Eduard Curc

Approximation of critical regularity functions on strati ed homogeneous groups

Let G be a strati ed homogeneous group with homogeneous dimension Q and whose Lie algebra is generated by the left-invariant vector elds X 1 ,...,X d 1 . Let 1 < p; q < 1, = Q=p and > 0. We prove that for any function f 2 _ F ;p q (G) there exists a function

where k is the largest integer smaller than min(p; d 1 ) and C is a positive constant only depending on . Here, _ F ;p q (G) is a Triebel-Lizorkin type space adapted to G. This generalizes earlier results of Bourgain, Brezis [4] and of Bousquet, Russ, Wang, Yung [6] in the Euclidean case and answers an open problem in [6].

Introduction

Let B R d (d 2) be a Euclidean ball. It is well-known that, if f 2 L p loc (B; R) with 1 < p < 1, then the equation div Y = f has a solution X 2 W 1;p loc (B; R d ). When p = d, this Y "almost" belongs to L 1 loc (B; R d ). A striking result obtained by Bourgain and Brezis (in [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF]) asserts that is possible to nd Y 2 W 1;d loc (B; R d ) \ L 1 loc (B; R d ). Their argument relies on a new type of approximation results.

This seminal work has been followed by a number of approximation results of similar type [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF], [START_REF] Bousquet | A limiting case for the divergence equation[END_REF], [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]. Our work is primarily motivated by two types of developments of the results in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] concerning functions in critical Sobolev spaces that barely fail the embedding in L 1 .

The rst of these results (Lemma 1.7 in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF]) deals with the extension of the approximation result given in [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF] (Theorem 11) in the Euclidean case, to the more general case of strati ed homogeneous groups. Somewhat informally this reads (see Section 2 for de nitions):

Theorem 1.1 Suppose G is a strati ed homogeneous group whose homogeneous dimension is Q and let X 1 ; :::; X n 1 be a minimal family of vector elds generating the Lie algebra of G. Then, for any Schwartz function f on G and any > 0 there exists a function F such that:

n 1 1 X i=1 kX i (f F )k L Q (G) kr b f k L Q (G) , kF k L 1 (G) + kr b F k L Q (G) C kr b f k L Q (G) ,
where C is a constant only depending on .

Here, r b f = (X 1 f; :::; X n 1 f ). Theorem 11 in [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF] corresponds to the Euclidean case. On the other hand, it was proved in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] (Theorem 1.1) that Theorem 11 in [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF] remains true, in the Euclidean case if we replace the critical Sobolev space _ W 1;d (R d ) by more general critical spaces such as _ F d=p;p q (R d ). More precisely, we have the following:

Theorem 1.2 Consider the parameters 1 < p; q < 1, = d=p and let k be the largest positive integer with k < min(p; d). Then, for every > 0 there exists a constant C > 0 only depending on , such that for every function f 2 _ F ;p q (R d ) there exists F 2 L 1 (R d ) \ _ F ;p q (R d ) satisfying the following estimates:

k X i=1 k@ i (f F )k _ F 1;p q (R d ) kf k _ F ;p q (R d ) , kF k L 1 (R d ) + kF k _ F ;p q (R d ) C kf k _ F ;p q (R d ) .
Note that here we have a somewhat unnatural technical condition on k, which does not seem to be optimal. Namely, we impose k < min(p; d) instead of only imposing k < d. (See [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] for a discussion on this assumption.)

The purpose of this paper is to nd a common roof to Theorem 1.1 and Theorem 1.2 and to give an a rmative answer to Open question 1.4 in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]. Our generalisation is an adaptation of Theorem 1.2 above to the strati ed homogeneous groups context of Theorem 1.1. In this case the role of the Euclidean dimension is played by the homogeneous dimension Q of the group and the critical regularity becomes, in this case, = Q=p. The role of the derivatives is played by the vector elds that generate the full Lie algebra of G.

The statement of our main result is:

Theorem 1.3 Consider the parameters 1 < p; q < 1, = Q=p and let k be the largest positive integer with k < min(p; d 1 ). Then, for every > 0 there exists a constant C > 0 only depending on , such that, for every function f 2 _ F ;p q (G), there exists F 2 L 1 (G) \ _ F ;p q (G) satisfying the following estimates:

k X i=1 kX i (f F )k _ F 1;p q (G) kf k _ F ;p q (G) , kF k L 1 (G) + kF k _ F ;p q (G) C kf k _ F ;p q (G) .
We will give in Section 2 precise de nition of the function spaces we consider on G. For the time being, let us mention that we cover the case of the more familiar anisotropic homogeneous Sobolev spaces _ N L m;p , de ned informally as containing the functions f on G for which r m b f 2 L p . Despite the fact that we also have the unnatural restriction k < min(p; d 1 ), as in the Euclidean case, this su ces for some applications to divergence-like systems. Basically, all the applications to such systems presented in [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF] can be easily adapted to the strati ed homogeneous group setting and higher order Sobolev spaces. We give one example, formulated for simplicity for spaces of integer regularity.

Theorem 1.4 Let m < Q be a positive integer. Suppose f 2 _ N L m 1;Q=m (G) and there exist functions v 1 ; :::; v d 1 2 _ N L m;Q=m (G) such that

X 1 v 1 + ::: + X d 1 v d 1 = f .
Then, there exist u 1 ; :::; u d 1 2 L 1 (G) \ _ N L m;Q=m (G) such that X 1 u 1 + :::

+ X d 1 u d 1 = f .
The paper is divided into two parts. The rst one (Section 2) deals with the construction of the Triebel-Lizorkin spaces on strati ed homogeneous groups. We mention that the Euclidean analogues of these spaces coincide with the classical ones and that in the general strati ed homogeneous group setting, they also satisfy similar interpolation and duality properties as their classical analogues.

Spaces of a similar kind were already de ned and studied for example in [START_REF] Bahouri | Paraproduit sur le groupe de Heisenberg et applications[END_REF], [START_REF] Hu | Homogeneous Triebel-Lizorkin Spaces on Strati ed Lie Groups[END_REF] and other works (see also [START_REF] Furioli | Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth[END_REF] for a construction of inhomogenous spaces in the more general context of Lie groups of polynomial volume growth). Our construction is very similar to the one given in [START_REF] Hu | Homogeneous Triebel-Lizorkin Spaces on Strati ed Lie Groups[END_REF] (it turns out that our spaces essentially coincide with the ones introduced in [START_REF] Hu | Homogeneous Triebel-Lizorkin Spaces on Strati ed Lie Groups[END_REF], as a consequence of our Proposition 2.10). While the construction in [START_REF] Hu | Homogeneous Triebel-Lizorkin Spaces on Strati ed Lie Groups[END_REF] is based on spectral decomposition of sublaplacians, our construction is based only on the relatively elementary technique developed in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] for obtaining a Littlewood-Paley decomposition for functions de ned on the group. (We also notice that our purpose is not to explore the properties of these spaces, but rather to prove a minimal number of their properties, required in the proof of Theorem 1.3.)

While in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] Littlewood-Paley decomposition is obtained by a Calder on reproducing formula with two convolutions, we will also need similar reproducing formulas with three convolutions (we will prove that all the de nitions of the spaces with two or more convolutions coincide). This allows us to prove the full analogue of the Littlewood-Paley inequality as well as other inequalities needed in the proof of Theorem 1.3.

The second part (Sections 3 and 4) is devoted to the proof of Theorem 1.3. We follow closely the proof in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]. Several relatively minor modi cations were made in order to simplify the exposition. Some more substantial adaptations were required in order to bypass the lack of commutativity of the vector elds. In some cases the arguments are easily adapted to the group setting, and in these situations we only sketch the arguments or refer to the proofs in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]. In the Appendix we recall the Calder on-Zygmund theory on strati ed homogeneous groups in order to give a direct proof of an inequality (Proposition A1) whose Euclidean analogue was proved in the Appendix of [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] by similar but more complicated means.

2 Function spaces on strati ed homogeneous groups

Basic facts on strati ed homogeneous groups

Here, we follow mainly Folland and Stein [START_REF] Folland | Hardy Spaces on Homogeneous Groups[END_REF] and Stein [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF]. We also present some auxiliary results, possibly known to experts, that we will need in order to develop the Littlewood-Paley theory of function spaces on strati ed homogeneous groups. We will consider homogeneous groups as de ned in ( [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF], p. 618). For such a group G, we write the following decomposition of its Lie algebra g:

g = V 1 V 2 ::: V `,
where V 1 ; :::; V `are vector spaces of left-invariant vector elds such that

(i) [V i ; V j ]
V i+j (making the convention that V `is not trivial and any V j with j > `is trivial), (ii) V 1 generates the whole algebra g (this is the so called H• ormander condition).

Dimension. We write d j := dim V j and d := d 1 + ::: + d `; the number Q = d 1 + 2d 2 + ::: + `dì s called the homogeneous dimension of G. As sets, we identify G with R d . In view of this identi cation, we consider the following dilation rule: if x = (x 1 ; :::; x d ) 2 G and > 0, then x = ( a 1 x 1 ; :::; a d x d ), where a := (a 1 ; :::; a d ) = (1; :::; 1; 2; :::; 2; :::; `; :::

; `) (2.1)
is the vector of the homogeneities, each j 2 f1; :::; `g appearing d j times. The dilations are known to be automorphisms of G and, with respect to them, the following "norm" on G is homogeneous:

kxk G := 0 @ X j=1 X d 1 +:::+d j 1 <i d 1 +:::+d j jx i j 2`! j 1 A 1 2`! . (2.2)
We have also the quasi-triangle inequality

kx yk G . kxk G + kyk G , for x; y 2 G.
Subgradient. We write X 1 ; X 2 ; :::; X d for the left-invariant vector elds forming the standard basis of g, with X 1 ; X 2 ; :::; X d 1 forming a basis of V 1 . We will call full gradient and subgradient respectively the following operators r := (X 1 ; X 2 ; :::; X d ) , r b := (X 1 ; X 2 ; :::; X d 1 ) .

Note that, whenever f is a Schwartz function on R d with r b f 0 then, thanks to the H• ormander condition, we get rf 0. Hence, in a sense, the subgradient encodes all the di erential information about f . We will always be concerned with the subgradient of functions rather than with the full gradient. We will consider for example the Sobolev-type space _ N L 1;Q , which informally is a space of functions on G whose subgradient is in L Q . Note that this space is not the same as _ W 1;Q on G seen as a manifold. Similar considerations hold for right-invariant vector elds. We will write X R j for the rightinvariant analogue of X j .

An important aspect is that, with the identi cation G = R d , we have that x y is a polynomial in x; y and (x y) k = x k + y k for any x; y 2 G as long as 1 k d 1 . Also we have x 1 = x for all x 2 G.

Balls and the maximal function. We consider balls on G de ned by the quasimetric on G, given by (x; y) = y 1 x G for x; y 2 G. The open ball centered in x and of radius > 0 is the set

B(x; ) = fy 2 Gj (y; x) < g ,
whose Lebesgue measure is jB(x; )j Q . For all balls B = B(x; ) and > 0 we will write B = B(x; ).

We also consider the Hardy-Littlewood maximal function M on G, de ned by

M f (x) = sup B3x 1 jBj Z B jf (y)j dy,
for all functions f 2 L 1 loc (G), where the supremum is taken over all balls B G containing x.

We recall the following classical facts (for proofs see Chapter 2 in [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF]):

Proposition 2.1 (i) If ' is a nonnegative decreasing function on [0; 1), such that C = R G '(kyk G )dy < 1 and is a measurable function on G such that j (y)j '(kyk G ) on G, then jf j . CM f on G,
for any Schwartz f . Here the convolution on G is de ned by

f (x) = Z R d f (y) (y 1 x)dy = Z R d f (x y 1 ) (y)dy.
(ii) M is of weak type (1; 1) and of strong type (p; p) for all 1 < p 1.

(iii) (the Fe erman-Stein inequality) Consider a sequence of Schwartz functions (f j ) j2Z . Then, for 1 < p; q < 1, we have kM f j k l q j L p . p;q kf j k l q j L p .

