Realization of homogeneous Triebel-Lizorkin spaces with p = \ and characterizations via differences Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 11. № 4 (2019). С. 114-129.

REALIZATION OF HOMOGENEOUS TRIEBEL-LIZORKIN SPACES WITH p = œ AND CHARACTERIZATIONS VIA DIFFERENCES

M. BENALLIA, M. MOUSSAI

Abstract. In this paper, via the decomposition of Littlewood-Paley, the homogeneous Triebel-Lizorkin space F^ is defined on Rra by distributions modulo polynomials in the sense that ||/1| =0 (|| ■ || the quasi-seminorm in F^) if and only if f is a polynomial on Rra. We consider this space as a set of "true" distributions and we are lead to examine the convergence of the Littlewood-Palev sequence of each element in First we use the

realizations and then we obtain the realized space F^ of

Our approach is as follows. We first study the commuting translations and dilations of realizations in F^, and employing distributions vanishing at infinity in the weak sense, we

construct F^. Then, as another possible definition of F^, in the case s > 0, we make

use of the differences and describe F^ as s > max(n/q — n, 0).

Keywords: Triebel-Lizorkin spaces, Littlewood-Paley decomposition, realizations.

Mathematics Subject Classification: 46E35.

1. Introduction

In this paper we study a realization of homogeneous Triebel-Lizorkin spaces on Rra. The spaces are defined by distributions modulo polynomials in the sense that || f s = 0 if and only if f is a polynomial on Rra, Some of their properties can be found in [12], [22].

The basic definition of is given via the Littlewood-Palev decomposition (abbreviated as LP decomposition). To recall this, we introduce some notations.

Bv p we denote an infinitely differentiate radial function obeying the estimates 0 ^ p ^ 1 such that

3

p(£) = 1 as |£| ^ 1, p(£) = 0 as |£| ^ -. We denote 7(£) := p(£) — p(2£). This function is supported in the annulus | ^ |£| ^ |, and

3

7(£) = 1 as - ^ |£| ^ 1, J] 7(2J£) = 1 as £ = 0.

jez

For m E N, the svmbol Vm stands for the set of all polynomials on Rra of degree less than m obeying V0 = {0}, By V^ we denote the set of all polvnomials. For m E N0 U {<^}, the set S^ of the tempered distributions modulo polynomials is the dual space of Sm, which is the orthogonal space of Vm in that is, Sm is the set of all / E S such that (u, f} = 0 for all u E Vm. For a tempered distributions f E S the svmb ol [f]m denotes the equivalence class of /modulo Vm.

M. Benallia, M. Moussai, Realization of homogeneous Triebel-Lizorkin spaces with p = œ

and characterizations via differences .

© Benallia M., Moussai M. 2018. Submitted October 11, 2018.

We define the operators Qj by the formula

QTf := 7(2-j(■))f ,, J e Z.

These operators are defined on S' as well as on S/,m sinee Qj f = 0 if and onlv if f e Vm. For instance, we have Qj (S) c S^. All these operators take values in the space of analytical functions of exponential type, see the Palev-Wiener theorem. Finally, we adopt the following convention: for f e S^, we define Qjf := Qjfi (or all fi e S' such that [fi]m = f. We turn to the LP decomposition; for all f e (or S'^) the identity

f = £ Qj f in ^ (or SU (1)

jez

holds; this is an easy application of Lemma 7 below. However, once we work in i7^ , it is possible to obtain the convergence of the series of the LP decomposition in S'^ for some integer

see (7) below. This leads us to the need to realize and to obtain the realized spaces by using the notion of realization. For a quasi-Banach distribution space E ^ S^, we need to find a continuous linear mapping a : E ^ S'm such that [a(f)]m coincides with f modulo polynomials in Vm for all f e E, cf. Definition 4 below. If in addition, E is a translation or a dilation invariant, that is,

IK/||B = 11/lis or \\hxf ||B = y 11/||B

with r e R where raf(x) := f(x — a^d h\f(x) := f(x/X) for all x,a e Rra and all A > 0, the existence of a such a commuting with translation or dilation operators, that is, obeying

ra o a = a o ra or h\ o a = a o h\,

is nontrivial.

We note that the realizations have been introduced by G, Bourdaud [3] for the homogeneous Besov spaces Bp q; the corresponding integer ^ was defined in [7]. In the same way, we know the realizations of both the homogeneous Triebel-Lizorkin spaces F*q with p < ro and the homogeneous Sobolev spaces W^m, and some of their properties, see, for instance, [2], [5], [6], [7], [16], [21]. Also, nowadays there are various papers presenting applications of the realizations to Navier-Stokes equations, pseudodifferential operators, wavelet, etc., see, for instance, [9], [15], [20] and in particular, a comment in [1],

On the other hand, the distributions vanishing at infinity play an important role to characterize such realization. We recall this notion.

Definition 1. We say that a distribution f e S' vanishes at infinity if

lim h\f = 0 in S'.

The set of all such, distributions is denoted by C0.

