SOME EMBEDDINGS RELATED TO HOMOGENEOUS TRIEBEL–LIZORKIN SPACES AND THE BMO FUNCTIONS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 13 (31), No 2, 2024, pp. 25-48

DOI: 10.15393/j3.art.2024.15111

25

UDC 517.98

B. Gheribi, M. MoussAi

SOME EMBEDDINGS RELATED TO HOMOGENEOUS TRIEBEL-LIZORKIN SPACES AND THE BMO FUNCTIONS

Abstract. As the homogeneous Triebel-Lizorkin space q and the space BMO are defined modulo polynomials and constants, respectively, we prove that BMO coincides with the realized space of 190,2 and cannot be directly identified with 190,2• In case p <8,

we also prove that the realized space of Fp/q is strictly embedded into BMO• Then we deduce other results in this paper, that are extensions to homogeneous and inhomogeneous Besov spaces, Bp and Bp q, respectively. We show embeddings between BMO and the classical Besov space B0,8 in the first case and the realized spaces of B0 2 and B0,8 in the second one. On the other hand, as an application, we discuss the acting of the Riesz operator on BMO space, where we obtain embeddings related to realized versions of B8 2 and B8,8.

Key words: Besov spaces, BMO functions, realizations, Triebel-Lizorkin spaces

2020 Mathematical Subject Classification: 30H35, 46E35

1. Introduction and the main result. The main result of this paper is the embeddings between the bounded mean oscillation space BMO and the homogeneous Triebel-Lizorkin spaces F^ in a certain sense. The spaces Fp defined by the Littlewood-Paley decomposition (abbreviated by LPd), in particular i78 2, as defined, e.g., in [10, (5.1)], are given by distributions modulo all polynomials; however, the space BMO is modulo constants, as defined, e.g., in [9]. We then observe that 190 2 cannot be identified with BMO, since for any polynomial f of degree ^ 1 it holds

that 11/IIpo = 0, while \\f\\bmo = 8. Concerning this identification, e.g.,

r 8,2

in [21, p. 243], the author replaced the space BMO-modulo constants by a

© Petrozavodsk State University, 2024

space modulo polynomials (denoting it by BMOp*q), which coincides with F0 2, we have BMO £ BMOp*q; we refer, e. g., to the short comment at the'end of [10, p. 70].

Another way to investigate the above identification is to introduce the realized space (Definition 5 below) of , which is a subspace of S'v (the collection of all tempered distributions modulo polynomials of degree < v) for some minimal values v, which depend on s — n/p, see (2).

The concept of realization was introduced by G. Bourdaud in [4] for the homogeneous Besov spaces Bp . This has an advantage in some fields, since there is no need to consider the spaces modulo polynomials. In general, realizations of homogeneous Besov and Triebel-Lizorkin spaces are defined up to a polynomial whose degree is less than an integer, denoted here by v, which plays a crucial role in studying the convergence of the series associated to the LPd of such functions, see, e. g., [6], [7], see also the comment below at the beginning of Subsection 2.2.2 just after formula (2).

Nowadays, we know a lot of concrete characterizations on realized spaces (see e.g. Remarks 2 and 3 below), and there are many papers in this subject, e.g., [13], [14]. There are also various works related to the realizations of certain homogeneous spaces, as, e.g., in Navier-Stokes, Hardy, and Gagliardo-Nirenberg type estimates, pseudodifferential operators, pointwise multipliers and wavelets, see, e. g., [2], [12], [15], see also [1], in which further references on these topics may be found.

Thus, we show:

Theorem 1.

(i) The identity BMO = F°8 2 holds with equivalent seminorms.

(ii) If 0 < p <8, then the embedding Fp'q ^ BMO is proper.

In relation with Theorem 1(i), the following statement (see, e.g., [5, Thm. VII.12, p. 147]) is proved:

Proposition 1. A function f e ¿2°° belongs to BMO if and only if

(i) J (1 + \x\n+l)'1 \f (x)| dx < 8,

R"

(ii) sup 2nk f V \Qjf (x)\2 dx <8.

yeR"'keZ J ^

\x-y\<2-k

The operators Qj are defined in Subsection 2.1 below. This assertion gives a characterization of BMO; its proof has a certain history:

• The authors of [9, Thm. 3] have previously proved it by using the

h

condition sup h—n t\VPtf(x)\2dtdx < 8, where Ptf is

yeR", h>0 J J

\x—y\<h 0

the Poisson integral of f, instead of (ii). Hence, it seems interesting

to see if F8,2 can be endowed with seminorms defined by the Poisson semi-group (probably, open).

• In [20, Sect. 2], the author has considered (ii) in a continuous form, that is, the formula (1.1) in this reference. The same situation is given, e.g., in [18, IV.4.3, p. 159].

We turn to the embedding given in Theorem 1 (ii); the authors in [19, Thm. 2(a)] proved: if 1 < p <8 then

\\Jn/pf\\bmo ^ Csupt\{x: \ f(x)\ > t}\1/p, >0

where Jn/P is the Bessel operator defined as Jsf := T 1 ((1 + \2)—s^2'f), s e R, and \{...}\ denotes the Lebesgue measure of the set {...}; the right-hand side can be easily estimated by c\\f \\p. Then (which is well known)

Hn/P ^ BMO (1 < p <8), (1)

where Hnp(1 < p < 8) is the Bessel-potential space defined as the set of all functions f satisfying \\f \\H«/P := \J—n/pj\\p < 8, but it is also well

known that Hnp coincides with the inhomogeneous Triebel-Lizorkin space Fpn2p; then the embedding properties of provide that ^ BMO

is satisfied for all 0 < p <8 and all 0 < q ^ 8. Hence, dealing with (0 < p < 8) presents the contribution of Theorem 1 (ii), and now we can

obtain (1) without using the operator Js, indeed we have Fpn/qp ^ Ftq since F^/q = Lp x Fpjq (see [15, Prop. 2.5]).

