Anisotropic, Mixed-Norm Lizorkin-Triebel Spaces and Diffeomorphic Maps
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.
First Sobolev embeddings and completeness of the scale were established in , using the Nikol’skiĭ-Plancherel-Polya inequality for sequences of functions in the mixed-norm space , which was obtained straightforwardly in . Then a detailed trace theory for hyperplanes in was worked out in , for example, with the novelty that the well-known borderline has to be shifted upwards in some cases, because of the mixed norms.
Secondly, our joint paper  presented some general characterisations of , which may be specialised to kernels of local means, in Triebel’s sense . 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 under coordinate changes. During the discussions below, the results in  are crucial for the entire strategy.
Indeed, we address the main technical challenge to obtain invariance of under the map when is a bounded diffeomorphism on . (Cf. Theorems 20 and 21) Not surprisingly, this will require the condition on that only affects blocks of variables in which the corresponding integral exponents are equal, and similarly for the anisotropic weights . Moreover, when estimating the operator norm of , that is, obtaining the inequality the Fourier analytic definition of the spaces seems difficult to manage directly, so as done by Triebel  we have chosen to characterise in terms of local means as developed in .
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 of Peetre-Fefferman-Stein type, which was established in [3, Theorem 2] for mixed norms and with uniformity with respect to a general parameter : Hereby the “cut-off” functions , 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 , which contains a serious flaw (as pointed out in ); this and other inaccuracies were corrected in .
A second adaptation of Triebel’s approach is caused by the anisotropy we treat here. In fact, our proof only extends to, for example, by means of the unconventional lift operator Moreover, to cover all , especially to allow irrational ratios , we found it useful to invoke the corresponding pseudodifferential operators that for are shown here to be bounded for all .
Local versions of our result, in which is only defined on subsets of , are also treated below. In short form we have, for example, the following result (cf. Theorem 22).
Theorem 1. Let be open and let be a -bijection on the form . When has compact support and all are equal for , and similarly for the , then and
This is useful for introduction of Lizorkin-Triebel spaces on cylindrical manifolds. However, this subject is postponed to our forthcoming paper . (Already this part of the mixed-norm theory has seemingly not been elucidated before.) Moreover, in  we also carry over trace results from  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 , we recall that the are relevant for parabolic differential equations with initial and boundary value conditions: when solutions are sought in a mixed-norm Lebesgue space (in order to allow different properties in the space and time directions), then -spaces are in general inevitable for a correct description of nontrivial data on the curved boundary.
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  on characterisation of -spaces by local means are recalled and used to prove an important lemma for compactly supported elements in . Sufficient conditions for to leave the spaces invariant for all are deduced in Section 4, when is a bounded diffeomorphism. Local versions for spaces on domains are derived in Section 5 together with isotropic results.
The Schwartz space contains all rapidly decreasing -functions. It is equipped with the family of seminorms, using for each multi-index with , and , or with The Fourier transformation for extends by duality to the dual space of temperate distributions.
Inequalities for vectors are understood componentwise; as are functions, for example, . Moreover, for .
For the space consists of all Lebesgue measurable functions such that with the modification of using the essential supremum over in case . Equipped with this quasinorm, is a quasi-Banach space (normed if for all ).
Furthermore, for we will use the notation for the space of all sequences of Lebesgue measurable functions such that with supremum over in case . This quasi-norm is often abbreviated to and when we simplify to . If sequences of -functions are dense in .
Generic constants will primarily be denoted by or and when relevant, their dependence on certain parameters will be explicitly stated. stands for the ball in centered at with radius , and denotes the closure of a set .
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 , where , on and some of its properties. Using the quasihomogeneous dilation for , is for defined as the unique such that (), that is, By the Implicit Function Theorem, is on . We also recall the quasi-homogeneity together with (cf. [1, Section 3])
The definition of uses a Littlewood-Paley decomposition, that is, , which (for convenience) is based on a fixed such that for all , if and if ; setting , we define
Definition 2. The Lizorkin-Triebel space with , and consists of all such that
The number is called the sum exponent and the entries in are integral exponents, while is a smoothness index. Usually the statements are valid for the full ranges , , so we refrain from repeating these. Instead we focus on whether is allowed or not. In the isotropic case, that is, , the parameter is omitted.
We will also consider the closely related Besov spaces, recalled using the abbreviation
Definition 3. The Besov space consists of all such that
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 ) and the quasinorm is subadditive, when raised to the power , Also the spaces do not depend on the chosen anisotropic decomposition of unity (up to equivalent quasinorms) and there are continuous embeddings where is dense in for .
The subspace of locally integrable functions is equipped with the Fréchet space topology defined from the seminorms , . By we denote the Banach space of bounded, continuous functions, endowed with the supremum norm.
Lemma 4. Let and be arbitrary. (i)The differential operator is bounded .(ii)For there is an embedding .(iii)The embedding holds true whenever .
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  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 for , while ; then the claim follows from the embedding . Part (iii) follows at once from [2, (3.20)].