Vector elds and polynomials. We remind the following elementary formula ( [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF], p. 621):

X j f (x) := @f (x y) @y j y=0 = @ j f (x) + X k>j q j;k (x)@ k f (x) (2.3)
where y = (0; :::0; y j ; 0; :::0) and q j;k are homogeneous polynomials of degree a k a j . Another elementary fact is that the integral of the functions of the form X j f , where f is a Schwartz function is, as in the Euclidean case, equal to 0. Here is a proof of this fact. For any y = (0; :::0; y j ; 0; :::0) 2 G, with y j 6 = 0, using the fact that the Lebesgue measure on R d is a bi-invariant Haar measure on G ( [START_REF] Folland | Hardy Spaces on Homogeneous Groups[END_REF], Proposition (1.2), p. 3), we have Z

R d f (x y) f (x) y j dx = 1 y j Z R d f (x y)dx Z R d f (x)dx = 0.
Using now the formula (2.3), the classical mean value theorem in the (Euclidean) R d and the dominated convergence theorem, we can pass to the limit when y j ! 0 in the above formula to obtain Z

R d X j f (x)dx = 0.
A similar formula holds for right-invariant vector elds. As an immediate consequence of this and the Leibniz rule we get the formula (see [START_REF] Folland | Hardy Spaces on Homogeneous Groups[END_REF]

, p. 21) Z R d (X j f ) gdx = Z R d f (X j g) dx (2.4)
whenever f and g are Schwartz functions or one of them is Schwartz and the other one is polynomial.

Before going to the next step let us x some notation. For a real valued function f su ciently smooth on G and a positive integer m, we write r m b f for the vector valued function whose components are

r b f := X 1 1 1 X 1 2 2 :::X 1 d 1 d 1 X 2 1 1 X 2 2 2 :::X 2 d 1 d 1 ::: X m 1 1 X m 2 2 :::X m d 1 d 1 f (2.5)
listed in the lexicographic order given by =

: N d 1 in N d 1 N
, we can de ne r b f by the above formula whenever j j < 1.

We will also use many times the notation r m b ' where ' = ' j j=m is a nite family of Schwartz functions. This has the following meaning

r m b ' := X j j=m r b ' . (2.6)
Let us see that high powers of the subgradient are able to anihilate low degree polynomials. More speci cally, The similar assertion for the right-invariant subgradient also holds.

Proof. It su ces to prove the statement when p is a monomial. Suppose p(x) = x = x 1 1 :::x d d for some = ( 1 ; :::; d ) 2 N d and consider the function q = r m b p. We can see from the formula (2.3) that q is a vector valued polynomial on R d . Writing x for the group dilation of x 2 G with the parameter > 0, we immediately see from the de nition of the subgradient that r m b (p( x)) = m r m b p( x). Also, we have

p( x) = ( a 1 x) 1 1 ::: ( a d x) d d = ha; i p(x).
From this we conclude that, for all x 2 G,

q( x) = (r m b p) ( x) = m r m b (p( x)) = ha; i m r m b p(x) = ha; i m q(x):
If cx is a monomial (c 6 = 0) of maximum degree in q, as before we get ( x) = ha; i x for all > 0. Choosing from these monomials one for which ha; i is maximum, we get by the above formula that ha; i = ha; i m and hence deg q = j j ha; i `j j m.

Let us next recall a fundamental formula that makes a connection between the derivatives on R d and the vector elds from g. More speci cally, for any 1 i d we have ( [START_REF] Folland | Hardy Spaces on Homogeneous Groups[END_REF], p. 25)

@ i = d X k=1 P k;i X k , (2.7)
where P k;i are homogeneous polynomials of degree a k a i . We will also need the following.

Proposition 2.3

We have that

@ i = d 1 X k=1 X k D k;i . (2.8)
where the operators D k;i are the adjoints of some operators of the form P p r R d for appropriate polynomials p and multi-indexes in a nite subset of N d . Proof. Since the vector elds X 1 ; X 2 ; :::; X d 1 are generating the full Lie algebra of the group, we can write each X j in terms of X 1 ; X 2 ; :::; X d 1 using commutators, which are linear combinations of expressions of the form r b = r 0 b X k for some 1 k d 1 and some indexes , 0 2 N d 1 N . Keeping the last vector eld from such an expression and using (2.3) to express r 0 b in terms of derivatives on R d and polynomials, we can rewrite (2.7) as

@ i = d 1 X k=1 D k;i X k , (2.9)
where each operator D k;i is of the form P p r R d for some polynomials p and in a nite subset of N d . Now, if f and g are arbitrary Schwartz functions we can write (see (2.4)):

Z R d f @ i gdx = Z R d (@ i f ) gdx = d 1 X k=1 Z R d (D k;i X k f ) gdx = d 1 X k=1 Z R d f X k D k;i g dx
and hence, by identi cation,

@ i = d 1 X k=1 X k D k;i ,
which proves the Proposition 2.3. The same is also true in the case of the right-invariant subgradient.

Remarks. (1) Since the assertion of (ii) in the above proposition remains true for any integer larger than m 0 , when applying this part of the proposition, we will assume for technical reasons that m 0 > m`.

(2) In particular, Proposition 2.4 gives the following (informally speaking): if ' 1 is a Schwartz family, then there exists another Schwartz family ' 2 such that:

r R b m 0 ' 1 = r m b ' 2 .
This property will be used several times.

Proof. Part (i) follows from Proposition 2.2 and by a repeated application of the formula (2.4). Part (ii) will be proved by induction on m. The case m = 0 is trivial (we take by convention m 0 = 0). Fix m 1 and suppose we have the statement of (ii) for m 1. Consider the number m 0 := (m 1) 0 + M + 2, where M is the maximum degree reached by a polynomial p entering in the expression of the operators D k;i that occur in (2.9). If R G pf dx = 0 for any polynomial p of degree at most m 0 , then we can use the well-known fact that in the Euclidean case there exists a collection of Schwartz families ( i )

1 i d such that f = d X i=1 @ i r m 0 1 R d i .
Using now formula (2.8) we can write:

f = d X i=1 @ i r m 0 1 R d i = d X i=1 d 1 X k=1 X k D k;i r m 0 1 R d i = d 1 X k=1 X k d X i=1 D k;i r m 0 1 R d i ! = d 1 X k=1 X k ~ k , where ~ k are the Schwartz functions ~ k = P d i=1 D k;i r m 0 1 R d i . It is easy to see that R G p ~ k dx =
0 for all polynomials p of degree at most (m 1) 0 . By the induction hypothesis, we get that for each k there exists a family of Schwartz functions

' k such that ~ k = r m 1 b ' k .
From this and the above formula, we get the conclusion.

Convolutions. We recall that, for two Schwartz functions f; g their convolution is de ned by the formula:

f g(x) = Z R d f (y)g(y 1 x)dy = Z R d f (x y 1 )g(y)dy.
It can be veri ed directly that the convolution is associative.

Concerning the interaction of vector elds with the convolution, it is known that ([8], p. 22): Proposition 2.5 For all Schwartz functions f; g we have:

X j (f g) = f (X j g) , X R j (f g) = X R j f g, (X j f ) g = f X R j g . (2.10)
We have also the following elementary fact.

Proposition 2.6 If 1 ; 2 are two Schwartz functions, then 1 2 is also Schwartz.

Proof. We can easily observe that, since each component of x y is a polynomial in x and y, we can nd a large number n G 2 N such that 1 + jx yj .

(1 + jxj) n G (1 + jyj) n G for all x; y 2 R d .
This implies that, for example, we have sup

x (1 + jxj) N j 1 2 (x)j sup x Z R d 1 + x y 1 y N 1 (x y 1 ) j 2 (y)j dy . sup x Z R d 1 + x y 1 N n G 1 (x y 1 ) (1 + jyj) N n G j 2 (y)j dy . Z R d (1 + jyj) N n G j 2 (y)j dy < 1.
More generally, the estimate of sup x (1 + jxj) N @ ( 1 2 ) (x) is reduced to the above calculation using the connection between the derivatives and the vector elds on G via (2.7) and (2.3).

The Littlewood-Paley decomposition

We introduce the following notation. Whenever is a Schwartz function on G and j is an integer, we write j for the function de ned by j (x) = 2 jQ (2 j x). Also, if f is another Schwartz function, we write j f = f j .

Proposition 2.7 Given m 2 N, there exist Schwartz families 3 (x)dx = 0 for all the polynomials P of degree m 0 (with m 0 as in Proposition 2.4) and such that for all Schwartz functions f we have

1 , 2 , 3 on R d such that R R d P (x) 1 (x)dx = R R d P (x) 2 (x)dx = R R d P (x)
f = X j2Z f 1 j 2 j 3 j = X j2Z 3 j 2 j 1 j f , (2.11)
the convergence being in any L p (R d ) for 1 < p < 1. In particular, according to Proposition 2.4 (ii), there exist families of Schwartz families ' i , i (i = 1; 2; 3)

such that i = O m b ' i = O R b m i
for each i = 1; 2; 3.

Remark. Some explanations are in order. The proposition literally states that there exist three nite Schwartz families i = ( i;a ) a2A (A is a nite set), i = 1; 2; 3, such that all the moments of order up to m 0 of each i;a are zero and

f = X j2Z X a2A f 1;a j 2;a j 3;a j = X j2Z X a2A
3;a j 2;a j 1;a j f .

The last assertion means that there exists 6jAj Schwartz families ' i;a , i;a such that

i;a = O m b ' i;a = O R b m i;a
for all a 2 A and i = 1; 2; 3 (see (2.6)). Since the use of the familly A leads to heavy notation, we prefer the form of the above proposition which turns out to be more convenient in the calculations that follow. This can be compared with the summation convention in geometry. We also note that the absolute value of expressions like j f , where = ( a ) a2A is a Schwartz family, will have the following meaning:

1 j f = X a2A 1;a j f .
Similarly, we set

2 j 1 j f = X a2A 2;a j 1;a j f ,
and so on. These conventions, together with (2.6), will enable us to estimate expressions involving Schwartz families as if they were functions. We will also abuse the notation in other situations, where the distinction between functions and nite families of functions will be clearly irrelevant (see also the conventions in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF]).

Proof. This proof follows the lines of Proposition 5.5 in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF]. We consider a radial Schwartz function with ^ 1 on B R d (0; 1) and supp ^ B R d (0; 2) (here B R d (0; 1) and B R d (0; 2) are Euclidean balls). We need now the easy argument of Proposition 5.1 from [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] which we reproduce below for the convenience of the reader.

Lemma 2.8 Let be a Schwartz function on R d such that R R d dx = 1 and x some 1 < p < 1. Then, for any Schwartz function f , we have f = X j2Z f ( j j 1 ),
the convergence being in L p .

Proof. We have, for any

N 2 N , X jjj N f ( j j 1 ) = f N f N 1 .
Hence it remains to see that f N ! f and f N ! 0 in L p when N ! 1. In order to prove the rst claim we write, using Minkowski's inequality,

kf N f k L p = Z R d f (x (2 N y) 1 ) f (x) (y)dy L p x Z R d f (x (2 N y) 1 ) f (x) L p
x j (y)j dy ! 0 (this can be seen by using the dominated convergence theorem).

In order to prove the second claim, again by Minkowski's inequality we have

kf N k L p kf k L 1 k N k L p = 2 N Q(1 1=p) kf k L 1 ! 0,
proving the lemma.

Proof of Proposition 2.7 continued.

The above Lemma applied to = (see Proposition 2.6) yields

f = X j2Z f ( ) j ( 1 1 1 ) j = X j2Z f ( 1 1 1 ) j = X j2Z f ( ( 1 ) + ( 1 ) 1 + ( 1 ) 1 1 ) j , (2.12)
the convergence being in L p (R d ) with 1 < p < 1. Since we have ^ ^ 1 0 in a neighborhood of 0, the function 1 is orthogonal to all polynomials. By applying Proposition 2.4 (ii) we can nd a Schwartz family ' such that 1 = r R b 2n 2 ', with n 2 := (2n 1 ) 0 where n 1 := (m 0 ) 0 .

Using (2.10) we can write schematically, abusing the notation,

( 1 ) = r R b 2n 2 ' = r n 2 b r R b n 2 ' = r R b 2n 1 ~ r R b n 2 ' = r n 1 b r R b n 1 ~ r R b n 2 ',
where ~ is a Schwartz family such that r R b

2n 1 ~ = r n 2 b
; this can be seen to exist thanks to Proposition 2.4 (see Remark ( 2)). The other terms in (2.12), namely

1 and ( 1)

1
1 can be handled in a similar way. We nd that each one of them is a nite sum in which each term is of the form Y

(m 0 ) 0 1 1 Y (m 0 ) 0 2 2 Y (m 0 ) 0 3
3 where i are Schwartz families and Y i is r b or r R b . This implies (2.11) via Proposition 2.4 (i), once we note that (m 0 ) 0 > m 0 `(see the Remark (1) after Proposition 2.4).

