Spaces of Sobolev type with positive smoothness on R n , embeddings and growth envelopes

We characterize Triebel-Lizorkin spaces with positive smoothness on ℝn, obtained by different approaches. First we present three settings Fp,qs(ℝn),Fp,qs(ℝn),ℑp,qs(ℝn) associated to definitions by differences, Fourier-analytical methods and subatomic decompositions. We study their connections and diversity, as well as embeddings between these spaces and into Lorentz spaces. Secondly, relying on previous results obtained for Besov spaces 𝔅p,qs(ℝn), we determine their growth envelopes 𝔈G(Fp,qs(ℝn)) for 0≺p≺∞, 0≺q≤∞, s≻0, and finally discuss some applications.


Introduction
In this article Triebel-Lizorkin spaces of positive smoothness on ℝ  are investigated.They were introduced independently by Triebel and Lizorkin in the early 1970s.For a detailed treatment together with historical remarks we refer to Triebel [18,19].The idea for this paper originates from its forerunner [12], where we studied corresponding problems for Besov spaces.Since the substantial theory of the Triebel-Lizorkin spaces is strongly linked with the theory of Besov spaces -in the sequel briefly denoted as Fspaces and B-spaces, respectively -the question came up whether those previous results could be carried over to the F-space setting.This paper aims at providing a rather final answer to this question.According to the well-known Besov spaces, Triebel-Lizorkin spaces inherited different characterizations, creating the task of comparing and -in the optimal case -identifying the resulting spaces.In the case, 0 <  ≤ ( 1 − 1), 0 <  < 1 , for a long time, it was only known that, say, two of the most prominent approaches -based on characterizations by differences on the one hand and by Fourier-analytical decompositions on the other hand -necessarily differ, but may otherwise share similar properties.Modern subatomic characterizations now admit new insights into the nature of these spaces.

𝑝
) ; in fact the second embedding holds in general as well.This result is further clarified in Corollary 4.1.Secondly, the paper is devoted to the study of the 'typical' singularity behaviour in these F-spaces in the sense of growth envelopes.This recently introduced concept originates from such classical ideas as the famous Sobolev embedding theorem [16].Basically, this characterizes the unboundedness of functions that belong to (classical) Sobolev spaces    (ℝ  ),  ∈ ℕ 0 , 1 ≤  < ∞, (and more general scales of spaces).By Sobolev's embedding theorem it is known that for  ≤   , 1 ≤  < ∞, there are (essentially) unbounded functions in    (ℝ  ), whereas beyond the 'critical line'  =   , i.e., for  >   (or  =  and  = 1) we have    (ℝ  ) →  ∞ (ℝ  ).In the past a lot of work has been done to refine Sobolev type embeddings in terms of wider classes of function spaces.We do not want to report on this elaborate history here; apart from the original papers assertions of this type are indispensable parts in books dealing with Sobolev spaces and related questions, cf.[1], [26], [14], [5].We study the growth and unboundedness of such functions (distributions) in terms of their growth envelope  G () = (ℰ  G (),   G ) , where  ⊂  loc 1 is a function space and its growth envelope function, and   G ∈ (0, ∞] is some additional index providing a finer description.Here  * denotes the non-increasing rearrangement of  , as usual.These concepts were introduced in [21], [8,9], the latter book also contains a recent survey of the present state-of-the-art (concerning extensions and more general approaches) as well as applications and further references.
Our second main result, Theorem 3.11, can now be formulated as ) , Moreover, globally we obtain Similarly for the spaces F  , (ℝ  ).This naturally extends results for   , (ℝ  ) below the line  =  max( 1  − 1, 0) which -though indispensable for spaces   , (ℝ  ) in order to admit an interpretation of  ∈   , (ℝ  ) as a regular distribution -is not necessary for the approaches   , (ℝ  ) and F  , (ℝ  ), respectively.Moreover, since the scale of F-spaces contains the (fractional) Sobolev spaces as a special case, i.e., our results admit new insights into the nature of these classical function spaces as well, cf.Remark 2.12 and Corollary 3.12.
The paper is organized as follows.In Section 2 we first present three different approaches to Triebel-Lizorkin spaces of positive smoothness and briefly discuss these concepts.We also extend well-known embedding results to all admitted values of positive smoothness.In Section 3 we recall the concepts of growth envelopes, collect some fundamentals needed below including basic examples.The main results in this context are contained in Section 3.2.Finally Section 4 contains two interesting applications of our results in terms of Hardy-type inequalities and criteria of sharp embeddings.