The paper is organized as follows: In Section 2, we collect the useful tools, in particular some characterizations of the realized spaces. Section 3 is devoted to the proof of Theorem 1. In the last section, we discuss two corollaries (Subsection 4.1) of the main result for the inhomogeneous Besov spaces and their realized counterparts, and give some applications (Subsection 4.2) related to the actions of Riesz operator on BMO.

Notation. We denote by N the set of all positive integers, N0 = N y {0}. We work in Euclidean space Rn, then one writes C8(Rn)

as C8, S(Rra) as S, etc. For s e R, [s] denotes its integer part. For a e R, we set a+ : = max(0,a). The symbol ^ means a continuous embedding. We denote by (k e Z,p e Zn) the dyadic cube 2-fc([0,1[n+p). By \\-\\p we denote the Lp quasi-norm. Ll°° denotes the space of functions in LP(Q) for any compact set ^ in Rra. V denotes the set of compactly supported functions in C8. The operators of translation ra (a e Rra) and of dilation h\ (A > 0) are defined by raf : = f (■ — a) and h\f : = f (a-1 •), respectively. For a measurable function f, m,Q f : = \Q\-:l $ f (x)dx is its

Q

mean value over the set Q. For f e Ll, the Fourier transform is defined by Tf (0 = №) := fe-^ f (x) dx,

the inverse by T-lf (x) : = (2n)-nf(—x). The operator T can be extended to the space S' of tempered distributions in the usual way. For m e N, we denote by Vm the set of all polynomials in Rra of degree < m, e.g., Vl : = {c: c e C}. We put V0 : = {0} and V8 the set of all polynomials in Rra. For m e N0 y {8}, the symbol Sm will be used for the set of functions p e S (the Schwartz space), such that (u, p) = 0 for all u e Vm, its topological dual is denoted by S'm. If f e S', then [f]m denotes its equivalence class modulo Vm. The constants c, cl,... are strictly positive, depend only on the fixed parameters as n, s, p, q,... and some fixed functions, their values may change from one line to another.

Throughout the paper, the real numbers , , satisfy e R and p,q e]0,8], unless otherwise stated.

2. Various function spaces.

2.1. Definition of Besov and Triebel—Lizorkin spaces. Throughout this work, we fix in C8 a radial function p, such that 0 ^ p ^ 1,

P(0 = 1 if \Z\ < 1 and p(0 = 0 if \Z\ > 3/2. We set -f(0 : = p(0 — p(20, which has support in the annulus 1/2 ^ ^ ^ 3/2 and 1(0 " 1 in 3/4 ^ \£\ ^ 1. We define the operators Sj and Qj (@j e Z) by

sTf(0 : = p(2-0f(0 and QTf(0 " l(2-0f(0, which are defined on S', take values in the space of analytical functions of exponential type (see the Paley-Wiener theorem), and are uniformly bounded in C(LP) (1 ^ p ^ 8) by virtue of the convolution Young inequality. We obtain the inhomogeneous LPd as follows:

For all f e S (resp. S') and all k e Z, we have f = Skf + X! Qjf in <S

j>k

(resp. 5'). Recall that p(2~fcf) + ^ 1 = 1 for £ e Rra.

j>k

We have the following definition: Definition 1.

(i) The inhomogeneous Besov space B^ is the set of all f e S1, such

that ||/\\Bs := \\Sof \\p + ( Z (2JS\\Q3 f \\P)q)1/q <

i>i

(ii) Let 0 < p <8. The inhomogeneous Triebel-Lizorkin space q is the set of all f e S1, such that

f\\F>,q := \\Sof\\p + f|)9)

i>i

1/9

< 00.

To extend the definition to S'8, we use the following convention:

If f e S'8, we define Qjf := Qjfi for any fi e S1, such that [/i]8 = f.

Thus, Qj are well-defined on S'CX), since Qj f = 0 (Vj e Z) if and only if f e Vc and:

For all f e (resp. S'CX)), we have f = ^ Qj f in (resp. S'CX)).

jeZ

Recall that £ 7(2j£) = 1 for £ ^ 0.

jeZ

We get the following definition: Definition 2.

(i) The homogeneous Besov space B^q is the set of all f e S'CX), such that \ \ f \ \

\ B s Bp, 1

:= ( \\ Q3f \\p)q) <».

jeZ

s

P,Q 1/Q

(ii) Let 0 < p <8. The homogeneous Triebel-Lizorkin space is the

1/9 '

set of all f e S^, such that \\ f \\ps := ( Z(2JsIQj f I)9)

p,q ^ jeZ '

< 00.

(iii) Let 0 < q <8. The homogeneous space FS,q is the set of all f e S'CX) such that

\ f\

F s

F CT

sup (<2kn f 2(P'IQj f (x)I)q dx]

keZ, ßeZn \ J ~~k /

.

P

P

(iv) For q = 8, we set Fc%, x = BS

The spaces B, F (resp. BB, F) are quasi-Banach for the above defined quasi-norms (resp. quasi-seminorms). Their definitions are independent of the choice of p, see, e.g., [16], [21, Sect. 2.3] and [10, Coro. 5.3]. In the above definitions, we can, also, change Qjf by Vj * f, where vj := 2jnT~1v(2j(-)) with v e P(Rn\{0}) be such that v ^ 0 and for b > 2a > 0, v(0 ^ c > 0 if a ^ ^ b; see [6, Lem. 2.1.2] and again page 46 and Corollary 5.3 of [10]. On the other hand, it holds that:

(P1) 5 ^B,F ^ S'. (P2) ^ ^B, F ^ S^.

(P3) Bmin(p, 9) ^ Pp,q ^ BS, max(p, ^ With 0 <P<8.