For instance, we have f e Co if f e Lp (1 ^ p < <x). If either f e L^ or f e C0 then dj f e C0 (j = 1,..., n). An easy statement is given by identity C0 ilV^ = {0} (see, for instance, [3]),

As usually, N stands for the natural numbers {1, 2,...} and N0 := N U {0}, All function spaces occurring in the paper are defined in the Euclidean space Rn. By || ■ ||p we denote the Lp quasi-norm for 0 < p ^ ro, For s e R the svmbol [s] denotes the integer part of s. For all m e N0, the standard norms in S are given by

(m(f):= sup sup (1 + Mr If(a)0*)|.

®ern |a|$m

The Fourier transform for a function f E L1 is defined as

Tf (0 = № = j e-i"? f (x) dx, £ E Rra

r"

The operator T can be extended to the whole S' in the usual way. In the same way we define the inverse Fourier transform T-i,

T-1f(x) := (2n)-nf(—x). For an arbitrary function f, we define the difference operators as

Ahf = Alf := r-hf — f, A£f := Ah(A™-1/), h E Rra, m = 2, 3,...

The constants c, c1; ... are strictly positive and depend only on the fixed parameters as n, s, q and probably on auxiliary functions, their values may vary from line to line. The notation A < B means that A ^ cB. The symbol E ^ F denotes that we have the embedding E C F and the natural mapping E ^ F is continuous. Throughout the paper, the real numbers s, q satisfy as s E ^^d 0 < q ^ unless otherwise is stated.

The paper is organized as follows. In Section 2 we recall the definitions and some properties of homogeneous Triebel-Lizorkin spaces q and of inhomogeneous ones F^ q. Section 3 is devoted to the realizations of F^ q. In Section 4, by means of the differences, we characterize the realized spaces of in the case s > max(n/q — n, 0),

2. Preliminaries

2.1. Homogeneous spaces F^q. By Pk,u (k E Z, v E Zra) we denote the dyadic cube with side length 2-k, left lower corner in the point 2-kv and sides parallel to the coordinate axes, that is,

Pk,v := {x E Rra : 2-kVj ^ Xj < 2-k(vj + 1), j = 1, 2,..., n}. The definition of i7^ was given by Frazier and Jawerth [12] as follows.

Definition 2. Let q e]0, The space F^,q is the set of f E S'^ such that ps := sup sup (2kn [ V 2jsq\Qj f (x)\qdx) <

q haz 7,i= z" v I /

kez vezn „ .. , Pk,v 3>k

Remark 1. For q = the set F^ ^ coincides with the Holder space B^ ^, see [14, Eq. (1.3)] and Lemma 3 below. We let

sup2^11^f < œ. jez

The space F^ g becomes a quasi-Banach with the above defined quasi-seminorm. On the one hand, its definition is independent of the choice of 7, see [12, Cor. 5.3]. On the other hand, by (1) and Lemma 7 below, we have Soo ^ i7^ ^ S^. We also have the following statements.

Lemma 1. There exist two constants c\, c2 > 0 such that the inequalities

* Hf11^, ^ flli^s,., ^ c4f H^, (2)

holds for all f E and all X > 0.

Proof. At the first step, we prove (2) with A := 2N, N E Z. Here by using the identity

Q3 (h-2 n f ) = Q3+n f (2-n (•)),

we obtain easily that

l|h2NfHF* =2-Ns\\fH*. .

II I I X n I I v I I X n

Fs

,

In the case of arbitrary A > 0, we introduce an integer N E Z such that 2N ^ X < 2N+1 Then we use the equivalent quasi-seminorm in F^ q defined by the function j1 := 7(2NA-1 and we get

\\f(AIb^ = 2Ns\\f(2-NA ■ .

Then it is not difficult to prove that

d\\f^ \\f (2-nA ■ ^ c2\\f H^ for some positive constants c1 and c2 independent of N, A and f. This completes the proof. □ The next lemma was proved in [11]. Lemma 2. There exists a constant c> 0 such that

sup ^(x) ^ c2n/q sup \M\Lq(PjiV) (3)

holds for all j E Z, u E Zn, and <p eS' with supp (p C {£ E Rn : |f | ^ 2j+1}.

Lemma 3. For all q > 0 we have F^ ^ F^^ = B^.

Proof. The identity is known, see, for instance, [12] and here we provide a proof of the embedding for more clarity.

Let f E F^ q. By Lemma 2 we have

IQj f (x)lq ^ C12n sup f IQj f (y)lq dy for all x E P3,v, ■nezn J

P

which is bounded by

d2-^n sup / £ 2lsqIQif (y)lqdy,

1eZ"PJ '

where the constant c1 is independent of f, j and v. This inequality implies that

IQjf(*)l < 2-s\\f \\f^ (Vx E P^v).

Then we get

If« = sup sup sup 2ks\Qkf (z)l < \\f

vez™ k'j zePj,v

The proof is complete. □

Remark 2. An inequality opposite to (3) can be easily proved, and for this, the assumption supp (p C {£ E Rn : l£l ^ 2J+1} is not needed.

Remark 3. In case 1 < q < x>, the space F^ q has another definition introduced by Triebel [19], which is compatible with the one of Frazier and Jawerth, see a comment in [12].