Triebel-Lizorkin spaces with positive smoothness on ℝ 𝑛
We use standard notation.Let ℕ be the collection of all natural numbers and let ℕ 0 = ℕ ∪{0} .Let ℝ  be euclidean -space,  ∈ ℕ, ℂ the complex plane.The set of multi-indices  = ( 1 , . . .,   ),   ∈ ℕ 0 ,  = 1, . . ., , is denoted by always to mean that there are two positive numbers  1 and  2 such that for all admitted values of the discrete variable  or the continuous variable , where {  }  , {  }  are non-negative sequences and ,  are nonnegative functions.If  ∈ ℝ, then  + := max(, 0) and [] denotes the integer part of .Given two (quasi-)Banach spaces  and  , we write  →  if  ⊂  and the natural embedding of  in  is continuous.All unimportant positive constants will be denoted by , occasionally with subscripts.Integration with respect to the n-dimensional Lebesgue measure in ℝ  is denoted by d, whereas || stands for the Lebesgue measure of a Lebesgue-measurable set  in ℝ  .As we shall always deal with function spaces on ℝ  , we may usually omit the 'ℝ  ' from their notation for convenience.

Different approaches.
In this section we discuss three different approaches to Triebel-Lizorkin spaces with positive smoothness.We first present these approaches separately before we come to some comparison.At the end we collect and extend some embedding results that will also be needed below.
Let for 0 < ,  ≤ ∞ the numbers   and   be given by (2.1) The classical approach: Triebel-Lizorkin spaces F  , (ℝ  ).If  is an arbitrary function on ℝ  , ℎ ∈ ℝ  and  ∈ ℕ, then For convenience we may write Δ ℎ instead of Δ 1 ℎ .Furthermore, for a function  ∈   (ℝ  ), 0 <  < ∞,  ∈ ℕ, the ball means are denoted by Remark 2.2.The approach by differences for the spaces F  , (ℝ  ) has been described in detail in [18] for those spaces which can also be considered as subspaces of  ′ (ℝ  ).Otherwise one finds in [22], Section 9.2.2, pp.386-390, the necessary explanations and references to the relevant literature.In particular, the spaces in Definition 2.1 are independent of  , meaning that different values of  >  result in (quasi-)norms which are equivalent.Furthermore, the spaces are (quasi-)Banach spaces (Banach spaces, if 1 ≤  < ∞, 1 ≤  ≤ ∞).Recall that we deal with subspaces of   (ℝ  ), in particular, we have the embedding There is a corresponding approach by differences for the classical Besov spaces B  , .The  -th modulus of smoothness of a function  ∈   (ℝ  ), 0 <  ≤ ∞,  ∈ ℕ, is defined by Let 0 < ,  ≤ ∞,  > 0, and  ∈ ℕ such that  > .Then the Besov space B  , (ℝ  ) contains all  ∈   (ℝ  ) such that (with the usual modification if  = ∞) is finite.Further information on the classical approach for B-and F-spaces -treated in a more general context -may be found in [11].
The Fourier-analytical approach: Triebel-Lizorkin spaces   , (ℝ  ).The Schwartz space (ℝ  ) and its dual  ′ (ℝ  ) of all complex-valued tempered distributions have their usual meaning here.Let  0 =  ∈ (ℝ  ) be such that (2.6) supp Remark 2.4.The spaces   , (ℝ  ) are independent of the particular choice of the smooth dyadic resolution of unity {  }  appearing in their definition.They are (quasi-)Banach spaces (Banach spaces for ,  ≥ 1), and (ℝ  ) →   , (ℝ  ) →  ′ (ℝ  ), where the first embedding is dense if  < ∞.An extension of Definition 2.3 to  = ∞ does not make sense if 0 <  < ∞ (in particular, a corresponding space is not independent of the choice {  }  ).The case  =  = ∞ yields the Besov spaces   ∞,∞ (ℝ  ).In general, the Fourier-analytical Besov spaces   , (ℝ  ) are defined correspondingly to the spaces   , (ℝ  ) by interchanging the order in which the (quasi-)norms are