Remarks. (1) We will use sometimes the function =

1 1

1 for which, as we can see in the above proof, we have the estimate j j f j

3 j 2 j 1
j f for all integers j and all Schwartz functions f . From (2.12), we have

f = X j2Z j f in L p , 1 < p < 1.
Schematically we write = 3 2 1 . We will also consider its weaker analogue,

1 := 2 1 . (2.13)
(2) It is easy to see that we can obtain decompositions of the form f = X j2Z k j :::

3 j 2 j 1 j f ,
with arbitrary k 1 and 1 ,..., k as in Proposition 2.7. It turns out that, for the estimates we need in this work, convolutions involving k 3 terms are in some cases very convenient. Note that a decomposition formula as above with k 2 convolutions implies a decomposition with k 1 convolutions. In this regard we note that even if in most cases a decomposition formula with two convolutions su ces (to de ne Triebel-Lizorkin spaces and to prove several of their properties), the proof of Theorem 1.3 relies on decomposition formulas with three convolutions (this will be used, for example to prove the Bernstein type inequalities (3.3)).

De nition of function spaces on strati ed homogeneous groups

Let s 2 R, p; q 2 (1; 1) and x m > jsj and some Schwartz families 1 , 2 whose moments up to order m 0 are zero (see Proposition 2.7 and the Remarks after) and such that we have the following decomposition formula with two convolutions:

f = X j2Z 2 j 1 j f ,
for any Schwartz function f . We de ne the spaces _ F s;p q and _ B s;p q as being the spaces of tempered distributions f on R d whose (semi)norms, respectively de ned as:

kf k _ F s;p q = X j2Z 2 sjq 1 j f q ! 1=q L p , kf k _ B s;p q = X j2Z 2 sjq 1 j f q L p ! 1=q
, are nite.

We notice that at rst sight these de nitions seem to depend on the families 1 , 2 . We will show however (Proposition 2.10), that the de nition of _ F s;p q (and of _ B s;p q ) does not depend on 1 , 2 . We will also show (Proposition 2.14) that, as expected, the space _ F n;p 2 with n a nonnegative integer, is the same as the more "classical" Sobolev space _ N L n;p .

Independence of the de nition. We will need the following simple lemma:

Lemma 2.9 Consider a sequence (f k ) k2Z of Schwartz functions such that all but a nite number of them are zero. Consider also an s 2 R, an integer m > jsj and two nite Schwartz families and for which all the moments up to the order m 0 are zero. Then, for 1 < p; q < 1, we have:

X k 2 skq k X j j f j q ! 1=q L p . X k 2 skq jf k j q ! 1=q L p . ( 2
j k = ( k j ) j = (r m b k j ) j = 2 m(k j) r R b m k j j , hence, j k = 2 m(k j) j r R b m k . (2.15)
In a similar way, we get

j k = 2 m(j k) ((r b ) m ) j ' k . (2.16)
Note that, if g, and are Schwartz and j, k are two integers, then

g j k . M (g j )
. M M g, where the implicit multiplicative constants only depend on and . Using this observation and (2.15), (2.16), we can write j k j f j j . 2 mjk jj M M f j .

Choosing 2 (0; 1) such that m > jsj, and using H• older's inequality, we can write:

X k 2 skq X j k j f j q . X k 2 skq X j 2 mjk jj M M f j ! q = X k 2 skq X j 2 (1 )mjk jj 2 mjk jj M M f j ! q . X k 2 skq X j 2 q mjk jj jM M f j j q = X j X k 2 skq 2 q mjk jj ! (M M f j ) q ,
where we had used, in the third line, the fact that

X j 2 q 0 (1 )mjk jj ! q=q 0 < 1.
We have now, for all j 2 Z, X k 2 skq 2 q mjk jj = X k j

::: + X k<j ::: = X k 0 2 sqj 2 (s m)qk + X k<0 2 sqj 2 (s+ m)qk 2 sjq
and, as a consequence of the above inequality,

X k 2 skq X j k j f j q ! 1=q . X j 2 sjq (M M f j ) q ! 1=q
.

Applying twice the Fe erman-Stein inequality we get (2.14).

Now we can see that the above lemma implies the independence of the de nition of the spaces of Triebel-Lizorkin type with respect to the choice of 1 , 2 . (The following statement is similar to Theorem 7 in [START_REF] Hu | Homogeneous Triebel-Lizorkin Spaces on Strati ed Lie Groups[END_REF].) Proposition 2.10 Given the parameters s 2 R, p; q 2 (1; 1), the space _ F s;p q does not depend on the auxiliary functions 1 , 2 .

Proof. Indeed, let s 2 R, p; q 2 (1; 1), and m 1 ; m 2 > jsj. Consider, as in the de nition of the Triebel-Lizorkin spaces, two couples of functions 1 , 2 and 1 , 2 corresponding to m 1 ; m 2 respectively. We can construct, using the rst and the second couples of functions, the spaces _ F s;p q and _ F s;p q respectively. Using Proposition 2.7 and Lemma 2.9 for = 1 , = 2

and f j = 1 j f for a Schwartz function f , we get, after a limiting argument that:

kf k ( _ F s;p q ) = X k 2 skq 1 k X j 2 j 1 j f q ! 1=q L p . X k 2 skq 1 k f q ! 1=q L p = kf k ( _ F s;p q ) .
Note that in a similar way we can obtain the converse inequality. Hence, by density, we have that _ F s;p q = _ F s;p q with equivalent norms.

Remarks.

(1) The same type of independence can be proved, in a very similar way, for the Besov spaces _ B s;p q . In this case the analogue of Lemma 2.9 is Lemma 2.11 Consider a sequence (f k ) k2Z of Schwartz functions such that all but a nite number of them are zero. Consider also an s 2 R, an integer m > jsj and two nite Schwartz families and for which all the moments up to the order m 0 are zero. Then, for 1 < p; q 1, we have:

0 @ X k 2 skq k X j j f j q L p 1 A 1=q . X k 2 skq kf k k q L p ! 1=q
.

Note that here we allow the values p = 1, q = 1. This is due to the fact that the Fe erman-Stein inequality is no longer needed.

(2) Lemma 2.9 can also be used to prove real and complex interpolation results for the Triebel-Lizorkin spaces with the same method of retract as for the classical spaces. In this case, the extension and retract operators E : _ F s;p q ! L p ( _ l q s ) and R : L p ( _ l q s ) ! _ F s;p q are de ned by Ef := ( 1 k f ) k2Z and R (f k ) k2Z := P j2Z 2 j f j . Lemma 2.9 is used to prove that R is well-de ned and bounded, while these properties are obvious for E. Similarly for Besov spaces, relying on Lemma 2.11.

Inspecting the above proof of Proposition 2.10, we can see immediately that, by a very similar reasoning, we get the following:

Corollary 2.12 Consider some parameters 1 < p; q < 1, s 2 R. Also consider an integer m > jsj and a Schwartz family ~ such that all its moments of order up to m 0 are zero. Then, for any Schwartz function f , we have:

X k 2 skq ~ k f q ! 1=q L p . kf k _ F s;p q .
The lifting property. Let us now see how Corollary 2.12 implies the lifting property for the spaces _ F s;p q (the following statement is similar to Corollary 21 in [START_REF] Hu | Homogeneous Triebel-Lizorkin Spaces on Strati ed Lie Groups[END_REF]).

Proposition 2.13 For any Schwartz function f , we have

kr b f k _ F s;p q kf k _ F s+1;p q .
Proof. Consider some Schwartz functions 1 j , 2 j for which all the moments of order up to m 0 are zero (s 2 R and the integer m > jsj being xed) and such that

f = X j2Z 2 j 1 j f ,
for any Schwartz function f . Combining the de nition of the Triebel-Lizorkin spaces, Proposition 2.5 and Corollary 2.12, we have

kr b f k _ F s;p q X j2Z 2 sqj (r b f ) 1 j q ! 1=q L p = X j2Z 2 sqj f r R b 1 j q ! 1=q L p = X j2Z 2 (s+1)qj f r R b 1 j q ! 1=q L p = X j2Z 2 (s+1)qj ~ 1 j f q ! 1=q L p . kf k _ F s+1;p q , where ~ 1 = r R b 1 .
For the opposite inequality, using Proposition 2.4 and the independence of the de nition (Proposition 2.10), we can assume that 1 for some Schwartz function , and then we have:

kf k _ F s+1;p q X j2Z 2 (s+1)qj f 1 j q ! 1=q L p = X j2Z 2 (s+1)qj f r R b j q ! 1=q L p = X j2Z 2 sqj f r R b j q ! 1=q L p = X j2Z 2 sqj r b f j q ! 1=q L p = X j2Z 2 sqj j (r b f ) q ! 1=q L p . kr b f k _ F s;p q .
Hence, for all Schwartz functions f we have

kr b f k _ F s;p q kf k _ F s+1;p q .
The identi cation _ F n;p 2 = _ N L n;p . The following statement is a generalisation of Proposition 5.7 in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF].

Proposition 2.14 Fix an m 2 N and consider Schwartz families 1 , 2 corresponding to m as in Proposition 2.7. Then, for any Schwartz function f we have

X j2Z 2 2nj 1 j f 2 ! 1=2 L p kr n b f k L p ,
for all n 2 N with n m 1 and 1 < p < 1. In other words, we have _ F n;p 2 = _ N L n;p with equivalent norms.

Proof. We follow the lines of Proposition 5.7 in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], which proves a similar statement in the case n = 1. The estimate "." easily follows by writing 1 = r R b n+1 ' for a Schwartz family ' and then applying Proposition 5.4 in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], whose statement is reproduced below in a simpli ed form (see also [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF], Chapter 13, section 5.3):

Lemma 2.15 If D is a Schwartz function such that R G Ddx = 0
, then for a xed 1 < p < 1 and any Schwartz function f we have:

X j2Z jD j f j 2 ! 1=2 L p . kf k L p .
Using this we immediately obtain:

X j2Z 2 nj 1 j f 2 ! 1=2 L p = X j2Z r n b f r R b ' j 2 ! 1=2 L p . kr n b f k L p .
For the reverse estimate we need to observe that, according to the proof of Proposition 5.5 in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], whenever we have a decomposition of the form f = P j f j j with and Schwartz and having zero integral, we get for any Schwartz function

f that kf k L p . X j2Z j j f j 2 ! 1=2 L p .
(2.17)

Before going further, we sketch, for the convenience of the reader the standard duality argument to prove (2.17). For all Schwartz functions g write, using the above Lemma 2.15,

hf; gi = X j h j j f; gi = X j j f; j g Z G X j j j f j 2 ! 1=2 X j j g 2 ! 1=2 dx X j2Z j j f j 2 ! 1=2 L p X j2Z j g 2 ! 1=2 L p 0 . X j2Z j j f j 2 ! 1=2 L p kgk L p 0 .(2.18)
We obtain (2.17) by taking, in (2.18), the supremum over g such that kgk L p 0 1.

Using (2.17) with = 1 and = 2 , replacing f by r n b f and using (2.10) together with Corollary 2.12, we obtain:

kr n b f k L p . X j2Z r n b f 1 j 2 ! 1=2 L p = X j2Z f r R b n 1 j 2 ! 1=2 L p = X j2Z 2 jn f r R b n 1 j 2 ! 1=2 L p = X j2Z 2 jn ~ 1 j f 2 ! 1=2 L p . kf k _ F n;p 2 X j2Z 2 jn 1 j f 2 ! 1=2 L p , where ~ 1 = r R b n 1 .
This proves the proposition.

3 Estimates of the auxiliary functions

Remark concerning the approximations

Following [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], our purpose is to prove the approximation property stated in Theorem 1.3. In the remaining part of the paper we will use decompositions formulas with three convolutions, as in Proposition 2.7.

It su ces to prove this approximation property for functions of a special form:

f J := X jjj J 3 j 2 j 1 j f = X jjj J j f ,
where 1 j , 2 j , 3 j and m > are xed. (This particular form of the functions f J will ensure, as we will see, that some expressions involving in nite sums and products are well-de ned.) Indeed, suppose that f is a xed Schwartz function and for each positive integer J we can nd an F J satisfying the estimates:

k X i=1 kX i (f J F J )k _ F 1;p q kf J k _ F ;p q , kF J k L 1 + kF J k _ F ;p q C kf J k _ F ;p q .
Note that Lemma 2.9 immediately implies that kf f J k _ F ;p q ! 0 when J ! 1. By the sequential Banach-Alaoglu theorem, we can choose a subsequence (J k ) k 1 such that F J k converges weakly star in L 1 to a function F 2 L 1 . Together with the last estimate and the above observation, this easily implies that F 2 _ F ;p q as follows. For any positive integer N and any compact set K G we have 0

@ X jjj N 2 qj 1 j F J k q 1 A 1=q L p (K) C kf J k k _ F ;p q . kf k _ F ;p q ,
where by . we indicate that the implicit multiplicative constany may depend on . Since, kF

J k k L 1 . kf k _ F ;p q we get 1 j F J k L 1 . kf k _ F ;p
q for all j. We also can see that

1 j F J k (x) ! 1 j F (x)
for every x 2 G. Hence, the above inequality and the dominated convergence theorem imply that 0

@ X jjj N 2 qj 1 j F q 1 A 1=q L p (K)
. kf k _ F ;p q , and from this we get the claim. Also we obtain that

kF k L 1 + kF k _ F ;p q . kf k _ F ;p q
and, in a similar way,

k X i=1 kX i (f F )k _ F 1;p q kf k _ F ;p q .
From now, we consider J is a xed positive integer.