(P4) if 0 < q,r ^ 8, si > s2, and 0 < p1 < p2 <8 are such that Si-n/p 1 = s2-n/p2 then B;i>q ^ B«q ^ B%;qn/P2, F;iq ^ b«p1 and F°lq ^ F;ir, see [11]. (P5) \s-n/p\\hxfH^ „ \\f\\ps for all f e F°q and all A > 0, if p <8; see, e.g., [21, Rem. 5.1.3/4], in case p = 8 see [1]. The same holds when F is replaced with BB.

(P6) ^ BBSo, c, see [1, Lem. 3]. Also, using the homogeneous Triebel-Lizorkin-type space Fs,q , the set of all f e S'CX), such that

\\f\\FsT := sup sup 2hnr (2(2jS\Qjf |)9)1/q

p'q hr.Z /»=Z" V 1 /

< 8,

Lp(Pk^)

where 0 < p < 8 and 0 ^ r < 8, see [22], we have F^q = Fp

and FpS',r ^ B^Ca n/p, see again [22, Prop. 4.1], then taking t = 1/p in the last embedding, we obtain the desired assertion.

Proposition 2. A member f of S'^ belongs to FpSq if and only if its first-

n

order derivatives 5ef, 1 = 1,... ,n, belong to FS~1. Moreover, \\dif \\ps-i

i=i

is an equivalent quasi-seminorm in FpS q. The same holds when F is replaced with BB.

Proof. See, e. g., [8, Prop. 5]. The same proof, given in [6, Prop. 2.1.1] for the case of BB, can be used to obtain the case of F, and also with p = 8, since Qj can be written as 2~jve(2~jD) o 5e, where ve e P(Rn\{0}) depends on 7. □

For further properties of the spaces B, F, B, and F, we refer the readers to, e.g., [10], [16], [21]. We also refer to the survey [22] in which the homogeneous Besov and Triebel-Lizorkin type spaces, B^ and F^^, are studied (see (P6)); these spaces coincide with q and F£ q, respectively, if t = 0. ' '

2.2. The realized spaces.

2.2.1. Generalities on realizations. We introduce the following definition and some remarks with respect to [6], [7], [13], [14].

Definition 3. Let m e N0 y {8} and k e {0,... ,m}. Let E be a vector subspace of S'm endowed with a quasi-seminorm, such that E ^ S'm holds. A realization of E in S'k is a continuous linear mapping

o : E ^ Sk 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.

For every f in E, the element a(f) is the unique representative of f in a(E); consequently, a is completely characterized by its range. We say that a realization a of E commutes with translations (resp. dilations) if ra o a = a o ra, a e Rra, (resp. h\ o a = a o h\, X > 0); this goes if and only if the range of a is translation (resp. dilation) invariant.

Remark 1. A subspace E of S'm has generally infinitely many realizations in S'k if k < m, in the case of k = m the identity is the unique realization. Of course, with additional conditions such as translations or dilations invariance, a realization of E in S'k for k < m has some chances to be unique.

We have the phenomenon that if a realization is known, we obtain other (see e. g. [7]):

Proposition 3. Let a0 : E ^ S'k be a realization. For any finite family (£a)k^\a\<m of continuous linear functionals on E, the following formula defines a realization of E in S'k:

a(f )(x) := a0(f )(x)+ ^ Ca(f) xc

k^\a\<m

Conversely, any realization of E in S'k is given in such a way.

Further information on difficulties arising in the study of translation or dilation commuting realizations is in, e.g., [7], [13]; see also Remark 4 and Subsection 2.2.3 below, for the F and B spaces, respectively.

2.2.2. Realizations of Triebel—Lizorkin spaces. To define the realized space of , we first fix the following natural number: to any 3-tuple (n,s, p) we associate:

j:

(\s — n/p\ + 1)+, if s — n/pR N0 or 1 <p ^ 8, s — n/p, if s — n/p e N0 and 0 < p ^ 1.

The number v characterizes the degree of the polynomials that define the realizations. In other words, if f e q, then the series Qjf (the

' jeZ

homogeneous LPd of f) converges in S'u, and there exist polynomials r-j in Vv, such that

f = Z(Qjf — r'j) in s

jeZ

see Propositions 4-6 below. For example, if v = 0 in the case of either ( s < n/p) or (s = n/p and 0 < p ^ 1), we have r-j = 0 (recall V0 = {0}) and S' coincides with SO; this case has been studied in several places, e.g., [16, pp. 55-56], [7, Prop. 4.6], [13, Thm. 4.1 and Rem. 4.3]. If v > 1 (Pv ^ {0}), there exists a function f e Fpq, such that the series j<0 Qjf diverges in S'v_1, see [6, Prop. 2.2.1] in which the proof given in Bp q can be adapted to Fp,q.

Finally, as mentioned in the Introduction, the number plays an important role in this work, and we refer to [1], [6], [7] for more information on this.

Second, we recall the following notion: Definition 4. A distribution f e S' vanishes at infinity if

lim h\ f = 0 in S'.

AjO

The set of all such distributions is denoted by C0. Here are two examples of such distributions:

(i) feCo iff e Lp (1 ^ p < 8);

(ii) de f eCo (I = 1,...,n)iff eLm or f e CO.

Before turning to some properties related to the number u, due to technical reasons, we introduce the definition of the realized spaces of , which can be found in [7, p. 483/Step 2] and [14, Sect. 2.3] if p <8, and in [1, Def. 5] if p = 8.

Definition 5. The realized space F^ is the set of all f e S'v, such that [f]cx> e and daf e C0 for all |a| = u, where v is defined in (2). This space is endowed with the quasi-seminorm \\ f \\ ~ := I I [f]œ I I ¿,s .