A local maximisation over a ball can be estimated in , at least for functions in certain subspaces of (cf. Lemma 4(iii)).
Lemma 5 (see ). When , then for each
Next we extend a well-known embedding to the mixed-norm setting. Let denote the Hölder class of order , which by definition consists of all satisfying whereby is the integer satisfying .
Lemma 6. For and with there is an embedding .
Proof. The claim follows by adapting the proof of [14, Proposition 8.6.1] to the anisotropic case, that is, The expressions in the Besov norm are for estimated using that has vanishing moments of arbitrary order, Using a Taylor expansion of order with chosen such that (or directly if ), we get an estimate of the parenthesis by Now we obtain, since , This bound can also be used for , if is large enough, so (21) holds for .
As a tool we also need to know the mapping properties of certain Fourier multipliers . For generality’s sake, we give the following.
Proposition 7. When for some has finite seminorms of the form then is continuous on and moreover bounded for all , with operator norm .
Proof. The quasi-homogeneity of yields that , and hence every derivative is of polynomial growth (cf. (12)), so is a well-defined continuous map on . 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 , but this is trivial because finitely many of the constants can enter the estimates.
2.3. Lift Operators
The invariance under coordinate transformations will be established below using a somewhat unconventional lift operator , , To apply Proposition 7, we derive an estimate uniformly in and over the set : while the mixed derivatives vanish, the explicit higher order chain rule in the Appendix yields Indeed, the precise summation range gives , so the harmless power results. (Note that this means that .)
Now has no zeros, and for it is analogous to obtain such estimates uniformly with respect to of , using the Appendix and the aforementioned. So Proposition 7 gives both that is a homeomorphism on (although ) and the proof of our lemma.
Lemma 8. The map is a linear homeomorphism for .
In a similar way one also finds the next auxiliary result.
Lemma 9. For any , the operator is a linear homeomorphism for all .
A standard choice of an anisotropic lift operator is obtained by associating each with , which is given the weights , and by setting This is in , as is so outside the origin. (Note the analogy to .) Moreover, is for each estimated by powers of , (cf. [15, Lemma 1.4]). Therefore, there is a linear homeomorphism given by In our mixed-norm setup, it is a small exercise to show that it restricts to a homeomorphism Indeed, invoking Proposition 7, the task is as in (27) to show a uniform bound, and using the elementary properties of (cf. [15, Lemma 1.4]) one finds for , When , then is the outcome on the right-hand side. But the uniformity results in both cases, since the estimates pertain to (.
We digress to recall that the classical fractional Sobolev space , for and , consists of the for which , with . If for all , then coincides (as shown by Lizorkin ) with the space of having in for all .
This characterisation is valid for with in view of the identification which by use of reduces to the case . The latter is a Littlewood-Paley inequality that may be proved with general methods of harmonic analysis (cf. [2, Remark 3.16]).
Remark 10. Traces on hyperplanes were considered for by Lizorkin  and for by Bugrov , who raised the problem of traces at for . This was solved by Berkolaĭko, who treated traces in the -scales for in, for example, . The range was covered on for and in , and in our forthcoming paper , we carry over the trace results to -spaces over a smooth cylindrical domain .
Remark 11. We take the opportunity to correct a minor inaccuracy in , where a lift operator (also) called unfortunately was defined to have symbol . However, it is not in for ; this can be seen from the example for with where [15, Example 1.1] gives the explicit formula Here an easy calculation shows that is discontinuous along the line , which is inherited by the symbol, for example, for . The resulting operator is therefore not defined on all of . However, this is straightforward to avoid by replacing the lift operator in  by the better choice given in (30). This gives the space in (32).
This section contains a pointwise multiplier assertion for the -scales. We consider the densely defined product on , introduced in [21, Definition 3.1] and in an isotropic setup in [22, Chapter 4], which is considered for those pairs in for which the limit on the right-hand side exists in and is independent of . Here is the function used in the construction of the Littlewood-Paley decomposition (in principle the independence should be verified for all 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.
Using this extended product, we introduce the usual space of multipliers equipped with the induced operator quasinorm
As Lemma 6 at once yields (a well-known result in the isotropic case) for , the next result is in particular valid for .
Lemma 13. Let and take such that also Then each defines a multiplier of and
Proof. The proof will be brief as it is based on standard arguments from paramultiplication, (cf.  and [22, Chapter 4] for details). In particular we will use the decomposition
The exact form of this can also be recalled from the below formulae. In terms of the Littlewood-Paley partition from Definition 2, we set for and . These are used in Fourier multipliers, now written with upper indices as .
Note first that , whence , which is useful since the dyadic corona criterion for (cf. [2, Lemma 3.20]), implies the well-known simple estimate Furthermore, since using the dyadic ball criterion for (cf. [2, Lemma 3.19]), we find that To estimate we first consider the case and pick . The dyadic corona criterion together with the formula and a summation lemma, which exploits that (cf. [15, Lemma 3.8]), gives Since implies and also holds, the above yields For the procedure is analogous, except that (43) is derived for , which is nonempty by assumption (37) on ; then standard embeddings again give (44).
