AND LIZORKIN-TRIEBEL SPACES

AND LIZORKIN-TRIEBEL SPACES

DOUADI DRIHEM AND MADANI MOUSSAI

Received 1 February 2005; Revised 22 February 2006; Accepted 4 April 2006

Under some sufficient conditions satisfied byF-space of Lizorkin and Triebel andB-space of Besov, we prove some embeddings of typesF·BF,F·FF, andB·BB.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction and preparations

In Besov spaces and Lizorkin-Triebel spaces, this paper is concerned with proving some embeddings of the form

F·B F, F·F F, B·B B, (1.1)

whereF andB, with three indices, will denote the Lizorkin-Triebel spaceF^s_p,q and the Besov spaceB^s_p,q, respectively. The different embeddings obtained here are under certain restrictions on the parameters.

In this introduction, we will recall the definition of some spaces and some necessary tools. In Sections2and3, we give the first contribution of this work. The theorems of Section 2will treat the caseF·BF where the first theorem is a generalization of the results of Franke [4, Section 3.2, Theorem 1, Section 3.4, Corollary 1] and Marschall [7].

The second theorem is in the sense of Johnsen’s works (see [5]).Section 3will contain a treatment of the embeddings of the typesF·FF andB·BBwhich presents an improvement of [3].

In the sense of [5, Theorems 6.5, 6.11], some limit cases are considered inSection 4, which constitute the second contribution of this paper.Section 5is an application of our results to the continuity of pseudodifferential operators on Lizorkin-Triebel spaces.

We will work on the Euclidean spaceRⁿ. If f ∈᏿, the Fourier transform is defined by the formula

Ᏺf(ξ)= f(ξ)=

Rⁿ f(x)e^−ix·ξdx ξ∈Rⁿ

(1.2)

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 76182, Pages1–18

DOI10.1155/IJMMS/2006/76182

andᏲ⁻¹f denotes the inverse Fourier transform of f; as usualᏲandᏲ⁻¹are extended from᏿to᏿.

Consider a partition of unity ψ(ξ) +

∞ j=1

ϕ2^−jξ=1 ξ∈Rⁿ

, (1.3)

whereϕ,ψ∈C^∞ are positive functions such that suppϕ⊂ {ξ∈Rⁿ: 1≤ |ξ| ≤3} and suppψ⊂ {ξ∈Rⁿ:|ξ| ≤3}. We define the convolution operatorsQj andΔkby the fol- lowing:

Q_jf =Ᏺ⁻¹ψ2^−j·

∗f (j=1, 2,. . .), Δkf =Ᏺ⁻¹ϕ2^−k·

∗f (k=0, 1,. . .),

(1.4)

and we setQ0=Δ0. Thus we obtain the Littlewood-Paley decomposition f =_∞

j=0Δjf (convergence in᏿).

Let us now recall the definitions ofF_p,q^s andB^s_p,q, where the general references include [1,9–13].

Definition 1.1. Letγ >0,−∞< s <∞, 0< p <∞(resp., 0< p≤ ∞), and 0< q≤ ∞. The spaceL^γp(_q^s) (resp.,^s_q(L^γp)) is the set of the sequences{fk}k∈N⊂Ssuch that suppfk⊂ {ξ∈Rⁿ:|ξ|< γ2^k}and

fk

k∈N|L^γp

_q^s= 2^ksfk

k∈N|Lp

q<∞, resp., f_kk∈N|_q^sL^γp= 2^ksf_kk∈N|_qL_p<∞

.

(1.5)

Definition 1.2. (i) Let 0< p <∞, 0< q≤ ∞, and−∞< s <∞, then F_p,q^s = f ∈᏿: 2^ksΔkf_k∈N|Lp

q<∞

. (1.6)

(ii) Let 0< p,q≤ ∞, and−∞< s <∞, then

B^s_p,q= f ∈᏿: 2^ksΔkf_k∈N|_qL_p<∞

. (1.7)

Remark 1.3. We introduce the maximal function

Δ^∗k^,afx=sup

y∈Rⁿ

Δkfx−y

1 +2^k|y|a (1.8)

for allx∈Rⁿ, f ∈᏿,a >0, andk=0, 1,. . . .Then, inDefinition 1.2(i) (resp., (ii)), we can replace Δkf byΔ^∗_k^,af witha >(n/min(p,q)) (resp., a > n/ p), (cf. see [13, Theo- rem 2.3.2]).