De nitions and properties of some auxiliary functions

For a real number and x 2 G we will write x := (2 x 1 ; :::; 2 x k ; x k+1 ; :::; x d ). Consider the functions S; E : G ! R de ned by:

S(x) := min(1; kxk Q 1 G
) and E(x) := exp (1 + kx k 2`! G ) 1=2`! , We will also consider the functions S j (x) := 2 jQ S(2 j x), E j (x) := 2 jQ E(2 j x) and set S j f = f S j . With this notation we introduce the new functions (where 1 was de ned in (2.13)):

! j (x) := Z R d S j 1 j f (2 j r)E(r 1 (2 j x)) p dr 1=p
, if jjj J and 0 otherwise.

Consider a smooth function : [0; 1) ! [0; 1] such that 1 on [0; 1=2] and 0 on [1; 1). Following [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], we de ne the functions j as follows:

j := ( 2 j ! j P k<j;k j(mod R) 2 k ! k , if P k<j;k j(mod R) 2 k ! k 6 = 0, 0 , otherwise,
where R is a large positive integer that will be chosen later. Using the j 's, we decompose a nite sum f J = P jjj J j f as follows:

f J = X jjj J j f = X jjj J (1 j ) j f + X jjj J j j f = X j h j + X j g j = h + g
where h := X j h j , with h j := (1 j ) j f if jjj J and 0 otherwise, g := X j g j , with g j := j j f if jjj J and 0 otherwise.

Then we put

h := X j h j Y j 0 >j (1 U j 0 ), with U j := (1 j )! j , g := R 1 X c=0 X j c(mod R) g j Y j 0 >j j 0 c(mod R) (1 G j 0 ), with G j := X t>0 t 0(mod R) 2 t ! j t .
The heart of the proof of Theorem 1.3 consists in establishing the fact that F J := h + g is a "good approximation" of f J = h + g.

Pointwise and integral estimates on ! j . Here we collect several useful estimates on ! j in which we will see an instance of the role played by the critical condition on the exponents:

p = Q.
In what follows we will need the following elementary approximation property proved in [13] (Proposition 3.6): Proposition 3.1 For any 2 R and x; 2 G we have:

k(x ) k G kx k G C k k G and k( x) k G kx k G C k k G .
In particular

jkx k G kxk G j C k k G and jk xk G kxk G j C k k G . Proposition 3.2 Let > 0.
With the above notation we have:

(i) ! j . E j S j 1 j f . 2 Q M M 1 j f for all j 2 Z; (ii) j j f j . ! j for all j 2 Z; (iii) k! j k L 1 . 2 k kf k _ F ;p q for all j 2 Z; (iv) kU j k L 1 . 2 k kf k _ F ;p q for all j 2 Z; (v) k2 j ! j k l q j L p . 2 Q kf k _ F ;p q .
Proof. It is not hard to see that there exist measurable pairwise disjoint sets M 1 ; M 2 ; ::: covering G, such that we have B i M i 3C B i for some balls B i of radius 1=3 in G, where 3C B i is the ball of the same center as B i and of radius 3C. (Here C > 1 is a constant such that (x; y) C ( (x; z) + (z; y)) for all x; y; z 2 G.) Indeed, let (x n ) n 1 be a C-net in G. That is, the balls (B (x n ; C)) n 1 cover G, and (x i ; x j ) C for all i 6 = j. We note that, if i 6 = j, then the balls B (x i ; 1=3) and B (x j ; 1=3) are disjoint. Now we put B i := B (x i ; 1=3) and

M 1 := B (x 1 ; C) n ([ j6 =1 B j ), and M k := (B (x k ; C) n (M 1 [ ::: [ M k 1 )) n ([ j6 =k B j ) for all k 2.
We observe that Proposition 3.1 implies that, for each x; 2 G with k k G . 1 we have E(x ) E( x) E(x) and S(x ) S( x) S(x). It follows, that

S j 1 j f (x ) = 2 jQ Z R d 1 j f (y)S 2 j y 1 2 j x 2 j dy 2 jQ Z R d 1 j f (y)S 2 j y 1 2 j x dy = S j 1 j f (x), for all x 2 G, provided k k G . 2 j .
If r i is the center of B i , then for all r in 2B i , and hence for all r in M i , we can write r = r i for some depending on r with k k G 2. Now, considering the above estimates and the decomposition G = S i M i we can write, since jM i j 1,

! j (x) = 1 X i=1 Z M i S j 1 j f (2 j r)E(r 1 (2 j x)) p dr ! 1=p 1 X i=1 S j 1 j f (2 j r i )E(r 1 i (2 j x)) p ! 1=p 1 X i=1 S j 1 j f (2 j r i )E(r 1 i (2 j x)) 1 X i=1 Z M i S j 1 j f (2 j r)E(r 1 (2 j x))dr = Z G S j 1 j f (2 j r)E(r 1 (2 j x))dr = E j S j 1 j f (x). (3.1)
Next we note that E(x) Ẽ(x) := exp( k2 xk G ) and therefore (using Proposition 2.1)

E j S j 1 j f Ẽj S j 1 j f . Ẽj M 1 j f . Ẽj L 1 M M 1 j f . 2 Q M M 1 j f . (3.2)
We obtain (i), from (3.1) and (3.2). Now we prove (ii). By the change of variables s 1 = r 1 (2 j x) we can write, as before,

! j (x) = 1 X i=1 Z M i S j 1 j f x (2 j s) E(s 1 ) p ds ! 1=p Z M 1 S j 1 j f x (2 j s) E(s 1 ) p ds 1=p S j 1 j f (x) .
To conclude we observe that, for all j 2 Z,

j j f j 3 j 2 j 1 j f = 2 j 1 j f 3 j 2 j 1 j f 3 j = 1 j f 3 j . 1 j f S j = S j 1 j f ,
where we used the fact that, since 3 is Schwartz, we have j 3 j . S and hence 3 j . S j . In order to prove (iii) we observe that, since p = Q,

1 j f L 1 . 1 j f 2 j L 1 . 1 j f L p 2 j L p 0 . 2 j 1 j f L p . kf k _ F ;p q , ( 3.3)
which together with (i), the fact that kE j k L 1 . 2 k and the Young inequality gives the estimate. Item (iii) and the de nition of U j immediately imply (iv).

In order to prove (v), we observe that

2 j 1 j f l q j L p = 2 j 2 j 1 j f l q j L p . 2 j M 1 j f l q j L p . 2 j 1 j f l q j L p = kf k _ F ;p q , ( 3.4)
which, again, together with (i) and the Fe erman-Stein inequality, gives the estimate.

Remark. Items (i), (ii) and (v) do not use the fact that = Q=p. In contrast, (iii) and (iv) require = Q=p.

Proposition 3.3 We have sup j2Z 2 j ! j L p . 2 k p kf k _ F ;p q .
Proof. We follow the proof in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] of Proposition 4.7. We have sup j2Z

2 j ! j (x) p = sup j2Z Z R d 2 j S j 1 j f (2 j r)E(r 1 (2 j x)) p dr Z R d E p (r 1 ) sup j2Z 2 j S j 1 j f (x 2 j r ) p dr Z R d E p (r 1 ) 2 j S j 1 j f (x 2 j r ) p l q j dr.
We note that, according to (3.4), 2 j 1 j f l q j L p

. kf k _ F ;p q , and hence, using Proposition A1

(see the Appendix) we get sup j2Z

2 j ! j p L p Z R d E p (r 1 ) 2 j S j 1 j f (x 2 j r ) l q j p L p x dr . Z R d E p (r 1 ) ln p (2 + krk G )dr kf k p _ F ;p q .
By a change of variables, we can write Z

R d E p (r 1 ) ln p (2 + krk G )dr = 2 k Z R d exp p(1 + kyk 2`! G ) 1=2`! ln p (2 + ky k G )dy . p 2 k ,
and we get the claim.

To make the notation more compact we introduce the functions I m (x) = 1 Am (x), where

A m := 8 < : y 2 R d 2 m ! m (y) > 1 2 X k<m; k m(mod R) 2 k ! k (y) 9 = ; , m 2 Z.
With this we have:

Proposition 3.4 k2 m ! m I m k l q m L p . R 2 k p kf k _ F ;p q .
Proof. Fix a j 2 f0; 1; :::; R 1g. Since ! m 0 for all but a nite number of m 2 Z, we can choose for each x 2 G, the largest integer m x j(mod R) with the property that x 2 A mx , i.e

2 mx ! mx (x) > 1 2 P k<mx; k mx(mod R) 2 k ! k (x)
. Using this, we can write

X m j(mod R) 2 m ! m (x)I m (x) 2 mx ! mx (x) + X k<mx; k j(mod R) 2 k ! k (x) 3 2 mx ! mx (x) 3 sup m 2 m ! m (x)
and hence,

k2 m ! m I m k l q m L p X m 2 m ! m I m L p R 1 X j=0 X m j(mod R) 2 m ! m I m L p 3R sup m 2 m ! m L p
. By using Proposition 3.3, we get the claim.

Estimates involving derivatives. Consider a smooth function u on G and > 0. If a := (1; :::1; 2; :::; 2; :::`; :::; `) is the vector of the homogeneities of G de ned in (2.1) and is a multiindex, then we easily see that

r (u( x)) = h ;ai (r u) ( x)
(where x ! x is the group dilation). If moreover u is homogeneous of degree 1, we have

(r b u) (x) = r b ( u(x)) = r b (u( x)) = j j (r b u) ( x)
and hence (r b u) ( x) = 1 j j (r b u) (x), for all x 2 G and > 0.

Then, for all x 6 = 0, writing x = where = kxk G , = x= kxk G , we get by the above formula that (r b u) (x) = 1 j j (r b u) ( ), which implies in particular that if kxk G 1 and j j 1 then

jr b u(x)j . kxk 1 j j G . 1. (3.5)
Let us also note that if t : R ! R and v : G ! R are some smooth functions, then

X j (t (v (x))) = t 0 (v (x)) X j v (x) for all 1 j d.
Iterating this, we get jr b (t (v (x)))j . X m 1 ;m 2 ;:::;mn 0 m 1 +2m 2 +:::+nmn=n t (m 1 +:::+mn) (v (x)) X

1 ;:::

; n j k j=k n Y j=1 r j b v (x) m j , (3.6)
for all multi-indexes 2 N d 1 N with j j = n.

These observations are the basis for proving the following proposition.

Proposition 3.5 For every 0 2 N k f0g d 1 k N and 2 N d 1 N with j j + j 0 j < 1 (see (2.5)), we have

r + 0 b ! j . ; 0 2 jj j 2 (j )j 0 j ! j .
Proof. Replacing G with R G, and considering

u(t; x) = k(t; x)k R G = (jtj 2`! + kxk 2`! G ) 1=2`! ,
we get by the above observation (3.5) that when t = 1, r b (1 + kxk 2`! G ) 1=2`! . 1 for all nite 2 N d 1 N , 6 = 0 as above.

By (3.6) we obtain

r b exp p(1 + kxk 2`! G ) 1=2`! . exp p(1 + kxk 2`! G ) 1=2`!
and as in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], by di erentiating the composition, r + 0 b E p (r 1 2 j x ) . 2 jj j 2 (j )j 0 j E p (r 1 2 j x ).

Consequently we have r + 0 b ! p j . 2 jj j 2 (j )j 0 j ! p j and by writing ! j = ! p j 1=p we get the estimate by using (3.6) again.

Proposition 3.6 For every 0 2 N k f0g d 1 k N and 2 N d 1 N with j j + j 0 j < 1, we have

r + 0 b j . ; 0 2 jj j 2 (j )j 0 j . (3.7)
Proof. Since the proof of (3.7) follows very closely the similar estimate in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], we only sketch the argument. We suppose j 6 = 0 and write j = (2 j ! j =v j ), where v j = P k<j;k j(mod R) 2 k ! k . From Proposition 3.5 we get r + 0 b v j . 2 jj j 2 (j )j 0 j v j .