*p,q гр,ч

Remark 2. The connections with the Lebesgue and homogeneous Sobolev spaces help us to understand the realized spaces. In this context, we recall that:

• F= Lp for 1 < p <8, see [13, Prop. 5.2],

• F™,2 " Lr xW™ for 1 < p <8, m = 1, 2,... and 1/r := 1/p — m/n > 0, and where W™ is the homogeneous Sobolev space endowed with the seminorm \\f\\:= \a\=m\\f{a)\\P, see [7, Thm. 5.3].

Remark 3. For convenience, in studying some analysis problems, the realizations can overcome some difficulties. For example, the pointwise multipliers in homogeneous Besov and Triebel-Lizorkin spaces, B^ and Fp,q, are not defined; for this reason, it is better to work with realized spaces, see, e.g., [3, Thms. 1 and 2].

We have the following property (see e.g., [7, Prop. 4.6], [13] if p <8 and [1] if p = 8):

Proposition 4. If f e F^q, the series Yjjez Qj f converges in S'v to an

- p,q>

element denoted by a(f ). The mapping а : F£ ^ S'u defined in such

P'Q

a way is a translation and dilation commuting realization of F^ q in S'v. satisfying [a(f)]8 = f and daa(f) e C0 for all |a| = v.

On the other hand, [7, Sect. 4.3] provides a construction of realizations of F£ in S' in case p,q ^ 1, which can be easily extended to p,q > 0, see [13, Thms. 4.1 and 4.5], see also the proof of Lemma 9 and Remark 5 in [1] for the case p = 8. Namely:

Proposition 5. For all f e F*q, define ai,u(f) (i = 1, 2, 3) by the following formulas:

ai,0(f) Qj f, if either (s < n/p) or (s = n/p and0 < p ^ 1), here v = 0; (3)

jeZ

°2,v (f) : = f - £ (Qi f ){a)(0)xa/«!), if either (a — n/p e R+\No)

jeZ \a\<v

or (s — n/p e N and 0 < p ^ 1), here v = 1, 2,...; (4)

(f) " StQjf + - E №){a){0)xa/a!),

j'1 340 \a\<v

if s — n/p e N0 and 1 < p 4 8, here v = 1, 2,.... (5)

Then ai>v is a realization of Fp,q in S', such that all above series converge in S', daaiyU(f) e CO (V|a| = u), [aitV(f)]m = f in S'8 and (/)]8|i9s = \\f\\ps . In (5) we can replace ^ and ^ with ^ and

p'9 P'9 j'1 340 j'm

for any m e Z, respectively.

j4m— 1

Proposition 6. The set aitV(F£ q) (aitV is defined in (3) -(5)) is the collection of all f e S', such that [f]8 e Fp daf e C0 (V|a| = u), and one of the following three conditions holds:

(a) There is no supplementary condition if either (s < n/p) or (s = n/p and 0 < p 4 1).

(b) f is of class Cl—1 and fpf)(0) = 0 for 3 4 v — 1, if either (s — n/p e R+\N0) or (s — n/p e N and 0 < p 4 1).

(c) f is of class Cv—1 and fpf)(0) = ^(Qjf )pf)(0) for ^1 4 v — 1, if

' 1

s — n/p e N0 and 1 < p 4 8.

The set ait V(F% q), also called the realized space of Fp , is endowed with the same quasi-seminorm, i.e., \\/\L. da \ : = \\[/]8\\fs . We also have

\r p, qq 1 p, q

o(F^q) = Flq and aitV(Fpq) £ Fap>q (i = 2, 3) if u ' 1.

Proof. Denote by M the set of all f e S' satisfying [f]8 e q, daf e C0 (V|a| = u) and one of the conditions (a) or (b) or (c). By definition, we have the embedding ai>l,(F% q) c M. Taking now f e M, we have ¿ — Oi,v([/]») e Vx, and da(f — aitV([f]m)) e CO if ^ = v. But as CO x Vcv = {0}, f — aiyV([/]») = afxf (with afi = 0 if v = 0), which

\f\<l

implies Bf(f — aitV([/]8))(0) = 3\af. Conditions (a)-(c) and (3)-(5) yield af = 0 for all m < u, and, consequently, f e ai>v(F% q).

We now prove aitl(F% q) £ Fpq (i = 2, 3) if v ' 1. The embedding follows from the Definition 5. To see that it is proper, let f e ai>v(F% q).

Set f-\_(x) := f(x) + xf /33!, then f1 e Fp q\ai,v(Fp q). Indeed, assume \f\<i ' ' ' first that the condition (b) is satisfied; we have fia)(0) = 1 (V|a| < v).

Assume now that the condition (c) is satisfied; we have Qjfi = Qjf (Vj e Z), then f[a)(0) = 1 + ^(Qjh)(a)(0) (VM < The proof is

complete. □

We add the following remark:

Remark 4. In connection with Remark 1, we have the following assertions, where we principally refer to [7, Sect. 4] and [13, Sect. 3]:

• The mapping o\, 0 defined in (3) commutes with translations, however, ai>v (i = 2, 3) defined in (4)-(5) are not.

• If s — n/p R N0, the mappings o\,0 and a2,v, defined in (3)-(4), commute with dilations; o\,0(f) and a2,v(f) are the unique representatives of functions from . If s — n/p e N0 and 1 < p ^ 8, the mapping as,v defined in (5) does not commute with dilations.

• If s — n/p e N0 and 0 < p ^ 1, F^ has infinitely many dilation commuting realizations a := â + XI ^a(')x", where â := o\,0

\a\-v

or a2,v (see (3)-(4)), and (Ca)\a\=v is a family of continuous linear functionals on satisfying Ca o h\ = X—vCa (VX > 0).

2.2.3. Realizations of Besov spaces. In the same way as in Definition 5, we define B^ q. Then we can take in Subsection 2.2.2 B and B

instead of F and F, respectively, by replacing in (2), Propositions 4-6, and Remark 4, the conditions 0 < p ^ 1 and 1 < p ^ 8 with 0 < q ^ 1 and 1 < q ^ 8, respectively.