taken, i.e., first using the   -norm and afterwards applying the ℓ  -norm -in view of (2.7).These B-spaces are closely linked with the Triebel-Lizorkin spaces   , (ℝ  ) via We shall later on return to this embedding.The theory of the spaces   , (ℝ  ) (and   , (ℝ  )) has been developed in detail in [18] and [19] (and continued and extended in the more recent monographs [21], [22]), but has a longer history already including many contributors; we do not further want to discuss this here.Note that the spaces   , (ℝ  ) contain tempered distributions which can only be interpreted as regular distributions (functions) for sufficiently high smoothness.More precisely, we have (2.9) cf. [17,Thm. 3.3.2].In particular, for  <   one cannot interpret  ∈   , (ℝ  ) as a regular distribution in general.The scale   , (ℝ  ) contains many well-known function spaces.We list a few special cases.
The subatomic approach: Triebel-Lizorkin spaces   , (ℝ  ).In this subsection we simultaneously give definitions for the Besov spaces   , (ℝ  ) and the spaces   , (ℝ  ).The reason for this is that later on we want to use results previously obtained for the spaces   , (ℝ  ) in [12], in order to now establish corresponding results for the spaces   , (ℝ  ).We complement our notation.Let  , with  ∈ ℕ 0 and  ∈ ℤ  denote a cube in ℝ  with sides parallel to the axes of coordinates, centered at 2 − , and with side length 2 −+1 .Besides, if  is a cube in ℝ  and  > 0, then  is the cube in ℝ  concentric with  and  times the side length of  .
Let  , be the characteristic function of  , and The subatomic approach provides a constructive definition for Besov and Triebel-Lizorkin spaces, expanding functions  via building blocks with suitable coefficients, where the latter belong to certain sequence spaces  , , and  , , , respectively.
for some fixed  > 0 and some fixed  ∈ ℕ, satisfying denote the building blocks related to  , .
We now define the related function spaces.
where the infimum is taken over all possible representations (2.19).
where the infimum is taken over all possible representations (2.21).
Remark 2.9.The definitions given above follow closely [22, Sect.9.2].The spaces   , (ℝ  ) as well as   , (ℝ  ) are (quasi-)Banach spaces (Banach spaces for ,  ≥ 1 ) and independent of  and  (in terms of equivalent (quasi-)norms).Furthermore, for all admitted parameters ,  , , we have as well as the following embedding for B-and F-spaces, (2.23) Proofs of the above assertions can be found in [22,Th. 9.8].In particular, the Besov and Triebel-Lizorkin spaces coincide if  =  , i.e., Concerning the convergence of ( We now discuss the coincidence and diversity of the above presented concepts of F-spaces and may restrict ourselves to positive smoothness  > 0 .In view of our Remarks 2.2, 2.4 and 2.9 concerning the different nature of these spaces, it is obvious that there cannot be established a complete coincidence of all approaches when  <   .It has been shown that such a characterization is also impossible if   <  <   (in particular, when 0 <  < ), cf.[22, Rem.9.15], based on [3] -so the situation is even more complicated.Nevertheless, under certain restrictions on the smoothness parameter , the above approaches result in the same F-space.Theorem 2.10.
Remark 2.12.In view of the results stated in Theorem 2.10 and Remark 2.4, where we noted that the (fractional) Sobolev spaces are contained in the F-scale as a special case, i.e., it makes sense to introduce new Sobolev-type spaces (2.27) In particular, for 1 <  ≤ 2, these spaces coincide with the (fractional) Sobolev spaces, i.e., The figure aside illustrates the general situation.