Since r + 0 b (v j =v j ) = 0, the Leibniz rule gives us,

v j r + 0 b 1 v j . X + 0 j j<j j+j 0 j r + 0 b v j r b 1 v j X 1 ; 2 0 j 1 j+j 2 j<j j+j 0 j r ( 1 )+( 0 2 ) b v j r 1 + 2 b 1 v j . ( 3.9)
The inequality (3.9) used in conjunction with (3.8) leads by a straightforward induction on j j+j 0 j, to

r + 0 b 1 v j . 2 jj j 2 (j )j 0 j 1 v j .
Using this, Proposition 3.5 and the Fa a di Bruno type formula (3.6) for the functions and 2 j ! j =v j , we can conclude as in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF].

Estimates of the approximation function 4.1 Estimates of the L 1 norm

In this subsection we are going to verify that the functions h and g are well-de ned and, under a smallness condition on kf k _ F ;p q ((4.2) below), obey the L 1 estimates:

h L 1 . 1, kgk L 1 . R. (4.1)
In the remaining part of the paper we assume that f satis es

kf k _ F ;p q , ( 4.2)
where is a su ciently small number (only depending on , R and ) that will be chosen later. We also assume that R > 1= .

In order to obtain the bounds (4.1), we will need the following observation. If (a k ) k2Z is a sequence of nite support, then we have the identity (Lemma 3.2 in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]):

X j 0 >j a j 0 Y j<j 00 <j 0 (1 a j 00 ) + Y j 0 >j (1 a j 0 ) = 1. (4.3)
An immediate consequence of this equality is that, whenever a k 2 [0; 1], we must have, for all j, X j 0 >j a j 0 Y j<j 00 <j 0

(1 a j 00 ) 1.

(4.4)

The boundedness of h. First of all we easily see that h is well-de ned (as a consequence of the fact that only a nite number of functions h j , ! j and U j are nonzero). Recalling the de nition of h j and using Proposition 3.2 (ii), we can write:

jh j j = 1 j j j f j . 1 j ! j = U j .
If f satis es (4.2) with small then, by Proposition 3.2 (iv), we get U j 2 [0; 1] for all j 2 Z and hence, by using (4.4) and the de nition of h, we get the estimate:

h X j jh j j Y j 0 >j (1 U j 0 ) . X j U j Y j 0 >j (1 U j 0 ) . 1.
The boundedness of g. Let us see rst that g is well-de ned. We have that all but a nite number of the functions g j are identically zero, hence it remains to discuss the nature of the products of the form Y

j 0 >j (1 G j 0 ). (4.5)
Following [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], we show that these products converge uniformly. Indeed, we have ! j 0 for all j > J. For small in (4.2), by Proposition 3.2 (iii), we have j! j j < 1 and thus we can write:

0 G j < X t>0; t j J t 0(mod R) 2 t min 2 R ; 2 (j J) 1 2 R .
If j is large, then we have G j . R 2 (j J) which proves the uniform convergence of (4.5).

Now we estimate the L 1 norm of g. When R > 1= , from the above inequality we get G j 2 [0; 1] for all j. By the de nition of j , we see that j (x) 6 = 0 only if

2 j ! j (x) X k<j k j(mod R) 2 k ! k (x).
Hence, jg j (x)j . j (x)! j (x) . X k<j k j(mod R)

2 (k j) ! k (x) = G j ,
and by using (4.4) and the de nition of g we obtain,

jgj R 1 X c=0 X j c(mod R) jg j j Y j 0 >j j 0 c(mod R) (1 G j 0 ) . R 1 X c=0 X j c(mod R) G j Y j 0 >j j 0 c(mod R) (1 G j 0 ) R.

Estimating h h

Our goal in this subsection is to prove the following estimates:

Proposition 4.1 Suppose , p, q and k are as in Theorem 1.3. Then we have

(i) k X i=1 X i (h h) _ F 1;p q . R 2 2 min(1; )+ k p kf k _ F ;p q + R 2 2 max(1 ;0)+(1+[ ]+ 1 p )k kf k 2 _ F ;p q ; (ii) d 1 X i=1 X i (h h) _ F 1;p q . R 2 2 k p kf k _ F ;p q + R 2 2 max(1 ;0)+(1+[ ]+ 1 p )k kf k 2 _ F ;p q .
(Here, [ ] stands for the integer part of .)

Before starting the proof, we note that, writing:

V j := X j 0 <j h j 0 Y j 0 <j 00 <j (1 U j 00 ),
and by using the de nition of h together with the identity (4.3) (as in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], p. 19), one obtains

h h = X j V j U j . (4.6)
In order to obtain Proposition 4.1, we rst collect some estimates satis ed by U j and V j .

Lemma 4.2 For every 0 2 N k f0g d 1 k N and 2 N d 1 N with j j + j 0 j < 1, we have

(i) r + 0 b U m . 2 mj j 2 (m )j 0 j ! m I m ; (ii) kr b U m k L 1 . 2 mj j 2 k kf k _ F ;p q .
Proof. As in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], this follows from Propositions 3.2, 3.5 and 3.6.

Lemma 4.3 For all m 2 Z, 2 N d 1 N with j j < 1 we have kr b h m k L 1 . 2 mj j kf k _ F ;p q .
Proof. This is a direct consequence of the de nition of h m , (3.7) and of the Bernstein type inequality (3.3), since we have

k j f k L 1 = 3 j 1 j f L 1 . 1 j f L 1 . kf k _ F ;p q , ( 4.7)
for all j.

Lemma 4.4 Under the smallness assumption (4.2), we have

(i) jV m j . 1, (ii) for all 2 N d 1 N with j j < 1, kr b V m k L 1 . 2 mj j 2 j jk kf k _ F ;p q .
Proof. We just follow the proof in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]. Item (i) follows directly from the construction and by using (4.4). The arguments are very similar to the ones used to prove (4.1). This item is also already proved in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] (the inequality (6.6)).

We prove now item (ii). By induction we can write (see [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] or [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF], Section 6)

r b V m = X m 0 <m r b h m 0 X 0< c 0 ; r b U m 0 r b V m 0 ! Y m 0 <m 00 <m (1 U m 00 ). (4.8)
This can be seen as follows. Suppose (A m ) m2Z and (B m ) m2Z are two sequences of smooth functions on G, such that for all integers m we have

A m = X m 0 <m B m 0 Y m 0 <m 00 <m
(1 U m 00 ) (4.9)

(also we assume "good" convergence properties for all the derivatives).

Then, if X is a left-invariant vector eld from the Lie algebra of G, we can write

XA m = X m 0 <m (XB m 0 ) Y m 0 <m 00 <m (1 U m 00 ) X m 0 m 0 <m B m 0 X m 0 < <m (XU ) Y m 0 <m 00 < (1 U m 00 ) Y <m 00 <m (1 U m 00 ) = X m 0 <m (XB m 0 ) Y m 0 <m 00 <m (1 U m 00 ) X <m (XU ) X m 0 m 0 < B m 0 Y m 0 <m 00 <m (1 U m 00 ) Y <m 00 <m (1 U m 00 ) = X m 0 <m (XB m 0 ) Y m 0 <m 00 <m (1 U m 00 ) X <m (XU ) A Y <m 00 <m (1 U m 00 ) = X m 0 <m (XB m 0 ) Y m 0 <m 00 <m (1 U m 00 ) X m 0 <m (XU m 0 ) A m 0 Y m 0 <m 00 <m (1 U m 00 ),
and hence, we get

XA m = X m 0 <m ((XB m 0 ) (XU m 0 ) A m 0 ) Y m 0 <m 00 <m (1 U m 00 ).
We observe that this equality is of the same form as (4.9); in the sense that, if we now de ne

A 1 m := XA m and B 1 m := (XB m ) (XU m ) A m , then A 1 m = X m 0 <m B 1 m 0 Y m 0 <m 00 <m (1 U m 00 ).
Appliyng this iteratively, using the de nition of V m , we get (4.8). Now, by using Lemmas 4.3 and 4.2,

kr b V m k L 1 . X m 0 <m kr b h m 0 k L 1 + X 0< 0 r b U m 0 L 1 r b V m 0 L 1 ! kf k _ F ;p q . X m 0 <m 2 m 0 j j + X 0< 2 m 0 j j 2 k r b V m 0 L 1 ! kf k _ F ;p q
and by induction on j j we get the inequality in item (ii). (Recall that we work under the smallness assumption (4.2).)

We are now in position to complete the proof of Proposition 4.1.

Proof of Proposition 4.1. We prove (i) in detail, following closely [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]. As in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], for all 1 k k, we write

X k (h h) _ F 1;p q = 2 ( 1)m 1 m X k (h h) l q m L p = 2 ( 1)m 1 m X k X j2Z V j U j ! l q m L p = 2 ( 1)m 1 m X k X r2Z V r+m U r+m ! l q m L p X r2Z 2 ( 1)m 1 m X k (U r+m V r+m ) l q m L p .
We split this last sum in three terms P r> , P r<0 , P 0 r .

(I) Estimate of P r> . Following [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] and using (2.10), we have:

2 ( 1)m 1 m X k (U r+m V r+m ) l q m L p = 2 ( 1)m (U r+m V r+m ) X R k 1 m l q m L p = 2 m (U r+m V r+m ) X R k 1 m l q m L p . k2 m M (U r+m V r+m )k l q m L p . k2 m (U r+m V r+m )k l q m L p . k2 m U r+m k l q m L p = 2 r k2 m U m k l q m L p . ( 4.10)
Recalling that U j = (1 j )! j and using Proposition 3.4 we get

k2 m U m k l q m L p . k2 m ! m I m k l q m L p . R 2 k =p kf k _ F ;p q ,
and summing up,

X r> ::: . X r> 2 r R 2 k =p kf k _ F ;p q . R 2 +k =p kf k _ F ;p q .
(II) Estimate of P r<0 . If a := [ ] then, as we have already seen, we can write

1 = r R b a '
for a Schwartz family ', and then 1 m = 2 ma r R b a ' m . Hence, if X k is a vector eld in a "good" direction, i.e. 1 k k, we have

2 ( 1)m 1 m X k (U m+r V m+r ) l q m L p = 2 ( 1)m 2 ma X k (U m+r V m+r ) r R b a ' m l q m L p = 2 ( 1)m 2 ma [r a b X k (U m+r V m+r )] ' m l q m L p . 2 ( 1)m 2 ma M r a b X k (U m+r V m+r ) l q m L p . 2 ( 1)m 2 ma r a b X k (U m+r V m+r ) l q m L p . 2 ( 1 a)r 2 ( 1 a)m r a b X k (U m V m ) l q m L p
, where we have used the Fe erman-Stein inequality in the third line.

As in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], using the Leibniz rule and Lemmas 4.2 and 4.4, we obtain

jr a b X k (U m V m )j . jV m (r a b X k U m )j + a X l=0 r l b U m r a+1 l b V m . 2 ma 2 m ! m I m + a X l=0 2 ml ! m I m 2 m(a+1 l) 2 k(a+1 l) kf k _ F ;p q . 2 m(a+1) 2 + 2 k(a+1) kf k _ F ;p q ! m I m .
Now we get, via Proposition 3.4,

2 ( 1)m 1 m X k (U m+r V m+r ) l q m L p . 2 ( 1 a)r 2 + 2 k(a+1) kf k _ F ;p q k2 m ! m I m k l q m L p . R 2 k p kf k _ F ;p q 2 ( 1 a)r 2 + 2 k(a+1) kf k _ F ;p q
and, summing up, X r<0 ::: . R 2

+ k p kf k _ F ;p q + 2 k(a+1+ 1 p ) kf k 2 _ F ;p q . (III) Estimate of P 0 r
. This is similar to the preceding estimate. Here, instead of taking a to be the integer part of , we consider a = 0. As above we conclude that

2 ( 1)m 1 m X k (U m+r V m+r ) l q m L p . R 2 k p kf k _ F ;p q 2 ( 1)r 2 + 2 k kf k _ F ;p q ,
and by summing up, X 0 r

::: . C ( ) R 2 + k p kf k _ F ;p q + 2 k(1+ 1 p ) kf k 2 _ F ;p q where C ( ) 1 if > 1, C ( ) if = 1 and C ( ) 2 (1 ) if < 1.
With this we have proved (i). The proof of (ii) follows the same lines as the one of (i). The main di erence is that since we are no longer restricted to the case of derivatives in "good" directions, we have to use, instead of Lemma 4.2 (i) applied with j 0 j = 1 (as in (II) and implicitly in (III) above), the weaker statement for the case j 0 j = 0. This will produce almost the same estimates, the di erence being that the coe cient 2 + k p of kf k _ F ;p q in the corresponding parts (I), (II) becomes 2 k p .