2.2.4. Realized spaces and the integrability. We formulate some

Fp!q's properties related to the integrability, which will be useful in what follows. Here v = 0 if 0 < p ^ 1 and v = 1 if 1 < p ^ 8. For brevity, we set

<?o(f) := ,o(f),

and

<Ji (f) := !(f)= 2 Qif + S (Qif — Q>f (0)) (m e Z);

j'm j^m— 1

see (3) and (5), respectively.

We need to apply the following technical lemma (the Bernstein inequality) proved, e.g., in [21, Rem. 1.3.2/1].

Lemma 1. Let 0 < p 4 q 4 8 and a e NQ . There exists a constant c> 0, such that

|| fPaq\\q 4 cff)a\+n/P-n/l\\ j'\\p

holds, for all R > 0 and all f, such that suppf ç : |£| 4 R}.

Note that all following properties are also valid by changing F to B with the necessary modifications.

Proposition 7. Let either (0 < p < 8) or (p = 8 and q = 2). Then

it holds Fn{q ^ L'°c x B8 8. In the case 0 < p <8, the embedding is proper.

Proof.

Step 1: proof of the inclusion. Let f e F^^. We have f—(rv([/]8) e Vv, ( v = 0,1). As L 8 ^ L11oc x B8 8 and ^ L8, it suffices to show a,([f]œ)eL10c xB8,8. '

Substep 1.1: proof of a, ([/]») e B8,8.

The case: p 4 1. We split a0([/]8) as <71 + g2, where <71 := Qjf and 92 := Z Qjf. Since [f]m e Fpjq , Fp{ ^ B° œ and SoQi = So(Qif), we

j40

have

\\S05,1 }8 4 1p\\ 1 \Qi/*\\œ 4 c|[/]œ\ijo

J8, 8

Also, as Fp{ ^ B8^ B8, 1, we get

\S092lœ 4 1p\\1 ^ \Qjf ¡8 4 c\[/]8\bo j40

On the other hand, as QkYiQj " Qk(Qk—1 + Qk + Qk+1) since

QkQj = 0 if ^ ' 2,

we have

\\QkMUU))\U 4 c\\[fUWq ^ (Vk e N).

Thus, all these facts yield \ \ <r0([/]8) \\ b8 8 <8, see Definition 1(i).

The case: 1 <p<8. We have [daf]8 e Fp,{q~1 ^ B818 (I = 1,...,n); here the integer associated with Fp/q 1 is v = 0. Then we split a0([dif]8)

as in the preceding decomposition gi + g2 by taking dfj instead of f. Thus, using Lemma 1, we obtain

\\So9i^ p||i\\Qi(Bef)\\ 8 ^ C| [f ]8 \\ B0

\ \ Sog2\\8 ^ \ \ VHi2\\ Qj(Bif)\\ 8 ^ C1 \ \ [f ] 8 \ \ /, 2 ^ c2 \ \ [f ] 8 \ \ B0

KO ' KO '

and

2 k\\Qk(a0([BifU))\\8 ^ c\\[fU\\¿om,c (@k e N). Hence, ao([Bef] 8) e B—8. "We use the formula:

Be o = a( 1 q+ o Be (I = 1,...,n),

proved in [7, Prop. 4.6]. We get Be o ox([/]8) = &o([Bef ] 8) e B8 8, which implies ([f ^ e B°8J 8.

The case: p = 8 and q = 2. We have [Bef]8 e F—2 ^ B8'8 (1 = 1,... ,n), where the integer associated with F812 is v = 0. Then we continue exactly as in the preceding case.

Substep 1.2: proof of av([f ]8) e Ll^c. The case 0 < p <8 can be done as, e.g., in [2, pp. 29-30]. Then, we assume that p = 8 and q = 2. We write ax([f]8) = 93 + 94, where g3 := Yj Qjf and

j'rn

g4 := X (Qjf — Qjf (0)), where m e Z is at our disposal, cf. Proposi-

j^m—1

tion 5. By Lemma 1, we first have

n

\ \ VQj f \ \ 8 ^ C^Yj \ \ B*(Qi f ) \ \ 8 ^ °223 \ \ Qj f \ \ 8 ^

^ C2* \\ [f]8j\\¿0m m, (@j e Z). (6)

We choose m ^ 0, then clearly

Mx)l ^ ClM £ \ \ VQ3f \\8 <

j^.m—1'm^.O

^ C2|x| \ \ [f]c8 \ \ ¡90 V2J ^ C3|x| \ \ [f]c8 \ \ ¡90 ,

1 J M^co, co 1 1 11 L J "^co,c

KO

hence g4 e . We turn to g3. Let Q be a compact set in Rra and let x0 a fixed point in Q; there exists an integer k := k(Q) < 0, such that Q is

contained in the ball B(x0, 2 k) c Rn centered at x0 of radius 2 k. Then we choose m := k < 0, and obtain

\g3 (x)\dx J \Y,Q3f(x)f dx)1/2. (7)

H B(xo, 2- k)

Now, since in the definition of F8 q we can replace the dyadic cubes Pk,^ with the balls Bk in Rn of radius 2~k, we continue as in [5, p. 153] (see also VII.13, p. 147 in this reference) to obtain that the right-hand side of (7) is bounded by c||[/]811 po , where c := c(Q) > 0.

Step 2. To prove that the embedding is proper, it suffices to test the locally integrable function u(x) := elXl, x := (x1,...,xn) e Rn. Since L8 ñ B8,8, it holds u e B8,8. An easy computation gives Qku = '"f(2~k, 0,..., 0)u for all k e Z, then Q0u = u; recall that = 1 in the annulus 3/4 4 \£\ 4 1. Now it is clear that [u]8 R Fp/q for any 0 <p <8. Indeed, if [u]8 e Fp/q then \\Q0u\\p < 8, which is impossible. □

Remark 5. We have F8 2 ñ L2oc; the proof is similar to that given for Proposition 7. Also, concerning Proposition 7 in the case p = 8 and q = 2, see Corollary 1 below.