2.2. Inhomogeneous spaces F^q. For each f E S (or f E S'), we use the inhomogeneous LP decomposition f = T-1p * f + J2j>0 Qj f in S (or S') and we obtain the inhomogeneous Triebel-Lizorkin spaces F^ q as introduced in [12].

Definition 3. The space F^ q is the set of f E S' such that

1/i

:= \\T-1p * f \U + sup sup (2kn ^ 2^lQjf(x)l9dx) "< <x>.

keN vezn ^ j ^k '

Also as above,

^ = 11/:= \\F-1P * f II« + sup 2S\\Q , f < œ,

j>0

cf. Lemma 3 and see also [19, Sect. 2.3.4, Rem. 3].

For some properties of , we refer to [12]. The case s > 0 is related with the case of the homogeneous space.

Lemma 4. Let s > 0. Then

(i) ^ Loo,

(ii) F^tq is the set of f E L^ such that [/]« E i7^. The expression \\f \L + lK/Ul^ is an equivalent quasi-norm in F^,q.

Proof. Proof of (i). This embedding can be found in [22], see in particular, Statement (iii) in Propositions 2.4 and Proposition 2.6 in the cited work as well as Remark 8 below.

Proof of (ii). Let f E L^ be such that [f E . Thanks to the convolution inequality

\\T-1p* f lU ^ \\T-1P||ill/\U

we have

Fi,, £ Wf II« + l\[/Ulfi^.

?s

For the opposite inequality, let f E F^^. By (i), we first have \\/H« < \\f Hfs . Then for all k ^ 0 and all v E Zra, we obtain

2kn f £ 2jsq\Qjf \qdx =2kn f ( £ + 2jsq\Qjf \qdx

r (4)

<Hf H^E 2jsq + 2kn Y,23sq\Qi f \q dx. K0 i ^

On the one hand, denoting by E(x) the vector ([x1],..., [x„\) E Zn for x E Rra, we get an elementary inequality

[21-fcVj] ^ 2xj < [21-kVj] + 1 + 21-fc, x E Pk,v, k ^ 0, j = 1,...,n,

and this yields

1+21-i

x E Pk,v ^ X E P1,E(21-ku)+rw0 ,

r=0

where w0 := (1,1,..., 1) E Zn. We then obtain

, 1+21-i

2jsq\Qj f \qdx ^ £ / £ 2jsq\Qj f \qdx

Pi 3>1 r=0 P J>1

:(2 + 21-k) sup I ^ 2i8qIQj f lqdx

^(2 + 2 ) OU.JJ I / ^ j J

^(2 + 21-fc) sup sup 2rn Î V 2i8qIQj f dx reNVez« J

^(2 + 21-fc )W/||fi

co, q

Finally, by inserting this inequality into (4), and taking into account that 2 kn(2 + 21 k) ^ 4 for

k ^ 0, we get

2kn i £ 2^ f rdx < \\ f \\I + \\ f \\F. q < \ \ f \\^ q, k ^ 0. (5)

Pk,v r'k

On the other hand, clearly for all k E N,

2 kn r ^ 2^lQ, f rdx ^ sup 2- i £ f l*dx ^ \\ ,,\ q

P{ j>k ren / '

Then this estimate and (5) yield the desired result. The proof is complete. □

The space can be described via differences. We recall the following statement. Lemma 5. Let m E N be such that

max (n/q — n, 0) < s < m. (6)

Then

(i) A function f belongs to F^ q if and only if f E L^ and

XS21 (/):= sup (*> t t-, sup iWf) 1 < ».

0 Pk,v

Moreover, the expression \\f + ) is an equivalent quasi-seminorm in F^ .

(ii) The same conclusion holds by replacing in (i) the term ) by

2i-fc

- - n

-^'2(/):= sup (2kn[ t-sq i (t-n i lAZf(x)ldhYdx^-)1,

keno,^ez" v .1 .1 v ./ / t /

0 Pk>v t/2K\h\<t

or

2

1-fc

^mq3(f):= sup (2kn[ t-sq i t-n i \A™/(x)r dhdx^) ?.

keno,^ez" v J J J t y

0 Pk}V t/2^\h\<t

Proof. We refer to [22, Rem. 4.8] if 0 < q < ro, and to [22, Cor. 4.3] as q = ro, in which the statement was proved for the Besov-type spaces , but = B^□

2.3. Definition of realizations.

Definition 4. Let m E N° U {ro} and k E {0,... ,m}. Let E be a vector subspace of S'm endowed with a quasi-norm such that the continuous embedding E ^ S'm holds. A realization of E into S'k is a continuous linear mapping a : E ^ S'k such that [a(f )}m = f for all f E E. The image set a(E) is called the realized space of E with respect to a.

Remark 4. In case k = m the identity is the unique realization.

If a realization is known, then it generates other realizations. We recall the following statement, see [6, Prop. 1].