Estimating g g

Our goal in this section is to prove the following counterpart of Proposition 4.1.

Proposition 4.5 Consider 1 < p; q < 1 and = Q=p. Also consider a 2 (0; ] such that a < = (1 ) if < 1 and a = 1 if 1. We have

kr b (g g)k _ F 1;p q . 2 Q R2 min(1; a )R kf k _ F ;p q + 2 ([ ]+1)Q R 2 2 min(1; a )R kf k 2 _ F ;p q .
We recall the de nition of G j :

G j := X t>0 t 0(mod R) 2 t ! j t .
The starting point is the identity (similar to (4.6))

g g = X j G j H j ,
where H j := X j 0 <j j 0 j(mod R) g j 0 Y j 0 <j 00 <j j 00 j(mod R)

(1 G j 00 ) and g j = j j f . Lemma 4.6 For all m 2 Z, 2 N d 1 N with j j < 1,

jr b G m j . 2 Q X t>0 t 0(mod R) 2 t 2 j j(m t) M M 1 m t f .
Proof. By the de nition of G m and Proposition 3.5,

jr b G m j X t>0 t 0(mod R) 2 t jr b ! m t j . X t>0 t 0(mod R) 2 t 2 j j(m t) ! m t .
Note now that, according to Proposition 3.2,

! m t . 2 Q M M 1 m t f , whence the estimate. Lemma 4.7 For all m 2 Z, 2 N d 1 N with j j < 1, jr b g m j . 2 j jm M 1 m f .
Proof. By Proposition 3.6 and the Leibniz rule, recalling the de nition of g m , we have

jr b g m j . X 0 0 2 j 0 jm r 0 b 3 m 1 m f . X 0 0 2 j 0 jm 1 m f r 0 b 3 m . X 0 0 2 j 0 jm 2 j 0 jm M 1 m f . 2 j jm M 1 m f
(since j 0 j = j j j 0 j when 0 0 ).

Lemma 4.8 For all m 2 Z, 2 N d 1 N with j j < 1, and under the smallness condition (4.2) on f , we have (i) jH m j . 1, (ii) jr b H m j . 2 j jQ P t>0; t 0(mod R) 2 j j(m t) M M 1 m t f .

Proof. Item (i) follows directly from the construction. Also, it is proved in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] (Section 11). Item (ii) is obtained following the strategy in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] (Lemma 6.5). The proof is similar to the one of Lemma 4.4. It is done by induction on j j and using Lemmas 4.6, 4.7.

Proof of Proposition 4.5. As in the estimate of h h, we can write

kr b (g g)k _ F 1;p q X r2Z 2 ( 1)m 1 m r b (G r+m H r+m ) l q m L p . Recalling that G r+m := X t>0 t 0(mod R) 2 t ! r+m t , we get kr b (g g)k _ F 1;p q X t>0 t 0(mod R) 2 t X r2Z 2 ( 1)m 1 m r b (! r+m t H r+m ) l q m L p = X t>0 t 0(mod R) 2 t X r>a t ::: + X t>0 t 0(mod R) 2 t X r 0 ::: + X t>0 t 0(mod R)
2 ( 1)m 1 m r b (! r+m t H r+m ) l q m L p . 2 (r t) k2 m ! m k l q m L p . 2 (r t) 2 Q kf k _ F ;p q .
Summing up we get:

X t>0 t 0(mod R) 2 t X r>a t ::: . 0 B @ X t>0 t 0(mod R) 2 t X r>a t 2 (r t) 1 C A 2 Q kf k _ F ;p q = 0 B @ X t>0 t 0(mod R) X r>a t 2 r 1 C A 2 Q kf k _ F ;p q . X t>0 t 0(mod R) 2 a t 2 Q kf k _ F ;p q . 2 a R 2 Q kf k _ F ;p q .
(II) Estimate of P r 0 . Let a 0 be an integer. As in the estimate (II) for h h we obtain

2 ( 1)m 1 m r b (! r+m t H r+m ) l q m L p . 2 ( 1 a)r 2 ( 1 a)m r a+1 b (! m t H m ) l q m L p .
In order to estimate the right hand side we recall that the following estimates hold (see Proposition 3.2, Proposition 3.5 and Lemma 4.8):

! m t . 2 Q M M ( 1 m t f ), r l b ! m t . 2 (m t)l ! m t ; jH m j . 1, r l b H m . 2 lQ P t>0 2 (m t)l M M 1 m t f
for all l 2 N. By using now the Leibniz rule we get:

r a+1 b (! m t H m ) . 2 (m t)(a+1) ! m t (4.11) +2 (a+1)Q X t 0 >0 a X l=0 2 (t 0 t)l 2 (a+1)(m t 0 ) M M 1 m t f M M 1 m t 0 f . Using (3.
3), we estimate the double sum from the right hand side as follows:

X t 0 >0 a X l=0 ::: . kf k _ F ;p q X 0<t 0 t 2 (a+1)(m t 0 ) M M 1 m t 0 f + X t 0 >t 2 (t 0 t)a 2 (a+1)(m t 0 ) M M 1 m t f ! 5 Appendix
We collect here some facts related to the Calder on-Zygmund theory on strati ed homogeneous groups for vector-valued functions. These results (Lemma A1 and Theorem A1) are well-known. However, since it is hard to nd the exact statements in the literature (see for example [START_REF] Benedek | Convolution operators on Banach space valued functions[END_REF] for a Euclidean version, or [START_REF] Coifman | Analyse harmonique non commutative sur certains espaces homog enes[END_REF] for similar considerations on spaces of homogeneous type) hence, we have chosen to present them here.

Consider a Banach space A. In what follows we deal with functions from the space L p A := L p (G; A) where 1 p 1.

A rst result is a Calder on-Zygmund decomposition of fuctions on G (see also Th eor eme 2.2 in [START_REF] Coifman | Analyse harmonique non commutative sur certains espaces homog enes[END_REF], Chapitre 3), obtained via the weak (1; 1) estimate for the maximal operator:

Lemma A1. Consider a function f 2 L 1
A and a number > 0. Then there exist a countable family of measurable sets ( n ) n 1 which are pairwise disjoint and a decomposition f = g + b = g + P n b n where g; b; b n 2 L 1 A for all n 1, and such that:

(i) kgk L 1 A . ; (ii) supp b n n , R b n (x)dx = 0 and kb n k L 1 A
. j n j for all n;

(iii) P n j n j . We set

1 := ~ \ C B 1 n [ j6 =1
B j ! and inductively we de ne

k := ~ \ C B k n [ 1 i k 1 i ! n [ j6 =k B j
! for all k 2. We see immediately that for all k 1 we have B k k C B k and this also give us that j k j jB k j jC B k j. By de nition the sets k are pairwise disjoint and ~ = S k 1 k . We can de ne the functions:

g(x) := f (x), if x = 2 ~ f k , if x 2 k and b k := (f f k ) 1 k for all k 1. Here, f k := j k j 1 R k f dx. To prove (i), we see that if x 2 k we have kg(x)k A = kf k k A 1 j k j Z k kf (y)k A dy . 1 jC B k j Z C B k kf (y)k A dy M kf k A (x 0 )
,

where x 0 is a point in C B k n ~ . (Such a point exists since 2 B k ~ and 2 B x C B k .) If x =
2 ~ , by the Lebesgue di erentiation theorem, which is a consequence of the weak estimate for the operator M , we can write kg(x)k A M kf k A (x) . To prove (ii) and (iii), observe that by the above inequality we have kkb k k A k L 1 j k j 1 j k j Z k kf (y)k A dy + kf k k A = 2j k j kf k k A . j k j , for all k, and, using the weak estimate for M ,

X k j k j . 1 X k=1 jB k j = 1 [ k=1 B k ~ . 1 kkf k A k L 1 .
We can also see from these inequalities that (ii) the operator T f = f K is well-de ned and bounded from L q A 1 to L q A 2 for some q 2 (1; 1). Then, T : L 1 A 1 ! L 1;1 A 2 is well-de ned and bounded. By real interpolation and duality we get that T : L p A 1 ! L p A 2 is well-de ned and bounded for any p 2 (1; 1).

X k kb k k L 1 A . X k j k j . kf k L 1 A . ( 5
(Here L(A 1 ; A 2 ) stands for the space of the bounded linear operators from A 1 to A 2 .)

Proof. We adapt again the proof in the Euclidean case. Using Lemma A1 we can write, for a given f 2 L 1 (A 1 ) and > 0, the decomposition at level : f = g + b. We next note that kT f k A 2 > 2 kT gk A 2 > + kT bk A 2 > . The size of the set kT gk A 2 > can be bounded using (ii) above and the Markov inequality: kT gk A 2 > q kT gk A 2 q L q . q kgk A 1 q L q = q kgk q A 1 L 1 q q 1 kgk A

1 L 1 = 1 kgk L 1 A 1 . 1 kf k L 1 A 1 .
To estimate the size of the set kT bk A 2 > we proceed as follows. Consider the sets k from the proof of Lemma A1; for each such k we denote by y B k the center of the ball B k k and we set k := (C 1 + C) B k k where C 1 > 0 is a large constant only depending on G and c. We write now

kT bk A 2 > [ k k + ( x 2 Gn [ k k j kT bk A 2 > ) . 1 kf k L 1 A 1 + 1 Z Gn S k k kT bk A 2 (x)dx,
and it remains to estimate the last term. For this purpose, we note that if x 2 Gn k and y 2 k , then (x;

y B k ) = y 1 B k x G (C 1 + C) R B k C 1 (C 1 + C) (y; y B k ) C 1 C 1 y 1 B k y G
(R B k is the radius of B k ) and thanks to the quasinorm property of k k G , we nd a constant C 2 > 0 depending on G only, such that ky , where we have used the condition (i) above and (5.1).

1 xk G = y 1 y B k y 1 B k x G C 2 y 1 B k x G ky 1 y B k k G (C 1 C 1 C 2 1)
Remark. We see from the proof that if kT k L q A 1 !L q A 2 1 then we have kT k

L p A 1 !L p A 2
. 1. Hence if the quantity in (i) is bounded by a number > 0 (instead of 1) and also kT k

L q A 1 !L q A 2
, then we have kT k

L p A 1 !L p A 2
. . Lemma A2. Suppose ' 2 L 1 (G) and: (i) R kyk G R j'(y)j dy . R 1 for any R 1; (ii) R G j'(x 1 y) '(y)j dy . kxk G for all x 2 G with kxk G 1.

If for r 2 G we de ne k j (x) := ' j (x 2 j r), where ' j (x) = 2 jQ '(2 j x) for all j 2 Z, then, for a constant c > 0 only depending on G, we have Z kyk G ckxk G X j2Z k j (x 1 y) k j (y) dy . ln(2 + krk G ).

Proof. We follow the proof in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF]. We decompose the sum under the integral as follows:

X j2Z k j (x 1 y) k j (y) = X We now estimate each term. Using (ii), we can estimate the rst term as follows

Z kyk G ckxk G I Z G X 2 j kxk G 1
2 jQ '( 2 j x 1 2 j y r) '( 2 j y r) dy

Z G X 2 j kxk G 1
'( 2 j x 1 y) '(y) dy . X

2 j kxk G 1 2 j kxk G . 1.
For the second term we have:

Proof. As we already saw, the function S satis es the requirements of Lemma A2. Let k j as in Lemma A2 with ' = S. We see directly that, for any Schwartz function f , we have f k j (x) = Z G f (y)S j (y 1 x (2 j r))dy = S j f (x (2 j r)).

Hence T (f j ) j2Z (x) := (f j ) j2Z K(x) = (f j k j ) j2Z (x) = (S j f (x (2 j r))) j2Z , the operator T being initially de ned for a sequence of Schwartz functions (f j ) j2Z . Considering the Banach spaces A 1 = A 2 = l q (Z) we can see that the statement of the Proposition A1 is equivalent to the fact that the operator T : L p A 1 ! L p A 2 is continuous, with its norm bounded by ln(2 + krk G ). This can be obtained as follows. Consider a sequence a in the unit sphere of l q (Z). We have that: K(x) K(y 1 x); a = X j2Z k j (x) k j (y 1 x) a j X j2Z k j (x) k j (y 1 x) q 0 ! 1=q 0 X j2Z k j (x) k j (y 1 x) , for all x; y 2 G. Hence kK(x) K(y 1 x)k P j2Z jk j (x) k j (y 1 x)j and thanks to Lemma A2 we get (using the same notation):

Z kxk G ckyk G K(x) K(y 1 x) dx Z kxk G ckyk G X j2Z k j (x) k j (y 1 x) dx . ln(2 + krk G ).
Also we can easily see that T : L q A 1 ! L q A 2 is bounded and of norm 1. These two last observations together with Theorem A1 and the Remark after, give us the claim.