Proposition 8. There exists a constant c > 0, such that the following estimate

Q J\Sk f (x)\dx 4 C||[/]»\\pn,v

Q

holds for all f e FnJvq, all k e Z, and all cubes Q in Rn.

Proof. By Proposition 7, there exists a constant c > 0, such that it holds:

\g(x)\ dx 4 c||[sr]8\ \ prv , @g e FnJvq.

\x\41

Let x0 e Rp and r >

0. Let f e fnp/Pq. Apply the last inequality to the function g := f(r(^) + x0). By homogeneity (see (P5)) and translation invariance of | | • | | ¿,n/P, it holds ||[/]8|Ln/p = ||[/(r(-) + x0)]8||^/p. Thus, we

"P, Q " P, Q " P, Q

have the existence of a constant c > 0 independent of x0, r, and f, such that

r-p J \f(x)\dx 4 c||[/]8|^P. (8)

Ix-xol^r-

We also have

r—nj \Skf(x)\dx ^ 2knj \T—1p(2ky)\(r—nj \f(u)\du)dy. (9)

\x—xo\^r Rn \u+y—xo\^r

Using (8), we see that the right-hand side of (9) is bounded by c1|| [f ]8 | | ^/p, where c1 := c||T —V||1. Let now Q be a cube in Rra, and let B be the smallest ball in Rra containing Q; then \B\ = c\Q\, where c := c(n) > 0. By the arbitrariness of x0 and r, applying (9) with B, and using ^ c\B\ — 1 ..., the result follows. □

Proposition 9. Let either (0 < p < 8) or (p = 8 and q = 2). If f e Fp{q, then it holds f (1 + \x\n+1)—1 \f (x)\ dx < 8.

Proof. Let f e FnJvq. As observed before, f — av([/]8) is equal to 0 if 0 < p ^ 1 and to a constant c if 1 < p ^ 8 (recall v = 0,1), and since $ (1 + \x\n+1)—1 dx < 8, then it suffices to give a proof for functions

Rn

f e av(Fpq).

Step 1: the case p > 1. As in the proof of Proposition 7 (Substep 1.2), f can be written as gs + g4, where gs :=X Qj f and g4 := X (Qj f — Qj f (0)),

j'm j^m—1

with m e Z that will be chosen later. We set

Uz := J (1 + \x\n+1)—1 \gz(x)\dx (i = 3, 4).

Rn

Estimate of U3. Assume that p <8 and choose m := 1. We introduce p1, such that max(1,p) < p1 <8, and set p[ := p1/(p1 — 1). By Holder's inequality, it holds

Us ^ £ I I Qj f 11 Pi ( [(1 + Wn+1)—p/l dx)11"'1 ^

^ ci \1 UU \1 Èmn V 2^ c2\ |[/]œ|\Èn/n

-°üi .œ ' * .co

-'pi ,co ' * Pi ,cc

1 j>i 1

we finish by using the embedding Fff/q ^ F£p\,8. Assume now p = 8. We have

U3 < £ ( i (1 + Mra+1)—2 dx)1/2 ( [ | £ Qj f (x)|2 dx)12 (10)

^eZn Pi, Pi, r'm

We also have

( f f(x)fdx)1/2 4 c ll[/]»l\f8 ,2, (11)

suP | 7j (x

V j'm

which is proved in [5, p. 153]. Indeed, in this reference the author used the balls instead of dyadic cubes, but if x e P1tp, then \x — 2-1^\ 4 2-1y/n< 2n, hence P1,^ is embedded in B(2-1^,, 2n): the ball in Rn centered at 2-1^ with radius 2n. Then we choose m := —n and obtain

Xj Qj"

2

Qif (x) dx 4 | | J] Qif (x)

n j'm , , j'-n

Pi,» B(2~V, 2-(-"))

2

cf. the formula (7) and the sentence just after. By inserting (11) into (10), and taking into account, first, that for x e P1,p., 1 + \n\n+1 4 c(l + \x\n+l) with a constant c := c(n) > 0 and, second, (1 + \p\n+1) 1 < 8, we

then get U3 4 c||[/]®\^o 2.

Estimate ofU4. Let 0 <b < 1. Using the estimate (6), then we have

\g4(x)\ 4 21-b ^ \ \ Qif \\\Qif (x) — Qjf (0)\6 4 (m : = 1 or m :=—n)

j4m— 1

4 C1\x\b \ Qjf \\8-b\\VQjf \\8 4

340

340

4 C2|x|fo l l [f]8 11 ^ 8 2 2jb 4 C3 |x|fc l l [f]8 11 ^,

Using the embedding F^q ^ B°8 8 and \x\6(1 + \x\n+1) 1dx < 8, we

finish.

Step 2: the case 0 < p 4 1. Let p2 be a number such that 1 < p2 <8. By the embedding F'pjq ^ FpiPq , and the fact that f e C0 implies dfj e C0

(I = 1,..., n), we have f e F^^q; recall that the integers associated with the spaces F^q and F^q are v = 0 and v = 1, respectively. Thus, an application of Step 1 with p2 instead of p gives the result. □

2.3. Definition of the space BMO. The space BMO is the set of

all f e Lloc such that

\ \ f \\bmo := sup \Q\-1 \f(x) — mQf \ dx < 8, Q J

Q

where the supremum is taken over all finite cubes Q in Mra; example of such a function: x ^ log \x\. We note that in connection with the definition of the Triebel-Lizorkin space, we cannot identify BMO with the set of functions f such that (X \Qjf\2)112 is bounded, see again [10, p. 70] or

jeZ

[5, p. 154]. We also note that BMO has several properties, in particular:

Proposition 10. Let 1 ^ p <8. Every element f of BMO belongs to Ll°c and satisfies (i) of Proposition 1.

