c K x R, whereby K c V is

compact, then f o a e x R) and

1=0 n-1

+ 11' <

fc=i |ß|<2d0

Using % on both sides (and omitting R" in the spaces), Lemmas 18 and 13 imply

F!'a (U x I m v

<c(supp /,a)||/ I F^l

Proof, We adapt the proof of Theorem 22: in Step 1 we take r e ]0,1[ so small that 6r is less than both dist(K, R"-1 \ V) and dist(<r' (K), R"-1 \U). Since the extension by zero eVxR/ is well defined, as K c V is compact, it is an immediate corollary to the proof of Lemma 18 that

lid d"-"ll lid 11/°^ HI <cHI% K (xi* ^ll

+ C £ ||% (^ (xi^o)} I •

:|<2d0

As % and differentiation commute on E'(U) 9 Lemma 4(i) leads to an estimate from above. But Lemma 18 applies since the supports are in W4r, so with ^ := ° r we find that the above is less than or equal to

I I M I I / \ I n5+r,fl||d

c||% faitfj 1 ^|| + c||% (Xi1

+ C||(fi • rvh) o a | (U)|| . Using Step 1 and Lemmas 13, 9, 8, and 18, this entails

f°a| F!'3(U)|| <c||(l-d2 A I F^'" I I +c||k|F!+r'i J M II II V *„/ I M y y I M ||

<cllA-/1| <i/|^ (v)| •

(107) □

This shows the local theorem for s < si.

There is also a local version of Theorem 21, with similar proof, namely the following.

Theorem 23. Let

V:, j = 1, ...,m, be C°

bijections, where U^, V^- c R > are open, When a, p fulfill (97)-(98) and when f e F!'a(U"1 x • • • x Um) has compact support, then (101) holds true for U = U1 x^ • •xUm and V = V1 x- • •xVm,

F!'a (V x Mv

Then the proof for s > s1 follows that of Theorem 22, with W:= (a'-1(K) +B(0,r)) x R.

For s < s1 we have eVxR/ = Arh for some h e F!+r'a(R")

(cf. Lemma 8). Hence (68) holds as an identity in x R) for #1 := (rVxR(1 - fh) o a and ^0 := (rVxRh) o

The o a are controlled using cut-off functions e C£° (U) with similar properties in terms of the sets Wr =

(a'-1(K) +1(0, r)) x R. Thus we obtain (104) in D'(U x R).

Now, as in (109) it is seen that / o <r and %xR(^ • / o a) have identical norms, so the estimates in Step 2 of the proof of Theorem 22 finish the proof, mutatis mutandis. □

5,2, Isotropic Spaces, Going to the other extreme, when also fln = a0 and = p0, then the Lizorkin-Triebel spaces are invariant under any bounded diffeomorphism (i.e., without (65)), since in that case we can just change variables in all coordinates, in particular in (88)-(89). Moreover, we can adapt Proposition 19 by taking = d0 and ^ = 0 in the proof; and the set-up prior to Theorem 20 is also easily modified to the isotropic situation. Hence we obtain the following.

Corollary 25. When a : R" ^ R" is any bounded diffeomorphism, then f ^ / o a is a linear homeomorphism of F*>(?(R") onto itself for all s e R,

This is known from work of Triebel [4, Theorem 4.3.2], which also contains a corresponding result for Besov spaces. (It is this proof we extended to mixed norms in the previous section.) The result has also been obtained recently by Scharf [23], who covered all s e R by means of an extended notion of atomic decompositions.

In an analogous way, we also obtain an isotropic counterpart to Theorem 22.

Corollary 26. When a : U ^ V is a Cm-bijection between open sets U,V c Rn, thenf°a e Fspq(U) for every f e Fsp^(V) having compact support and

foa\ FSM (U)|| < c (supp f a) \\f \ F*>(? (V)\\. (110)

Appendix

The Higher-Order Chain Rule

For convenience we give a formula for the higher order derivative of a composite map

R" -U Rm -iU C.

Namely, when f, g are Ck and x0 e Rn, then for every multiindex y with 1 < \y\ < k,

dr (g°f)(xa)

= X da0(f(*o))

1<|«|<|y|

, X „ n M*^

H. H r \L neil\ ß'!

■J i<lß'l<\y\

•i,ß> ß>ß

Hereby the first sum is over multi-indices a = (a1, which in the second are split arbitrarily

a1 = X nP '■■■'am = X nPm

1<\p1\<\r\ 1<\P"\<\r\

into integers np > 0 (parametrised by = ) in

N", with upper index j) that fulfil the constraint

y=X X np,ß>

J=1 1<lß>l<\y\

Formula (A.2) and (A.4) result from Taylor's limit formula: g(y + y*o) = Z\«\<fc caya + °(\y\k) that holds for y ^ 0 if and only if ca = (1/a!)dag(y0) for all \a\ < k. (Necessity is seen recursively for y ^ 0 along suitable lines, sufficiency from the integral remainder.)

Indeed, k = \y\ suffices, and with y = f(x + x0) - f(x0) Taylor's formula applies to both g and to each entry fj (by summing over an auxiliary multi-index e N"),

g(f (x + xo))

= x-^dag(f(xo))y^" ■■■yamm+o(\y\k)

|«|<k

= Xdag(f(xo))

n¿1 X M+'W

j=1ar\ 1<|ß>\<k(ß )!

Here the first remainder is o(\x\k) since o(\y\k)l\x\k = o(1)(\f(x + x0) - f(x0)\l\x\)k ^ 0. Using the binomial formula and expanding n^p the other remainders are also

seen to contribute by terms that are o(\x\k) or better; whence a single o(\x\k) suffices.

Hence we will expand (■ ■ ■ )a> using the multinomial formula. So we consider arbitrary splittings a^ = £ n^, with integers np > 0 in the sum over all multi-indices e N0 with 1 <\fi}\ < k. The corresponding multinomial coefficient is al/npi(npi)!, so (A.5) yields

g(f(x + xo))

= Xdag(f(xo))

|«|<fc

m i / ß'

n I n n1-! i^f' ('o)

j=1 aj=X"ßi 1<P |<k

Calculating these products, of factors having a choice of a^ = Znp, for each j = 1,...,m, one obtains polynomials xw associated with multi-indices w = =1 X1<\^i\<k npifi1.

For \w\ > k these are o(\x\k) and hence contribute to the remainder. Thus modified, (A.6) is Taylor's formula of order k for go f, so that dy (g o f)(xo)/y! is given by the coefficient of xw for w = y, which yields (A.4) and (A.2).

This concise proof has seemingly not been worked out before, so it should be interesting in its own right. For example, the Taylor expansions make the presence of the obvious, and the condition y = ^jpj npjpj is natural. Also the constants yl/Hnpi! and (^j)l-nP' lead to easy applications. Clearly dag(f(x0)) is multiplied by a polynomial in the derivatives of f1fm, which has degree 1 np = Ijaj = \a\.

The formula (A.2) itself is well known for n = 1 = m as the Faa di Bruno formula (cf. [24] for its history). For higher dimensions, the formulas seem to have been less explicit.

The other contributions we know have been rather less straightforward, because of reductions, say to f, g being polynomials (or to finite Taylor series), and/or by use of lengthy combinatorial arguments with recursively given polynomials, which replace the sum over the pj in (A.2), such

as the Bell polynomials that are used in, for example [25, Theorem 4.2.4].

Closest to the present approach, we have found the contributions [26, 27] in case of one and several variables, respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work Supported by the Danish Council for Independent Research, Natural Sciences (grant no. 11-106598).

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