In closing, we remark that as required the product is independent of the test function appearing in the definition. Indeed for this follows from Lemma 12, which gives the coincidence between this product on and the usual one, hence by density of (cf. (18)) and the above estimates, the map extends uniquely by continuity to all . For the embedding for 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 diffeomorphisms. An extensive treatment of characterisations of mixed-norm spaces in terms of quasinorms based on convolutions, in particular the case of local means, was given in , which to a large extent is based on extensions to mixed norms of inequalities in . For the reader’s convenience we recall the needed results.
Throughout this section we consider a fixed anisotropy with and functions that fulfil Tauberian conditions in terms of some and/or a moment condition of order ( means that the condition is void), 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 .
We henceforth change notation, from (15), to which gives rise to the sequence . The nonlinear Peetre-Fefferman-Stein maximal operators induced by are for an arbitrary vector and any given by (dependence on and is omitted) Later we will also refer to the trivial estimate
Finally for an index set , we consider , , where the satisfy (47) for some independent of , and also that fulfil (45)-(46) in terms of an . Setting for , we can state the first result relating different quasinorms.
Theorem 14 (see ). Let , , and . For a given in (49) and an integer chosen so large that , we assume that where the maxima are over such that or , respectively, over with . Then there exists a constant such that for ,
It is also possible to estimate the maximal function in terms of the convolution appearing in its numerator.
Application of Theorem 16 yields a useful result regarding Lizorkin-Triebel spaces on open subsets, when these are defined by restriction:
Definition 17. Let be open. The space is defined as the set of all such that there exists a distribution satisfying We equip with the quotient quasinorm given by ; it is normed if .
The Besov spaces on can be defined analogously. The quotient norms have the well-known advantage that embeddings and completeness can be transferred directly from the spaces on . However, the spaces are probably of little interest, if 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 be open and . When has the infimum quasinorm given by derived from the local means in Theorem 16 fulfilling , and holds for some with compact support, then In other words, the infimum is attained at for such .
Proof. For any other extension the difference is nonzero in and . So by the properties of , Since , there is some such that , and hence on an open set disjoint from . This term therefore effectively contributes to the -norm in (55) and thus , which shows (58).
4. Invariance under Diffeomorphisms
The aim of this section is to show that is invariant under suitable diffeomorphisms and from this deduce similar results in a variety of setups.
4.1. Bounded Diffeomorphisms
A one-to-one mapping of onto is here called a diffeomorphism if the components have classical derivatives for all . We set .
For convenience is called a bounded diffeomorphism when and furthermore satisfy In this case there are obviously positive constants (when denotes the Jacobian matrix) For example, by the Leibniz formula for determinants, .
Conversely, whenever a -map fulfils (60) and that , then is (as , if denotes the adjugate, each is in if is so) and using, for example, the Appendix it is seen by induction over that also (61) is fulfilled. Hence such a is a bounded diffeomorphism.
Recall that for a bounded diffeomorphism and a temperate distribution , the composition denotes the temperate distribution given by It is continuous as the adjoint of the continuous map on : since is in , continuity on can be shown using the higher-order chain rule to estimate each seminorm , cf. (7), by (changing variables, can be estimated using the Mean Value Theorem on each ).
We need a few further conditions, due to the anisotropic situation: one can neither expect to have the same regularity as , for example, if is a rotation, nor that when . On these grounds we first restrict to the situation in which
To prepare for Theorem 20, which gives sufficient conditions for the invariance of under bounded diffeomorphisms of the type (65), we first show that it suffices to have invariance for sufficiently large .
Proposition 19. Let be a bounded diffeomorphism on on the form in (65). When (64) holds and there exists with the property that is a linear homeomorphism of onto itself for every , then this holds true for all .
Proof. It suffices to prove for that
with some constant independent of , as the reverse inequality then follows from the fact that the inverse of is also a bounded diffeomorphism with the structure in (65).
First is chosen such that is a natural number. Setting and taking such that , we have that .
Now Lemma 8 yields the existence of such that , that is, Setting and , we may apply the higher-order chain rule to for example (using denseness of in and the -continuity of composition in (63), the Appendix extends to ). Taking into account that , and letting prime indicate summation over multi-indices with , where and the are functions containing derivatives at least of order 1 of , and these can be estimated, say by . Composing with and applying Lemma 4(i) gives for , when denotes the -norm, According to the remark preceding Lemma 13, the last sum is finite because . Finally, since and , the stated assumption means that and are bounded, which in view of and Lemmas 8 and 9 yields proving the boundedness of in for all .
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 .
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 and in Theorem 16 are for the proof of Theorem 20 chosen (as we may) so that in the definition of fulfils and so that both are even functions and The set in Theorem 14 is chosen to be the set of matrices that, in terms of the constants in (62) and (60), respectively, satisfy
Splitting , we set for some (chosen later) and define where is identified with , which obviously belongs to (for each ).
To verify that the above functions , , satisfy the moment condition (47) for an such that the assumption