'0 8, 2-

Proof. See [18, IV. 1.3, p. 144]. □

3. Proof of Theorem 1. Step 1: proof of (i). Let f e F\ Remark 5 and Proposition 9 allow application of Proposition 1; then f e BMO. Hence, the embedding in one direction is obtained. Conversely, let f e BMO. By Propositions 1 and 10 we have [f]8 e F'c

0

8, 2-

We also have f — a1([f]8) e V8, where a1 := a3> 1 (see (5)). By condition (1 + \x\n+1)—1\f (x)\ dx < 8 and Proposition 9 applied to a1([f]8)

(since a1([f]8) e F^2), we obtain f — a1([f]8) " c e C; indeed, any non-constant polynomial G satisfies (1 + |x|ra+1)—1|G(x)| dx = 8 (an

easy exercise). The desired result follows.

Step 2: proof of (ii). Let us begin with some preparations. By the embedding Fp/q ^ Fp/^ with max(p, 1) < p1 < 8, the estimate \ \ f \\bmo ^ ^l/Wpn/p! (see (1) and the comment that follows just after) and

the fact that \ \ f \\p + \ \ [f ]8 \ \ ¿,n/p is an equivalent quasi-norm in FP1^, see, e. g., [15, Prop. 2.5], for all 0 < p <8 it holds

\ \ f\\bmo < c(\\f\\p + \\[/]8\\jp,i,p), @f e Fp/p.

In this inequality, replace f by h\f for any A > 0. Using the property (P5), the fact that \ \ h\f\\bmo = \\f\\bmo, and by letting A ^ 0, we get

\ \ f \\BMO < 4 UU\pn/p , @f e F^/p. (12)

Now we turn to the embedding, and limit ourselves to 1 < p <8. The case of 0 < p ^ 1 can be obtained as in Step 2 of the proof of Proposition 9.

Take f e Fnv'vq (recall that Vm $ Fp/p if m ' 2) and set

fk := £ Q,f for k = 0,1,....

f — k

The sequence (fk) has the following properties:

• by [8, Prop. 4] it holds that \ \ [fk]8 \ \ pn/p ^ c||[f]8\\pn/p for all k e No,

e Lp; indeed

\[f]8\\Pn/P ( £ 2-*/p) € c2kn/p\\[/]8\\pm, 1 p,8 \ ' * / 1 p,8

P € \\[J ]8\\p^/p\ ¿J 2 ' " j € C 2 ' '\\[J ]8\\pn/p .

By applying (12) to fk, we get

\ \ fk\\bmo € 4 [fU\pn/p , @k e No . (13)

гp, q

As f = X Qj f in S'8J, it holds that f - fk = X Qj f is in S'8. Then,

jeZ j<-k

by [8, Prop. 4] again, we obtain

[f -fk ]8 \ \ pn/p = II £ Qj f

1 p, q

■ , €c 1 p

1/1

@k e No

j<-k "p,q j<-k ' P

with the change sup 2:'n<lp\Qjf \ in the inner norm \ \ • \\ P when q = 8. j<-k

Set

Vk := ( £ (2np\Q,fI)*)PP\ k = 0,1,...,

<- k

(resp. taking sup ... if q = 8). The positive sequence (vk) satisfies

<- k

lim vk = 0 a. e. on Rn, since

k^8

( £ {2n/p\Q3 f (x)\y)P/q € \\[/]»\^ ( £ 2^n'P\PPq € <- k <- k

€ c2-kn\\ [f]8\\P,n/p,

(resp. we have the bound c 2 \\ [f]8 \\ Pn/p if q = 8). Also, as the as-

Pp,8

sumption on f yields vk € ( Yi(2j'np\Qjf\)9)pq e L\ (resp. also if q = 8).

jeZ

Thus, by dominated convergence theorem applied to the inequality

[ f - fk]« I I *L/p 4 c vk (x)dx,

we deduce that lim \ \ [f — fk]8 \ \ ¿,n/P = 0, which yields

fk —f in F^l,

(see Definition 5). On the other hand, by Proposition 7, we get fk — f in L11oc. Then, classically, there exists a subsequence (fkj) of (fk), such that

fkj — f a. e. on Rn as j — 8.

Thus, for all finite cubes Q in Rn, we have fkj — mQf — f — mQf a. e., which implies

\ f(x) - mQf \ dx = lim \ fk (x) - mQf \ dx 4 J ]

Q Q

4 lim \ fkj (x) - mQ fkj \dx + \Q\ lim \mQf - mQ fkj\. (14)

I J—>8

Q

By the Fatou lemma and using (13) with (fkj), we get

liminf \ fkj (x) - mQfkj \ dx 4 \Q\liminf || fkà ^mo 4

j—8 j—8 Q

4 c\QU| [f]811pn/P. (15)

1 P,Q

For the last term in (14), it is clear that

\mQ f - mQfkj \ 4 \Q\-1 J\S-kj-i f(x)\ dx>

Q

which gives, in view of Proposition 8, \mQ f -mQfkj \ 4 c\\[f]811pn/p for all j e N0 and all finite cubes Q in Rp. Inserting both the last estimate and (15) into (14), dividing by \Q\ and taking the supremum over all Q, we get | | f 11 bmo 4 c\\[f]811 pn/p; which is the desired result.

We prove that the embedding is proper. Let f (x) := log \x\, x e Rn, and let pi > p. The function f coincides with c|£\-n on Rn\{0}. Set P(g) := \£\-n7(£), which satisfies ^ e S8 and | | Qjf}}Pl = 2-jn{pi\\Pl. This implies | | [f ]8| | ¿n/P1 = 8 since p <8. But as Fp{ ^ B^p, 0 < q ^ 8,

we have [f ]. R Fp{. □

4. Some remarks.

4.1. BMO functions and the Besov spaces. An application of the main result yields the following assertions for the inhomogeneous Besov spaces and their homogeneous and realized counterparts.