Lemma 6. Let a° : E ^ S'k be a realization. For all finite families (£«)k^\a\^ N of continuous linear functionals on E, the following formula defines a realization of E in S'k :

a(f )(x):= a°(f )(x) + ^ £a(f) xa (modulo Vk). And vice versa, each realization of E modulo Vk is given in such a way.

1-k

3. Realizations of ,q

In what follows, to any space F^ , we associate a number ^ E N0 defined by:

p := max(0, [s] + 1). (7)

We shall employ the following lemma, a classical consequence of Taylor formula, see, for instance, [16, Prop. 2.5].

Lemma 7. Let 0 < p ^ ro and N E N0. There exist c1,c2 > 0 and m1,m2 E N0 such that

(i) HQjP < C12-nCmi(T-17)(mi(p) for all y E S and all j E N0.

(ii) \ \ Q3<p\\p ^ C22jN(m2 (T-17)(m2 (p) for all <p E SN and all j E Z \ N.

Our main aim is to prove the following result.

Theorem 1. Let f E F^,q. Then the series J2jeZ Qj f converges in S'^. Let us define a(f) as the its sum belonging to S'^. Then the mapping a : q ^ S'^ is a translation and a dilation commuting realization of F^,q into S'^. The element a(f) is the unique representative of f in S'^ satisfying [a(f)]« = f in S^ and daa(f) E C0 for all \a\ = p,. Moreover,

l l k(/)]~11 p. = H/ll*. .

Proof. Step 1. Let f E F^ . We introduce a radial and positive function 7 E £*(Rn\{0}) such

that 77 = 7. Then we define a sequence of operators (Qj) as (Qj) by taking 7 instead of 7. Let g E S^. We begin with the inequality

\{Qif,Qsg)\ ^ 2'\ \ Q,f \\ U2-3Sl l Qig11 1).

Then by Lemma 7 with p =1, p := g and an arbitrary N and F^ ^ B^,^ we get:

\{Qi f,Qi9)\< 2-smin(2-N, 2^)(m(g)\\f\ ^, j E Z, (8)

where an integer m depends only on N and p,. We choose N such that N + s > 0, and by the definition of ^ we have ^ — s > 0. Then by the identity (Qj f, g) = (Qj f, Qjg) we get

- co, q

f,g)l < Ug) |1 f |1 . (9)

jez

Step 2. Inequality (9) yields

Fs

c, q

sup ),g)l <

gesß, Cm(s)<i

for all f G F^ q. Then a is a realization of F^ q into S'ß. Step 3. The identity [a(f)]« = f in S'^ is implied by (1).

Step 4. Let |a| = ß, X > 0 and g g5. We introduce an integer r such that 2--1 < A ^ 2-Then suppT(h\(Qj-rf(a))) is contained in the annulus 2j-1 ^ |£| ^ 3 ■ 2j, and

T(Qkhx(Qj-r f(a))) =0 as k - j ^ 3 or k - j ^ -2.

Hence,

(hx(Qj-rf(a)),g) (hx(Qj-rf(a)),Qj+kg). k=-2

By Bernstein inequality we have

| | hx(Qj-rf^)|1 « < 2(j-r)|a|| | Qj-rf |1 « < 2j(M-1 | f 11 6.

on the one hand. On the other hand, by Lemma 7(i) and the fact that \\Qj+k9\\1 < IMk, for some N E N° and m := m(N) E N° we have

l(hx (daa(f)) ,g)l < Xp-S\\f Wi,^^ ((m(g) £ 2^--N) + \M\1 £ ^^).

j>° j<°

Choosing N such that N + s — ^ > 0, and taking into account that ^ — s > 0 for all s E R, we pass to limit as A tends to 0 and arrive at daa(f) E C°.

Step 5. Let fi E , i = 1, 2, satisfy the identity [f1 = [/2]^ = f and dafi E C° for all l^l = p,. Then

/1 — /2 EV„ and da(f1 — /2) E C° nVoo = {0} for all M ' p. Hence, /1 — /2 E Pp.

Step 6. Since each operator Qj commutes with the mapping ra for all a E Rn, the realization a commutes also with ra.

Let A > 0. Since F^ q is dilation invariant, that is, h\f E F^ q, see Lemma 1, it follows that a(h\f) = J2jeZ Qj(h\f) E . We define the operators Qj,x as Qj replacing 7 by hx^f. It is easy to see that Qj(h\f) = hxQj,xf in S' since Qj<p(\(-)) = Qj, x(hx-ip) for all p E S; recall that Qj(S) C Soo. We now define the realization ax(f) := J2jeZ Qj,xf of F^ q into . Then

(a(hxf),<p) = Y.^hxQ,,xf,<fi) = Y,{Qj,xfMK))) = ^(«xU),¥>(A(-))>

eZ eZ

for all p E Sp. Hence,

o(hxf ) = hxax(f) in sp. (10)

As above, we also obtain that for ax, the arguing in Steps 1-5 hold true. Then

W)]« = kx (J)]« = f,

and a(f) — ax(f) E P^. But da(a(f) — ax(f)) E C° nP^ = {0} if M ' ^ and hence, °(f) — ax(f) E Pp. This implies hx(a(f) — ax(f)) E Pp. Therefore,

hx a(f ) = hxax(f) in S'^. (11)

Now, by (10) and (11) we obtain that a(hxf) = hxa(f) in . Step 7. It is clear that QrQjf = 0 as [j — t\ ' 2. Then

Q \1/Q

ax

\\[v(f )Uf. = supsup(>/ £ 2J" £ QrQ3f q dx)

00" ieZ»eZnK Jv % ,-1^+1 }

sup sup (> / V H V Qm+3Q3 f qdx) leZ "eZ"K J I'i ¿=-1 7

We let

1

71 := £ 7(2-m -b,

m=— 1

and define the operators Qjt1 as

Q~Jf := 71(2-0)/.