Proof.
Step 1.We want to establish (i), the so-called Franke-Jawerth embedding.J. Vybíral  where 0 < ,  ≤ ∞, which is the desired result.The second embedding follows immediately from the monotonicity of the ℓ  sequence spaces, i.e., ℓ  → ℓ  for  ≤  .
Step 3. We want to prove (iii).Using (2.39) together with an embedding for Besov spaces proved in [12], namely (which follows immediately from the monotonicity of the ℓ  spaces) we see that where 0 <  1 ,  2 ≤ ∞, and  1 ,  1 are chosen such that Step 4. In order to establish (iv) we again make use of (2.39) and the following embedding for Besov spaces established in [  We finally add what is known for the spaces F  , in terms of embeddings results.
We begin with the left-hand embedding in (2.45).Recall that we need only consider the case  < , that is, min(, ) =  .In view of (2.50), (2.51) it is sufficient to show that We make use of the generalized triangle inequality for integrals, (2.53) Then the left-hand side of (2.52) can be written as and an application of (2.53) yields that is, the right-hand side of (2.52).

𝑝
and the monotonicity of the ℓ  sequence spaces.The first embedding is clear using (i) and corresponding assertions for the Besov spaces B  , , cf. [12, Th. 1.16(i)].We see that which yields the desired embedding.
Step 3. Now (iii) follows from (ii) and the Franke-Jawerth embedding as stated in Theorem 2.16(i).We obtain where the last embedding follows from the monotonicity of the ℓ  sequence spaces and holds if  ≤  .
Step 4. The proof of (iv) follows from (ii), (iii), and the corresponding assertion for Besov spaces, cf.[12, Th. 1.16(iii)], and where for the same restrictions on the parameters.Furthermore in Proposition 2.19(iii) we only established the right-hand side of the well-known Franke-Jawerth embedding.Until now we were unable to prove that the left-hand side holds in general for the full range of parameters.But when  ≤  Theorem 2.10(i) together with Theorem 2.16(i) yields for 0
This concept was introduced and first studied in [21, Ch. 2], [8], see also [9].For convenience we recall some properties.In view of (i) we obtainstrictly speaking -equivalence classes of growth envelope functions when working with equivalent (quasi-)norms in  as we shall usually do.But we do not want to distinguish between representative and equivalence class in what follows and thus stick at the notation introduced in (i).Concerning (ii) we shall assume that we can choose a continuous representative in the equivalence class [ℰ  G ], for convenience (but in a slight abuse of notation) denoted by ℰ  G again.It is obvious that (3.2) holds for  = ∞ and any  .Moreover, one verifies that ) 1   (d) ) 0   (d) Assume for their growth envelope functions for some  > 0 .Then we get for the corresponding indices G .This result coincides with [9,Props. 3.4,4.5].

Remark 3.3. For rearrangement-invariant Banach function spaces 𝑋 with fundamental function
where In contrast to the local characterization in Definition 3.1(ii) it turned out recently, that sometimes also the global behaviour of the envelope function, denote the wellknown Lorentz spaces, consisting of all functions  for which the quantity ) , for  → ∞.

Growth envelopes for F-spaces.
We now turn to Triebel-Lizorkin and Besov spaces and first collect what is known.We will make use of our previous results obtained for growth envelopes in Besov spaces   , (ℝ  ), cf.[12].
As explained, the above concept is interesting only for spaces  ∕ →  ∞ ; in case of F-spaces this reads as follows.