Corollary 1. The embedding BMO ^ B. 8 is proper.

Proof. The inclusion follows by Proposition 7 and Theorem 1(i). To prove the embedding is proper, it suffices to observe that B. 8 $ L^. Recall that B.>q c Lx°c if and only if 0 < q ^ 2, cf. [17, Thm. 3.3.2]. □

In [19, Thm. 2(b)], it was proved that if s > 0 then

| | Jsf 11 < 4f 11 bmo for all f e BMO x (Li + L.),

Js is defined in the Introduction; this implies BMO x(Li + L8) c B8,8 ?8 8 isomorphically onto Bs8 8

since Js maps B8,8 isomorphically onto Bs8, 8 and the expression I I ^sfIIB°œ,œ is an equivalent to | | f \\8, see e.g., [16, p. 67] or [21, Thm. 2.3.8]. Hence, the improvement given now by Corollary 1 removes the assumption L\ + L8 in this embedding.

Corollary 2. It holds that È8,2 ^ BMO ^ È8,8.

Proof. This is immediate by B8,2 ^ F8 2 ^ B8,8, which can be easily obtained by B8 2 ^ F8 2, the property (P6) and the definition of v, see (2). We note that the second embedding in this corollary is given in [12, Thm. 10.1]. □

In the same way, in [16, p. 169, Lines 2-3 p. 252] with a small modification, and in [5, p. 154], it was proved: if f e BMO, then [f ]8 e B8 8. This can now be obtained easily by applying Theorem 1(i) and the property (P6).

4.2. Applications related to the Riesz operator. In order to investigate actions of the Riesz operator X8 (defined as X/ f := T-i(\£\-// f ),

ft e R) on BMO, we recall that Tp takes S8 to itself (an easy proof). It is therefore consistent to define Tp : S8 ^ S'8 as: if f e S'8 then

(T8f, <p) := {h,Tn<p)

for all p e S8 and all f1 e S', such that [f1]8 = f.

4.2.1. The F-spaces. We have Tp maps Fp, 0 isomorphically onto

Fp^ß and \\Ißf\\ps+ß „ ||f\\ps , see, e.g., [21, Thm. 5.2.3/1]. Then we get:

Proposition 11. If f e Fpq then T88([f] e Fp8q8, and there exists a function g e FppP, such that Tp([/]8) = [g]8 in S'8. In particular, if f e BMO then Tp([f]8) e F8,2.

Proof. It suffices to take g := ai>v(Tp([/]8)), where ai>v is defined in Proposition 5. □

On the other hand, using the embedding F^2 ^ F02 we obtain:

Proposition 12. Let 0 < p < q < 8. Put ft := n/p — n/q. If f e 2,

then there exists a function g e F0 2, such that Tp([/]8) = [g]8 in S'8

and | | g|| ~0 4 c|| f|| ~0 . The positive constant c depends only on n, p, and

^ 1, 2 ^ p, 2

Q.

Proof. The existence of g is obtained as in Proposition 11. For the estimate we apply the above embedding. □

Remark 6. Clearly, the problem now arises: to see if Tp takes Fp,q to Fp^Jp; it seems to be open.

4.2.2. The B-spaces. An analogue of Proposition 11 holds in the case of Besov spaces. We have

B^2 ^ Tp(BMO) cB8,8 (P e R). (16)

These inclusions are proved in [20, Thm. 3.4], at least for ft > 0, with spaces B8,q endowed with the functional

MS

dh \ l/i

S,mU) := ( {\h\-s\\KfW»)9

\h\'

for m e N and 0 < s < m, where A™ is the m-th difference operator (A* f := T_hf — f and A™ := A{ o A^1, m = 2, 3,...). Here the spaces

138 are defined modulo polynomials of degree ^ s, cf. the last line in [20, p. 552]. Note that in B.as defined by the LPd (cf. Definition 2), | | • 11 Bs and Msgm(-) are not equivalent, since for any polynomial f of degree ^ m, 11[f]811b^ = 0 while Ms^m(f) = 8 (e.g., f(x) := xi then A^f (x) = h\). However, with q = 2 or 8 and s > 0 we have:

(i) u = [s] + 1 for B.q,

(ii) | | . 11 Bs and M's'm() with m ^ v, are equivalent in B, see [14, Thm. 1.1]. 8,9

In this sense, we can view (16) as follows:

• if f e BMO, there exists a function g e £>8,8 such that Iß([/]8) =

= [gU in S8,

• if f e B82, then I-ß([f] 8q e B8, 2; as B8, 2 ^ i>8, 2, by ProPosi" tion 11 we obtain the existence of a function h e BMO, such that I-ß([f]») = [h]8 in S8, this implies [f]8 = Iß([h]8).

Remark 7. As in Remark 6, we also have Iß : Bp q ^ Bsp+ß as an open question, cf. [15, Rem. 2.11].

Acknowledgment. We thank both the General Direction of Education and Training MESRS (Projet de recherche C00L03UN280120220005), and the General Direction of Scientific Research and Technological Development DGRSDT Algeria. We also thank the referees for their valuable comments and suggestions.

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Received November 03, 2023.

In revised form, February 28, 2024.

Accepted March 23, 2024.

Published online May 07, 2024.

Bochra Gheribi

Laboratory of Functional Analysis and Geometry of Spaces,

Faculty of Mathematics and Computer Science,

University of M'sila,

PO Box 166 Ichebilia, 28000 M'sila, Algeria

E-mail: bochra.gheribi@univ-msila.dz

Madani Moussai

Laboratory of Functional Analysis and Geometry of Spaces,

Faculty of Mathematics and Computer Science,

University of M'sila,

PO Box 166 Ichebilia, 28000 M'sila, Algeria

E-mail: madani.moussai@univ-msila.dz