Then we get

q \ 1/(i dx

(12)

y^ Qm+jQj = Qj,1 for all j E Z. (13)

m=— 1

We have

( 1 3 ï 3

supp 71 c| e G :- ^ |e| ^ 2] and ^^ ^ 1 as 3 ^ ^ ^ 1

since 71 (£) ^ 72(C), see the definition of 7 in Section 1. Then 71 satisfies equations (2.1)-(2.3) in [12] and owing to equation (5.1) and Corollary 5.3 in [12], we can replace the operators Qj by Qjt1 in Definition 2 to obtain

Ï., < sup sup ( 2

TC,q lez uezn

(2n J Y, 2sq\ è Qm+jQjf\qdx)1/q <

tel m=-l

Fs

co,q

Hence, it follows from (12) that \ \ [a(f)]« \\ ps = \\/\ljps

Finally, for this identity for quasi-seminorms, we can add the following observation. Let f1 E S' be such that [/1]^ = [a(f )]«. We have

l l № )]« 11 = \ \ [/1]« \ \ p.,., , .

Let f2 E S' be such that [f2]<x = f. By Step 5, f1 — f2 is a polynomial; we denote f1 — f2 =: f. But Qj([a(f)]«) = Qjf1 = Qjf2 since Qjf = 0; we also have Qjf1 = Qjf2 in the sense of functions, since both Qj f1 and Qj f2 are smooth functions of exponential type, see Paley-Wiener theorem [13, Thm. 1.7.7]). We again arrive at the desired identity. The proof is complete. □

Remark 5. For all s E R, if f E , the series Qj f converges in S'. Indeed, the inequality (8) becomes

\(Qif,Qsg)\ < 2-(N+s)Ug)11 f 11

for all g E S and all j E N0; here Qj is the same as in Step 1 in the proof of Theorem 1.

The next lemma characterizes the number the proof of this lemma is similar to that given by G. Bourdaud for Besov spaces [4, Prop. 2.2.1].

Lemma 8. Let s ^ 0. Then there exists a function f E F^ q such that the series J2j<0 Qj f diverges in S'^t-1.

Proof. We briefly outline the proof, since in case q < ro we do not have the same spaces as in [4]. We denote m := ^ — 1 = [s]. Let p E 'D be such that f <^(x)dx = 1. As d™^ E Sm, we

r"

split the sum j<0(Qj f, dinto /1 + I2, where

h := (—1)m W {d?Qj f (x) — d?Q3 f (0))lp(x)dx, I2 := (—1)m £ d?Q3 f (0). j<0 r" j<0

It is sufficient to construct a function f E F^,q such that \/i\ < ro and \I2\ = ro. For this purpose, let g E S be such that

g EV, g ^ 0, supp £ c{ £ :3 < \£\ < 1,6 ^ ^ .

We let

f (x) := Y 2 k(s+m)/2g(2-kx). k'^0

Clearly, we have

Qjf (x) = 2-(s+m)/2g(2x) if j < 0, Qjf (x) = 0 if j > 1,

since 7(2 )g(2kfa) = 0 if k = —j and ^g = g; we recall that 7(fa) = 1 as 3 ^ lfal ^ 1. It is also clear that for all j ^ 0 the identities hold:

d?Q3f(0) = (2n)-nim2(m-s)/2 j dfa,

Rn

n „

ldTQ, f (x) — d^Qj f (0)l ^ (2n)-n2j(m-s+2)/2 £ lxkl &l &g(£) d£.

Then

k-l r„

| £ d?Q3f (0)| = œ, J] \\VdTQ,f 11« < œ.

3^0 j^O

It remains to prove that [f ]« G Since

j lg(2jx)^dx ^ 2-jn\\g\\\

Pk,v

and s - m ^ 0, that is, 2jq(s-m)/2 ^ 1 for all j ^ 0, we first have

2 kn £ 2j('(s-m)/2lg(2jx)l('dx ^ \M\? £ 2(k-»n < \M\? (14)

p O^j^k O^j^k

for all k E Z \ N. Therefore, by taking the supremum over k E Z \ N and v E Zn in (14), we get

\\[f<1.

The proof is complete. □

Without use the LP decomposition, we define the realized space of F?

«,q~

Definition 5. The realized space of F^q denoted by Fsooq is the set of all f E S' such that

s and f(a) «, q

[/]« G Q and f ^ G Co for all H = ^

We should be sure of the identity q) = F^ q, where the mapping a was defined in

Theorem 1. The direct embedding is by the definition; let us prove the opposite one.