Remark. Proposition A1 is reminiscent of an inequality due to Bourgain (see for example [START_REF] Hyt• Onen | Foundations of vector-valued singular integrals revisited-with random dyadic cubes[END_REF]).

Proposition 2 . 2

 22 Suppose p 2 R [x 1 ; :::; x d ] is a polynomial and consider m 2 N . Then r m b p is a vector valued polynomial with deg r m b p `deg p m. In particular, if m is such that m > `deg p, then we have that r m b p 0.

Proposition 2 . 4

 24 Let m 2 N and f be a Schwartz function. (i) If f = r m b ' for a family of Schwartz functions ', then for any polynomial p with deg p < m=`we have R G pf dx = 0. (ii) There exists an m 0 2 N only depending on m and G such that if we have R G pf dx = 0 for any polynomial p with deg p m 0 , then there exists a family of Schwartz functions ' such that f = r m b '.

  .14) Proof. From the assumptions on and , and Proposition 2.4, we know there are some Schwartz families and ' such that = r m b and = r R b m '. With compact notation,

  of P r>a t . Using the fact that kH m k L 1 . 1 and Proposition 3.2 we have (as in (4.10)) :

1 kf k L 1 A. 1 ,x j ~ 1 kf k L 1 A

 111 Proof. We adapt the standard proof in the Euclidean case. Consider the open set ~ := fx 2 Gj M kf k A (x) > g. For each x 2 ~ we consider a ball B x centered in x and such thatB x ~ , but 2 B x ~ (here if c > 0 and B is a ball in G centered in x B of radius R B , then c B is the ball in G ofcenter x B and of radius cR B ). Notice that, by Proposition 2.jB and hence, the balls B x have uniformly bounded radii. Using the Vitali covering lemma (which has the same proof in G as in the Euclidean case), we can nd a countable subfamilly of balls (B k ) k 1 of the familly (B x ) x2 ~ , which are pairwise disjoint and such that ~ = S x2 ~ B x S k 1 C B k , where C > 2 is an absolute constant only depending on G.

. 1 ) 1 A with kgk L 1 A+ kbk L 1 A. kf k L 1 A.

 11111 This proves in particular that the series de ning b is absolutely convergent in L 1A and b; g 2 L Theorem A1. Suppose A 1 and A 2 are two Banach spaces and K 2 L 1 loc (Gn f0g ! L(A 1 ; A 2 )) has the following properties:(i) there exists a constant c > 0 such that R kxk G ckyk G kK(x) K(y 1 x)k dx 1 for all y 2 G;

y 1 kB k x 1 G 1

 111 y G , where we used the equality a 1 = a on G. If C 1 is su ciently large, we deduce ky1 xk G c ky 1 y B k k G = c (y 1 x) y 1 Bn x) op dx ! kb n (y)k A 1 dy X n Z n kb n (y)k A 1 dy . kf k L 1 A 1

2 j kxk G 1 :

 1 :: + X 1<2 j kxk G <2+krk G ::: + X 2 j kxk G 2+krk G ::: =: I + II + III.

Acknowledgements

The author thanks Petru Mironescu and Emmanuel Russ for useful discussions and suggestions. He was supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit e de Lyon, within the program \Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Part of this work was completed while the author was visiting the Institute of Mathematics of the Romanian Academy. He thanks the Institute for its hospitality.

where B a; (t) = P 0<t 0 t 2 ( 1 a)t 0 . Finally, we obtain

If we choose now a = [ ] and we observe that in this case we have B a; (t) . 1, we can write

(III) Estimate of P 0 r a t . Using the estimate (4.12) above with a = 0, we get

where A (t) .

<

:

and B 0; (t) .

<

:

Now summing up we get three possible bounds:

(1) if < 1, we have a < 1 and

(2) if = 1, we have a = 1 and

(3) if > 1, we have a = 1 and

Now from the above estimates, since 0 < a < , we have

Toghether with (I) and (II), this gives Proposition 4.5.

Proof of Theorem 1.3

Now we can estimate the Triebel-Lizorkin norm of f J F J = (h h) + (g g). By Proposition 4.1 (i) and Proposition 4.5, we have

where D R; is a large constant depending on R and .

As in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], for 2 N, we set

If > 0 is xed, then it is easy to see that (using the fact that k=p < min(1; )) for a large enough, we have

Hence, for a large D we have

and since we assumed that kf k _ F ;p q is small (see (4.2)), then we may take D kf k _

In a similar way, using Proposition 4.1 (ii) and Proposition 4.5 we get

and hence, as above,

provided that kf k _ F ;p q is small enough. From (4.14) and the lifting property (Proposition 2.13) of the Triebel-Lizorkin norm, we get

Now (4.13) and (4.15) together with the L 1 estimates (4.1) give Theorem 1.3 under the smallness assumption on kf k _ F ;p q (observing that the bounds proved do not depend on J and taking J ! 1). We complete the proof of Theorem 1.3 via the homogeneity of the norms.

Remarks. (1) Following the same lines, it is also possible (and easier) to prove a version of Theorem 1.3 for the Besov spaces introduced in Subsection 2.3: Theorem 4.9 Consider the parameters 1 < p < 1, 1 < q 1, = Q=p and let k be the largest positive integer with k < min(p; d 1 ). Then, for every > 0 there exists a constant C > 0 only depending on , such that for every function f 2 _ B ;p q (G) there exists F 2 L 1 (G) \ _ B ;p q (G) satisfying the following estimates:

(2) To mention one application of Theorem 1.3, we state the following generalisation of Theorem 1.8 in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] concerning the Hodge systems on the (2n + 1)-dimensional Heisenberg group H n . Note that in this case d = 2n + 1, d 1 = 2n and Q = 2n + 2. Theorem 4.10 Consider 1 < p; q < 1, := (2n + 2) =p and let r be an integer with 1 r < min(p; n). For any

.

(See [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] for notation.) This is proved by using Theorem 1.3 to approximate in an e cient way the coe cients of the form ' and then to conclude by using an iteration argument. Since the proof is very similar to the one given in [START_REF] Wang | A subelliptic Bourgain-Brezis inequality[END_REF] and its Euclidean analogue in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] (Theorem 1.2), we omit it. Theorem 1.4 can be proved following the same lines.

To estimate the third term we use (i), which yelds

where the constant c 1 > 0 in the rst line is obtained by using the quasinorm property of k k G .

Summing up these estimates we get the claim.

In what follows we will need to apply the above lemma to the function ' = S. It is easy to verify that the function S(x) = min(1; kxk Q 1 G ) satis es the conditions (i) and (ii) required by Lemma A2. Indeed, by a change of variables, we can write for all R 1, Z

which proves that (i) is satis ed. To verify (ii), we recall that jkb ak G kak G j C kbk G for all a; b 2 G (see Proposition 3.1) and note that if kyk Proposition A1. Suppose p; q 2 (1; 1). Then, for every sequence (f j ) j2Z in L p (G; l q (Z)) and for every r 2 G we have S j f j (x 2 j r ) l q j L p x . p;q ln(2 + krk G ) kf j k l q j L p .


Yves Gagne

Bernard Castaing

Christophe Baudet

Yann Male

Reynolds dependence of third-order velocity structure functions

We study the experimental dependence of the third-order velocity structure function on the Taylor based Reynolds number, obtained in different flow types over the range 72рR р2260. As expected, when the Reynolds number is increasing, the third-order velocity structure functions ͑plotted in a compensated way͒ converge very slowly to a possible Ϫ4/5 plateau value according to the Kolmogorov 41 theory. Actually, each of these normalized third-order functions exhibits a maximum, at a scale close to the Taylor microscale . In this Brief Communication, we show that experimental data are in good agreement with the recent predictions of Qian and Lundgren. We also suggest that, from an experimental point of view, a log-similar plot suits very well to study carefully the behavior of the third-order velocity structure functions with the flow Reynolds number.

͗͑␦u͑r͒͒ 3 ͘ӍϪ4/5͗⑀͘r ͑1͒ ͑r and ⑀ are the inertial range separation and the mean dissipation rate, respectively͒. Even though the previous relation is strictly valid for Reynolds number tending to the infinity, experimental data seemed to verify it as soon as a conspicuous power-law scaling range exists ͑i.e., for R у500), whatever the flow type. To emphasize this feature, the third-order longitudinal velocity structure functions are usually compensated by the opposite of the Kolmogorov scaling term ͓defined as Ϫ͗(␦u(r)) [START_REF] Sreenivasan | The phenomenology of small-scale turbulence[END_REF] ͘/(͗⑀͘r) and hereafter noted S 3 (r)], and are plotted in log-log coordinates ͑cf. Refs. 3, 4, and references therein͒. Actually, the difference between S 3 (r) and 4/5 is experimentally very difficult to determine. Indeed, the identification of S 3 (r) with 4/5 is very often the most accurate way to experimentally determine ͗⑀͘ at large R .

However, some experiments have shown a very slow convergence towards the above infinite Reynolds result when R is raised. For instance, in a ''Von Karman'' flow, Moisy 5 showed that S 3 (r) tends to 4/5 like R Ϫ6/5 . By another way, Mydlarski and Warhaft, [START_REF] Mydlarski | On the onset of high Reynolds-number grid-generated wind tunnel turbulence[END_REF] found a slow convergence of the slope of the energy spectrum measured in a grid turbulence towards Ϫ5/3 as R increases. For the peculiar scale rϭ, Pearson and Antonia 7 found that S 3 () does not exactly scale as the Kolmogorov prediction for Reynolds number up to R Ӎ1000, reflecting the large scale anisotropy effects on the inertial range.

On the theoretical side, in some recent papers, [START_REF] Qian | Inertial range and the finite Reynolds number effect of turbulence[END_REF][START_REF] Qian | Slow decay of the finite Reynolds number effect of turbulence[END_REF] Qian examines this problem, dicussing systematically the influence of the energy input. Qian predicts the slowness of the convergence, but points out noticeable differences between various types of power injection. Lundgren 10,[START_REF] Lundgren | Kolmogorov turbulence by matched asymptotic expansion[END_REF] goes back to the Karman-Howarth equation to propose a solution in the decaying self-similar case which seems not easily compatible with the possibility of intermittency.

In the present state, it is difficult to decide if an experimental discrepancy with the Lundgren prediction is due to ͑i͒ a peculiar injection regime as discussed by Qian, ͑ii͒ an intermittency effect, ͑iii͒ or, simply, an experimental error in the determination of R . In this Brief Communication, we show that a suitable diagram allows to illustrate how the various theoretical predictions overlap, and to decide among the above questions.

Let us summarize the Qian and Lundgren results. Based on the additive property of the energy input and dissipation, the main prediction of Qian ͓Eq. ͑24͒ in Ref. 9͔, valid for ӶrӶᐉ 0 , is

S 3 ͑ r ͒ϭ4/5ϪC 1 ͑ r/ᐉ 0 ͒ m ϪC 2 ͑ r/ ͒ Ϫ4/3 ͑2͒
with C 1 Ͼ0 and C 2 Ӎ5.26K 0 where K 0 is the Kolmogorov constant, ᐉ 0 the large scale, and the viscous scale. The second term of the right-hand side represents the large-scale effect and the last term the viscosity effect. At the particular scale where S 3 (r) is maximum, the above relation leads to

␦ϭ4/5Ϫ͓S 3 ͑ r ͔͒ max ϭC ␦ R Ϫ , ϭ 6m ͑ 3mϩ4 ͒ . ͑3͒
From this Reynolds dependence of ͓S 3 (r)͔ max , Qian has deduced several predictions of the exponent depending on the types of flow. For example, he found 2/3рр1 in the homogeneous shear-flow turbulence. For some types of large-scale forcing turbulence, he predicted an exponent equal to ϭ6/5, in agreement with the Moisy experiments. [START_REF] Moisy | Kolmogorov equation in fully developed turbulence[END_REF] In the interesting case of a decaying self-preserving turbu-lence, Qian's prediction is ϭ2/3. In this case, Lundgren has also derived an equation for S 3 (r) ͓Eq. ͑45͒ of Ref. 10͔ by using a method of matched asymptotic expansions. In terms of , it can be written as

S 3 ͑ r ͒ϭ4/5Ϫ 3.34 ͱ15 ͑ r/ᐉ 0 ͒ 2/3 Ϫ8͑r/ ͒ Ϫ4/3 ͑4͒
which leads to

͓S 3 ͑ r ͔͒ max ϭ4/5Ϫ8.45R Ϫ2/3 . ͑5͒
Note that the coefficients of Eqs. ͑4͒ and ͑5͒ are compatible with those of Eqs. ͑2͒ and ͑3͒ when K 0 ϭ1.5-1.8 ͑see Table II of Ref. 9͒.