(i) Then
(ii) Furthermore, Proof But also in the limiting case the argument relies on the result for -spaces.Assume  / , →  ∞ , then for all 0 <  < , Theorem 2.16(i) gives Conversely, if 0 <  ≤ 1, we may choose  with  <  < ∞, such that (3.10) yields  / , →  / , →  ∞ .This completes the proof.□ Remark 3.6.Note that we did not use the identity (2.25), that is in the above proof, which is only clear for  >   .This implies that for  =   these spaces may differ if  < min(, 1).However, as verified above, they are both embedded into  ∞ .
(i) Then (ii) Assume  <   or  =   , 1 <  < ∞ and  > min Proof.The proof of (i) follows immediately from Theorem 2.10(i) and Proposition 3.5(ii), since for  >   or  =   and 0 <  ≤ 1 we have Concerning (ii), if  <   we proceed indirectly assuming F  , →  ∞ .Choosing  <  <   we see that which gives the desired contradiction according to Proposition 3.5(ii).
If  =   , 1 <  < ∞, Theorem 2.10(i) yields from which we see -again using Proposition 3.5(ii) -that On the other hand, if In view of Proposition 3.5 we claim that "if" in Proposition 3.7(i) could probably be replaced by "if, and only if", i.e., when 0 <  ≤ ∞ Proposition 3.7(ii) can be generalized to ) .
We now focus on growth envelopes for the spaces   , .In the diagram aside we have shaded the area corresponding to the remaining cases (assuming that  ≥ 1 ) apart from (3.9), where the lower right triangle refers to our new result in Theorem 3.11 below, extending the already known situation repeated below.Recall that by (2.9) and Proposition 3.5(i) only smoothness parameters 0 ≤  ≤   are of interest for the local behaviour of ℰ G ().
(iii) We have Proof.
Step 1.We show that ℰ where Step 2. In order to show that for the additional index = , we use the Franke-Jawerth embedding from Theorem 2.16(i), where we may choose  0 ,  1 and  0 ,  1 such that  0 >  >  1 > 0, and But then the growth envelope functions for all these spaces turn out to be equivalent.This observation together with Proposition 3. Step 3. We prove (ii).Choosing  1 <  <  2 according to Theorem 2.16(i) such that we see from Propostion 3.10(ii) that But then Proposition 3.2(i),(iii) yields ) .
Step 4. We establish for  → ∞.Concerning the global behaviour -using the same argumentation as in Step 1 together with Proposition 3.10(iii) -we see that which completes the proof.□ In terms of the Sobolev-type spaces introduced in Remark 2.12 the results read as follows.
(iii) We have Proof.Note that (ii) is simply a consequence of Theorem 2.10(i) and Proposition 3.11(ii) as well as the embedding for parameters Furthermore we had similar assertions as in Proposition 3.5 for the spaces F  , .But this is not yet verified by our arguments.

Applications
We briefly present two typical applications of the preceding envelope results: sharp embedding criteria and Hardy-type inequalities.
This completes the proof of (i).
Step 2. Now we turn our attention towards (ii).Assume that  ≤  .Using the embedding obtained in Theorem 2.16(iv) yields where the second embedding is well-known for Lorentz spaces and can be found in [2, Prop.4.2.].
In order to prove sufficiency, again we gain from corresponding results for Besov spaces.In [12] we proved that if, and only if, q≤  .But then, using this together with (2.39) yields This finally shows that  ≤  .□ In terms of the spaces F  , the sharp embedding results read as follows.Corollary 4.2.Let  >  > 0 , 0 < ,  < ∞ and 0 <  1 ,  2 ≤ ∞.
(ii): Let where 0 <  < ∞, 0 <  ≤ ∞, and  > 0 ; that is, we have by (2.42) the embedding   , →  , only, whereas the corresponding envelopes even coincide.This can be interpreted as  , being indeed the best possible space within the Lorentz scale in which   , can be embedded continuously.On the other hand this can also be understood in the sense that  , is 'as good as'   , -as far as only the growth of the unbounded functions belonging to the spaces under consideration is concerned, whereas (additional) smoothness features are obviously 'ignored'.
if, and only if, Proof.
Step 1.The necessity, i.e., that (4.4) implies (4.3) is covered by Theorem 2.16(i).It remains to show the converse implication.This is done in two steps: first we use our envelope results for small smoothness parameters, that is, when 0 <  <   ; secondly we combine Proposition 2.14(i),(iii) with the identity (2.25) in Theorem 2.10(i).
Step 2. First we assume 0 <  < In view of Proposition 3.10(i) and Theorem 3.11(i) this is just (4.4). Step

Proposition 2.14.
), where the parameters satisfy (2.35).We want to prove corresponding results for spaces of type   , and F  , .
Proposition 3.2.(i)Let  ,  = 1, 2 , be some function spaces on ℝ  .Then  1 →  2 implies that there is some positive constant  such that for all  > 0 , . Part (i) coincides with [6, 2.3.3(iii)],so we are left to prove (ii).Clearly the limiting case  =   is of interest only, in view of Theorem 2.16(i) and the corresponding result for the Besov spaces in [12, Prop.2.5], (3.10)   , (ℝ  ) →  ∞ (ℝ  ) if, and only if, Compared to the other approaches associated to the spaces   , and   , , our results for the spaces F  , are not as complete.The reason for this is mainly the lacking left-hand side of the Franke-Jawerth embedding, as already mentioned in Remark 2.20.If (2.59) was true in general, we immediately obtained for the additional index that