Let f E Fs^q, then f — a([f ]«) is a polynomial. Since C° nV^ = {0} and f(a) — daa([f]«) E

C° for all l^l ' p,, we conclude f — a([/]«) E Vp, that is, f = a([/]«) in S'.

7?S • \ \ LJ J« IIPs

F%o,q 00 A

The space F^^ is equipped with a quasi-seminorm defined as

.Hi

Of course, one has to justify this definition. If [f = [/1 and [f ]« = [f2]<x, then f\ — f2 E Voo,

but Qj(/1 — f2) = 0, which is a sufficient argument. In the case s ' 0, F^ q can be characterized in S'. This is done in the next lemma; for the case s = 0 see Remark 6 below.

Lemma 9. Let s > 0. Then F^ is the set of f E S' such that [f ]« E F^ q, and f ^ E C° for all l^l = ^, and moreover:

(i) if sE N, then f E Cp-1 and f (a)(0) = 0 for all H ^ p — 1,

(ii) If s E N, then f E Cp-2 and f(a) (0) = 0 for all H ^ ^ — 2 with ^ = s + 1 ' 2.

Proof. The proof is similar to the proofs of Proposition 4.8 in [7] and of Theorem 4.5 in [16]

1 * ^ B « . «,q «,«'

thanks to the embedding ^ B^ ^; let us briefly outline this.

Proof of (i). We first define F«>q in S' by replacing each Qj f by a polynomial of degree less than ^ in a(f), see Theorem 1. Then we get a realization denoted a1. Since any realization on

i7«^ is a surjective mapping, then if f E F« q, there exists g E i7«^ such that [/]M = g, and it is sufficient to take f := a1(g).

Construction of a1. Let g E F^ q. Then the series

' )(«)(0)

°i(9) := E fa3 - E ^^

• • a!

jez \a\<n

converges in S'. The mapping a1 : F^ q ' is a realization of F^ q into S', where a1(f) is the unique representative of g in S', of class C^-1, daa1(g)(0) = 0 for all \a\ < ^ — 1, daa1 (g) E C0 for all \a\ = ¡J, and H^sO]«!^, = IMIf.,.

We now present the role of the assumption s E N: by the Bernstein inequality

H(Qj^)(a)H« < v^llQig\\« < 2i(\*\-')llg\\ b^,

we get

,__ Ta ,__ ITI M

Q,g(x) - E (Qj9){a)(0) ^ + E ^T3)(a)H

jy ll « ,

a\

la\<p |a| Kp— l

(2-* + 2j(p—1—s\l + Mr-1L) Hgll6Soœ ,x G , j G No.

<—

On the other hand, by the Taylor formula we have

Q.ig(x) - E (Qj9){a](0) ^ | K » E ^f /(1 - tr—1l(Qi9){a)(tx)l dt

M<p H=p 0

Therefore,

M^)l <{ E fa3S + 2(p—1—s)(1 + lxir—L) + E s) Mp}lMl

3>o j<0

ps ■ , q

Thus, thanks to assumption s E R+\N0, we get the convergence of above series with ^ — 1 — s = [s] — s < 0 and ^ — s > 0.

Proof of (ii). As in the previous step, we consider the mapping:

Q39 — E (Qj9)(a)(0) for all g E F^q, (15)

3>0 j<0 \a\<ti !

where a2(g) is the unique representative of g in S', and a2 is also a realization of F^ q into S' satisfying daa2(g) E C0 for all \a\ = ^ and H[^2(^)]«Hj?, = IMIf* . If in addition s > 0, then a2(g) is of class C^-2.

Owing to Lemma 6, if f E F^^, there exists g E F^ q such that [f = g and it is sufficient to take

f := *2(g) — E (E(^i3)(") (0)) C.

For the realization a2 we refer to [7, Rem. 4.9]. In case s > 0, for \[3\ < ^ — 2, we have \(5\ — s < ^ — 2 — s = —1, and then

E ll(Q>^)(/3)H« < II^Hj^.,,, E2m-a)3 < llyHj^.,,,;

3^0

DC

1

the estimate for the sum

El5" {^ 9 — E (Qj 9){a](0) }l z—' z—' a!

i<° \a\<p

can be obtained as in [16]. The proof is complete. □

Remark 6. If f E then f = a2(g), where a2(g) is defined in the above proof, see (15).

o,q

Remark 7. Clearly, we can not identify F° 2 with BMO, where the space BMO is as defined in [10], since \\ [f ]oo \\ fo =0 for all polynomials, while one can easily find a polynomial f E 'P1

F co ,2

such that [ (1 + lxln+1)-1lf (x)ldx = ro, see [10].

rn

4. Characterizations by differences

We now present a characterization of realized spaces F^ g by means of differences. In view of Lemmata 4 and 5, one could think that the scales N;^""'1 (f), i =1, 2, 3, are other equivalent quasi-seminorms in F^ q. But this is not the case since for any polynomial f of degree m we can have i(f) = 0, while \\[/= 0; for instance f (x) := x", then A™/(x) = m\h"

and ■N'^"",1(f) = m\2m-s(q(m — s))-1/q, which tends to infinity as s t to; the kernel of A? is V '