In order to test these theoretical predictions, we have used three different types of air flows: a round jet, a grid turbulence and in the return channel flow of the ONERA-Modane wind tunnel. In the laboratory flows, measurements of the longitudinal velocity fluctuations were performed in classical conditions, respectively, on the axis of the jet at x/dϭ38 downstream the nozzle ͑with a diameter d ϭ12 cm), and at x/M ϭ40 downstream a square bars grid ͑with a mesh size of M ϭ4 cm and a solidity of 0.36͒. At Modane, the probe was located on the axis of the 24 m diameter return channel. In such an industrial facility, the flow generates a ''chunk'' turbulence. Further details on these experimental conditions are given in Ref. 12. Velocity signals were obtained using the CTA hot wire technique with a home made single-wire probe. The hot wires used in the laboratory flows and in the industrial flow, were 0.3 mm and 0.5 mm in length and 2.5 m and 3.5 m in diameter, respectively, and were operated on DISA 55M01 anemometers. The real time duration of data records ͑of about 75 mega-samples͒ was always greater than 12 000 times the integral time scales. The time-space conversion of signals has carried out a ''corrected Taylor hypothesis'' using the instantaneous velocity. [START_REF] Pinton | Corrections to the Taylor hypothesis in swirling flows[END_REF][START_REF] Gledzer | On the Taylor hypothesis corrections for measured energy spectra in turbulence[END_REF] Table I lists the relevant data for the records analyzed here, estimated with the usual homogeneous and isotopic definitions:

͗⑀͘ϭ15͗u 2 ͘/ 2 , ϭ( 3 /͗⑀͘) 1/4 , 2 ϭ͗u 2 ͘/͗(ץu/ץx) 2 ͘.
Figure 1 shows compensated third-order longitudinal structure functions S 3 (r) plotted against the separation distance r in linear coordinates. In addition, we have reported the data obtained from an atmospheric turbulence, that we have extracted by hand from Fig. 1 of the paper of Dhruva et al. [START_REF] Dhruva | Transverse structure functions in high Reynolds number turbulence[END_REF] Following the authors, R is about 10 4 Ϫ1, 4ϫ10 4 . In Fig. 1, no plateau is observed, the third-order velocity structure function does not strictly behave as a power law, even in the middle of the inertial range of one of the highest Reynolds number flow never investigated. Note that the accuracy of measurements does not affect the shapes of the S 3 (r) ͑and cannot lessen this finding͒, it only affects the level value of the S 3 (r) which drastically depends on the squares of the velocity gradients calculated at the Kolmogorov scale . As expected, these maxima continuously converge to the horizontal line 4/5 as R is increasing. This monotonous evolution suggests that data have been obtained with a rather good accuracy despite the difficulty to measure the small scale velocity gradients. The lack of a true ''Ϫ4/5 plateau'' has been studied in detail by Danaı ¨la et al. [START_REF] Danaı ¨la | A generalisation of Yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence[END_REF] in the case of a grid turbulence. From the complete form of the Kolmogorov equation involving both the viscous and the nonstationary terms, this Brief Communication shows how the latter term acts as a negative production term at large scale. The authors showed that experimental data verify the Kolmogorov equation if one accounts for the nonstationarity of the second-order moments. The aim here is not to study this interesting finding in others types of flow, but rather to compare behaviors of the functions S 3 (r) towards the flow type and the Reynolds number value.

A way to highlight the gap between ͓S 3 (r)͔ max and 4/5 studied by Qian 8,[START_REF] Qian | Slow decay of the finite Reynolds number effect of turbulence[END_REF] and Lundgren, [START_REF] Lundgren | Kolmogorov two-thirds law by matched asymptotic expansion[END_REF][START_REF] Lundgren | Kolmogorov turbulence by matched asymptotic expansion[END_REF] is to plot the quantity (1Ϫ(5S 3 (r)/4))R 2/3 versus the separation r. Such a stringent diagram is shown on Fig. 2, where the separation r has been rescaled with the Taylor microscale ͑and not with as in Fig. 1͒. We observe ͑as in Ref. 5͒ that all the minima of the experimental curves ͑corresponding to ͓S 3 (r)͔ max ) occur at a unique scale close to . To our knowledge, there is no physical explanation of this feature. Even though the minima values agree ͑more or less͒ with the previous theoretical predictions ͓ln͑8.45͒ϭ2.13͔, these curves diverge each other both at large and small scales.

In fact, due to the existence of the two scales ᐉ 0 and , the problem is that neither r/ᐉ 0 nor r/ can be considered as the good variable. A characteristic scale in turbulence is not identified by its ratio to ᐉ 0 but by the way this ratio scales with the Reynolds number. If this scale is almost equal to the Taylor scale , while its ratio with ᐉ 0 scales like R Ϫ3/2 , it must be identified with and not . In this context, as recognized in previous studies, [START_REF] Male ´cot | A statistical estimator of turbulence intermittency in physical and numerical experiments[END_REF][START_REF] Chanal | Intermittency in a low temperature gaseous helium jet[END_REF] the good variable is of kind ln(r/ᐉ 0 )/ln R , in such a way that characteristic scales having the same behavior with R converge towards the same abcissa when R tends to infinity. Faster convergence is obtained using ln(R /R * ) with R * ϭ28, [START_REF] Male ´cot | A statistical estimator of turbulence intermittency in physical and numerical experiments[END_REF][START_REF] Chanal | Intermittency in a low temperature gaseous helium jet[END_REF] which ensures that ln(ᐉ 0 /) remains close to [START_REF] Sreenivasan | The phenomenology of small-scale turbulence[END_REF] 2 ln(R /R *

) down to moderate R . We then propose such a plot. As we want to emphasize the role of , we take the abcissa ␤ ln(r/) with ␤ ϭ1/ln(R /R * ). To maintain the scaling laws, we have to divide the logarithmic ordinates by the same factor ln(R /R *

). We thus take as the ordinate ␤ ln(1Ϫ(5S 3 (r)/4)). In such coordinates, Fig. 3 shows the experimental curves ͑the same as those of Fig. 2͒, while Fig. 4 displays the results obtained from Eq. ͑4͒.

While none of the two plots show perfect merging, this presentation has numerous advantages. First, the ordinate Ϫ of the minimum gives the Reynolds dependence of the maximum of S 3 (r) that we define as ͓S 3 (r)͔ max ϭ4/5 ϪA(R /R * ) Ϫ , analogously to the predicted Eqs. ͑3͒ and ͑5͒. Second, the slopes close to zero ordinate give the power law dependences of the outer scale (ϳᐉ 0 ) and inner scale ͑ϳ͒ corrections to the Ϫ4/5 law as expressed in Eqs. ͑2͒ and ͑4͒. Last, the distance between the two points of intersection of each curve with the zero ordinate is bounded by R (Ӎ3/2ϩ͓3/ln(R /28)͔). The different data converge towards the infinite Reynolds curve is characteristic of the coefficients of these laws. For instance, it can be seen that choosing R * ϭ28 gives A close to 1, both for Eqs. ͑3͒ and ͑5͒ and for real data. Indeed, this choice comes from an independent work ͑cf. Ref. 12͒ and this coincidence has to be remarked. Figure 4 shows how curves degenerate in two straight lines when R →ϱ. The position of the maximum of S 3 , its dependence with R , the power law dependences of the correction to S 3 , and even their coefficients, all are probably linked. For instance, if we write, as Lundgren did 1 Ϫ(5S 3 /4)ϭ(R /R * ) Ϫ␣ (A 0 (r/) ␥ 0 ϩA i (r/) ␥ i ), we must have ␥ 0 Ϫ␥ i ϭ2 as one of the terms comes from the integral of ͗␦u(r) 2 ͘, and the second from its derivative. Then, to have the maximum ͓S 3 (r)͔ max at an abcissa r close to , even at moderate R , we must have A 0 ϭ2/3 and A i ϭ1/3 if ␥ i ϭϪ4/3.

Further remarks can be discussed in relation to the questions addressed in the beginning of this paper.

The two slopes plotted in Fig. 3 ͑equal to Ϫ4/3 and 2/3 according to the theoretical predictions͒ seem to be reached only for R Ͼ1000 ͑note that an error on ␤ does not affect these slopes͒. On the large scale side, the two highest Reynolds data behave as nearly as ϳR 2/3 even though they correspond to two very different flows. Moreover, on the large scale side of experimental curves, we observe that the correction slope ͓noted m in Eq. ͑2͔͒ never exceeds the value 2/3, whatever the flow type. In fact, the different power input types ͑grid turbulence, jet͒ are hardly noticeable, contrary to the theoretical expectations.

With regard to the accuracy of the Reynolds number determination, such coordinates clearly reveal that in Ref. 15 data, R has been grossly overestimated. This underestimates ␤ and causes the diagram to shrink.

Other differences between the theoretical predictions and the real data are within the experimental uncertainty. This was to be expected, in particular, the intermittency effect cannot probably be detected in such a diagram, as the following calculus shows.

In his matched asymptotic expansion, Lundgren derives an outer and an inner expression for the second moment ͗␦u(r) 2 ͘(R L ϭR 2 /15) such as ͗␦u(r) 2 ͘ϭU 2 b ll 0 (x 0 ) with x 0 ϭr/L, and ͗␦u(r) ) as small parameter, he writes b ll

0 ϭb ll 0,1 ϩR L Ϫ1 b ll 0,2 ϩ¯and b ll i ϭb ll i,1 ϩR L Ϫ1/2
b ll i,2 ϩ¯. But, as remarked by Barenblatt,[START_REF] Barenblatt | Scaling laws for fully developed turbulent shear flows[END_REF] incomplete selfsimilarity can result in the absence of limit for infinite R L for one of these functions. The general case writes b ll i ϭR L Ϫ␦ b ll i,1 ϩ¯. Then, Eq. ͑27͒ of Lundgren, which ensures the matching of the two expressions, becomes ϭC K x i 2(1ϩ2␦)/3 . By the comparison between these two latest equations and the result of intermittency studies, the value of ␦ should be of order ␦Ӎ0.02. The two exponents ␥ 0 and ␥ i ͑which are the limit slopes in the log-similar diagram͒ are ␥ 0 ϭ2/3 ϩ4␦/3 and ␥ i ϭϪ4/3ϩ4␦/3. The minimum occurs at ␤ ln(r/)ϭ␦/2 and its value is Ϫ2/3Ϫ␦. The very small value of ␦ explains why the difference with the Lundgren prediction is not visible.

U 2 R L Ϫ1/2
In summary, the present experimental data confirm the very slow convergence of S 3 (r) towards an asymptotic re-gime, whatever the flow type and/or the Reynolds number. Also, the pertinent scale of ͓S 3 (r)͔ max seems to be close to the Taylor microscale but with a different Reynolds scaling. From an experimental point of view, the log-similarity plot suits to account for the detailed behavior of S 3 (r) in the whole inertial range when the Reynolds number tends to infinity. The question of intermittency is not relevant here and must be studied by other means.

FIG. 1 .

 1 FIG. 1. S 3 (r) in lin-log coordinates.

FIG. 2 .

 2 FIG. 2. Compensated S 3 (r) according to the Qian and Lundgren predictions.

TABLE I .

 I The basic parameters of the data sets.

		Grid turbulence	Jet		ONERA
	R	72	144	350	695	2260
	U (m/s)	4.95	19.6	1.68	6.3	20.75
	uЈ (m/s)	0.17	0.64	0.44	1.55	1.58
	͑mm͒	0.4	0.15	0.33	0.14	0.30
	͑cm͒	1.02	0.35	1.23	0.67	2.80
	L (cm)	4	4	20	20	Ӎ410
	͗⑀͘ ͑m 2 /s 3 ͒	0.0093	7.7	0.29	11.7	0.95
	10 5 (m 2 /s)	1.51	1.55	1.52	1.53	2

  2 ͘ϭU 2 R L

	ϭR L 3/4 x 0 .	Ϫ1/2 b ll i (x i ) with x i ϭr/
	Now, remarking that the von Karman equation in the
	outer ͑respectively, inner͒ variables has R L Ϫ1 ͑respectively, R L Ϫ1/2

1070-6631/2004/16(2)/482/4/$22.00 © 2004 American Institute of Physics

Phys. Fluids, Vol. 16, No. 2, February 2004 Gagne et al.

Phys. Fluids, Vol. 16, No. 2, February 2004 dependence of third-order velocity