' m-

Lemma 10. Let (6) be satisfied. Then there exists a constant c> 0 such that the inequality N (f) ^ c\\[f ]oo Hii's holds for all f E F^ q, where N := N^""'1. The same holds if we replace

N^"/ by N^"/ with i = 2,3. Proof. Lemmata 4 and 5 we have

X (f) < \\f \\oo + \\[f ]o\\f.

for all f E F^ . Replacing f by fx := f (A(-)) arbitrary A > 0 in this inequality and using Lemma 1, we obtain:

lim A-M(fx) ^ c \\[f]o\\f,s for all f E F^>q. (16)

x^-oo

Let now A > 1 and N E N be such that 2 N ^ A < 2N+1. By the elementary inequality

Vx E Pk,v : [2 NX-1 ] ^ 2 k+NX-1Xj < [2nX-1v3] + 2, j = 1,...,n recall that 2-1 < 2NA-1 ^ 1, we obtain

X E Pk,v ^ X 1X E Pk+N,E(2Nx-1 v) U Pk+N,E(2Nx-1 v)+wo ,

where w° := (1,1,..., 1) E Zn and we have employed the notation E(x) = ([x1 ],..., [xn]) E Zn, x E Rn. As A"f (x) = A"x-1h)fx(X-1 x), with the change of variables y := X-1x, r := X-11 and u := X-1h, we get:

21-fc

2 kn i t-sq sup i lAm/(x)\* dx^ J 2<\h\<i J 1

° 2 Pk,v

21 —( k + N ) (17)

< A-'*£2(k+N)n i r-sq sup i lAmix(y)lqdy-.

l=°o i ^H< J r

° Pk+N, E(2N X—1v) + lwo

We assume that k E N0 and this allows us to bound last term in (17) by

21 —

cX-sq sup sup 2jn f r-sq sup i \A™fx(y)\q dy—, (18)

jeNoveZ" J r/2<\u\<r J r

0 Pj,v

where c is independent of k. Calculating the supremum over k E N0 and v E Zn in (17), and taking (18) into consideration, we obtain M(f) < c\-sN(f\). Finally by (16), we complete the proof. □

Here our second main result is as follows.

Theorem 2. Let m E N be such that (6) is satisfied. Then N^1 (f), i = 1, 2, 3, define equivalent quasi-seminorms in F^^.

Proof. We consider only (f), since the estimates of (f), i = 2, 3, can be obtained

in the same way. To simplify the notations, in the proof we write N(f) instead of ^^^(f). The proof of \\ [f ]«\\< cN(f), for all regular tempered distribution f obeying N(f) < ro

can be done as in [18, Subs. 4.1] and we omit the details.

The opposite inequality is similar to that given in [18], and we present only the needed

changes. Let f E F^^. We denote fk := Y1 -k<j<ks Qj f, where k E N0. We also define ks := 0 as s E N and ks = k as s E R+\N. Then the function fk belongs to F^ q. Indeed, the inequality 11 fk\ \« < c|| [f]«Hj?. with a constant c := c(k) > 0, can be obtained by the assumption on s and the following estimate:

\Qj f (x)\ < c 2-sHf || j?. , j E Z, x E Rra. (19)

, q

In order to prove (19), it is sufficient to employ the embedding F^,q ^ F^,« = B^,«. Now we are goin to prove that

HL/fcU^,, < c||[f]«\\p^q (20)

with a constant independent of f and k. We proceed as in Step 7 in the proof of Theorem 1. Then similar to (12) recalling that QrQj f = 0 as \j — r\ ^ 2, we get

fc]«Hi?. = sup sup (2ln f V| V Qr Qj fq2sqdx]l/q

00,q iez vez"\ J ^ I /

p f^i —K<r<Ks

n,v k-jj<1

: sup sup Î2(l—N )n f y | V Qr Qj f q 2 sqdx]l/q

j .iï—N^ —fcKKfcs y

(21)

for all N E Z. Since here the supremum is taken over all I E Z, it is translation invariant in Z. The last identity is trivial but is useful for the next computation. On the one hand, in the sum \i.-J\<1... we have at most three terms corresponding to r E {j — 1,j,j + 1}, and hence

£ Qr Q3 f | ff< 22( q-1 E I Qr Qj f I ff. (22)

-k<r<k. — k<r<k.

\ - \<1 \ - \<1

On the other hand, by the following elementary inequalities

if — k < r < ks and \r — j\ < 1 ^ —k — 1 < j < ks + 1,

if — k — 1 < j < ks + 1 and \r — j\ < 1 ^ — k — 2 < r < ks + 2,

by the fact that

{r E Z : — k < r < ks} c {r E Z : — k — 2 < r < ks + 2},

and by using (22), we obtain

£ I E QrQi f 23sq < ^ £ | QrQj f |

j'l-N - k<r<ks j'l-N -k<r<k3

k-iK1 \r-j\<1 (23)

<c E E I ^f I q23sq.

j'l-N \r-;/\<1 -k-1<j<ks+1

Choosing the integer N := Nk,i such that —k — 1 ' I — Nk,i, we bound the last term in (23) as follows:

C E 1 Qj+mQj f1 q23sq with m := r — j.

j'l- Nk,i \m\<1

Substituting this bound into (21), letting 1 := I — Nk,i, and taking the supremum over all 1 E Z,

we get

\\[/k]o\\ p> < c V sup sup (Vn / V iQ+mQjf 92i8qdx)1/q (24)

00" №1 leZv eZ"K J

for all k E N°. We continue by letting 7m := 7(2-m(-))7, and this function possesses the following properties:

1 3 3

supp7° C fa E Rn : - < lfal < - , 7°(fa) ' 1 as - < lfal < 1,

jn

- 2 -=1^.1 - 2y ^ 4

1 3 9 11

supp 7-1 C<j fa E Rn :2 < lfal < 3 , 7-1(0 > 0 as - < ^ < -.

Hence,

9 11

7-1(0 ' c > 0 on fa E Rn :- < lfal < - , c := q min ii 7(2^)7(v).

I 16 16 J A<|„,< 11

16 ^^ 16

The next property is

-\Fg Rn :1 < lfal < ,, , „ _ ^ ^

2 ' 8 "" 8

3 9 11

supp 71 ci fa E Rn : 1 < lfal < , 71(e) > 0 as 9 < lfal < 11,

and hence,

71(0 ' c > 0 on ifa E Rn :8 < lfal < -j , c := q min^ 7 (2) 7 (v).

< > g <M< -g-

Then we define the operators Qj,m as Qj,mf := 7m(2-J(-))/, and as in (13), this yields

Qm+jQj = Qj,m for all j E Z.

We replace the operators Qj by Qjm with m E { — 1, 0,1} in Definition 2 and we denote by \\ ■ W^ the associated quasi-seminorms. By [12, Cor. 5.3], we have:

Fco ,q

\\[fU\fs < c \\[/UU,

where c is independent of f. But from (24), we also have

1

\\[fkU\F»oq < c £ \\[/^«H^ for all k E Z.

^ , 1 *-* F co ,q

m= - 1

This proves estimate (20).

Applying now Lemma 10 to fk, we obtain

M(fk) < c ||[/]«Hi^q for all k E N0, (25)

the constant c is independent of k, see (20). On the other hand, letting

Ti(*):= Y, (Qlf )(°'(°)

- ^ a,

\a\<ii

and recalling that ^ = [s] + 1, cf. (7), we obtain that the sequence (fk — -k<j<ks rj)k>0 converges uniformly on each compact subset of Rra to a limit denoted v, see [18, (22), Subs. 2.2] for B^«. At the same time, F^ q ^ B«,« cf. Lemma 3. By applying twice the Fatou lemma in (25), we get

< C H[/]«Hi?.,q. (26)

In case s E N, we add the following inequality:

^(Ef) < c H[/]«Hi^.,q, (27)

>0 , q

that is, £J>0 Qj f E F^ q. The latter can be obtained by Lemma 10 since we can apply (19) thanks to s > 0, see (6), and to obtain

llEQif ll« < ll[/U^

3>o

and similar to Step 7 in the proof of Theorem 1, we also have

llE Qj f lL < ll[/]«L s .

j>o

, q

We let g := v + J2j>0 Qj f if s E N and g := v if s E R+\N. We have f — g E V^ and = {0}; recall that A™(xa) = 0 for all \a\ < m, and by assumption m > ^ > s. Then it follows from (26) and (27) that

*(/) < M(/ — g)+ N(g) < H[/]«||^,q.

The proof is complete. □

Remark 8. Of course, the statement of Lemma 4 is certainly known and in particular (i) is classical, but now this can be deduced from Theorem 2 at least for q > 1. Indeed, the difficult part in the proof of Lemma 4 is H[/]«Hi?. < II/Hf., , where now, we get

H [/]«Hi?.,q < (I) < ^(Z) + H/H« < HZ IF., q

if q > 1 and m E N is such that 0 < s < m.

Conclusion

The realized spaces F^^ of the homogeneous Triebel-Lizorkin spaces F^ q are now characterized by quasi-seminorms in discrete and continuous (if s > 0) forms. Our next step will be

the extension of the study on F^ g to:

• the pointwise multiplication as in e.g. [2],

• the composition operators as in case of the realized homogeneous Besov spaces, see e.g. [8, Thm. 4] or [17, Thm. 5.1],

• the pseudodifferential operators as in e.g. [15].

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Mohamed Benallia,

Laboratory of Functional Analysis and Geometry of Spaces, Mohamed Boudiaf University of M'Sila, 28000 M'Sila, Algeria E-mail: benalliam@yahoo.fr

Madani Moussai,

Laboratory of Functional Analysis and Geometry of Spaces, Mohamed Boudiaf University of M'Sila, 28000 M'Sila, Algeria E-mail: mmoussai@yahoo.fr