The product f ·gis defined by f ·g=lim

j→∞Q_jf ·Q_jg ∀f,g∈᏿ (1.9) if the limit on the right-hand side exists in᏿(see [10, Section 4.2]), and we have

Δk(f ·g)= ∞ j,=0

Δk

Δjg·Δf=

Πk,1+Πk,2+Πk,3

(f,g), (1.10) where

Πk,1(f,g)=ΔkΔkf·Q_k+1g, Πk,2(f,g)=Δk

Q_k+1f·Δkg,

Πk,3(f,g)=^∞

j=k

Δk

Δjf ·Δjg, (1.11)

withΔk=_k+4

j=k−2ΔjandΔk=_k+1

j=k−1Δj.

In the below proofs of the different cases of type (1.1), written as G1·G2G3, to see f ·gbelongs toG3, (f ∈G1,g∈G2), it suffices to an estimate of terms of the form {Πk,i(f,g)}k∈N|L^γp(^s_q)and{Πk,i(f,g)}k∈N|^s_q(L^γp),i∈ {1, 2, 3}.

Now we recall some lemmas which are useful for us.

Lemma 1.4. (i) Let −∞< si<∞, 0< pi<∞(resp., 0< pi≤ ∞), and 0< qi≤ ∞(with i=0, 1). If

s0> s1, p0=p1, (1.12)

or

s0≥s1, s0− n

p0 =s1− n p1

q0≤q1for Besov space, (1.13)

then it holds

F^sp⁰0,q0 Fp^s11,q1

resp.,B^sp⁰0,q0 B^sp¹1,q1

. (1.14)

(ii) Let−∞< s,s_i<∞, 0< p, p_i<∞, and 0< q,q_i≤ ∞(withi=0, 1) such thats0− n/ p0=s−n/ p=s1−n/ p1. If

s0> s > s1, q0≤p≤q1, (1.15) or

s0=s=s1, q0≤min(p,q), q1≥max(p,q), (1.16) then it holds

B^s_p⁰0,q0 F^s_p,q B^s_p¹1,q1. (1.17)

(iii) Let−∞< s <∞, 0< p <∞(resp., 0< p≤ ∞), and 0< q≤ ∞. If s >n

p, (1.18)

or

s=n

p, 0< p≤1 (resp., 0< q≤1), (1.19) then it holds

F^s_p,q L_∞ resp.,B^s_p,q L_∞. (1.20) Lemma 1.5. Let 0< γ <1 and 0< q≤ ∞. Let{εk}k∈Nbe a sequence of positive real num- bers such that {ε_k}k∈N|_q =A <∞. Then the sequences δ_k=_k

j=0γ^k−jε_j and η_k= _∞

j=kγ^j−kεjbelong toq, and the estimate δk

k∈N|q+ ηk

k∈N|q≤cA (1.21)

holds. The constantcdepends only onγandq.

Lemma 1.6. Let 0< p≤ ∞andγ >0. Let{fj}j∈N⊂Lpbe a sequence of functions such that suppfj⊂ {ξ∈Rⁿ:|ξ| ≤γ2^j}. Then the estimate

Δkf_j|L_p≤c2^(j−k)ρf_j|L_p

k≤j <∞,ρ=max

0,n p⁻n

(1.22) holds. The constantcdepends only onn,p, andγ.

Lemma 1.7. Let 0< p <1 andγ >0. Let{fj}j∈N⊂Lpbe a sequence of functions such that suppfj⊂ {ξ∈Rⁿ:|ξ| ≤γ2^j}. Then the estimate

∞ j=0

fj|B^ρp,∞

≤c 2^jρfj

j∈N|Lp

_∞

ρ=n

p⁻n

(1.23)

holds. The constantcdepends only onn,p, andγ.

Lemma 1.8. Let 0< p≤q≤ ∞andγ >0. Then there exists a constantc=c(n,p,q)>0 such that for all f ∈Lpwith suppf⊂ {ξ∈Rⁿ:|ξ≤γ}, one has

f |L_q≤cγ^{n(1/ p−1/q)}f |L_p. (1.24)

ForLemma 1.4, we can see [11, Sections 2.3 and 2.8] and [12, Section 2.7].Lemma 1.5 follows from Young’s inequality in_q. The proof ofLemma 1.6is given in [4, Section 2.4, Theorem 1(iii)] andLemma 1.7in [7, Lemma 3]. For the proof ofLemma 1.8, we can see [14, Proposition 2.13], 1≤p≤q≤ ∞, it is the classical inequality of Bernstein.

2. Multiplication of mixed type

The following results give an extension of the sufficient hypotheses used in [5, Theo- rem 6.1].

Theorem 2.1. Let 0< p,p1,p2<∞, 0< q,q2≤ ∞,−∞< s <∞, andr >0 be such that

−r+ max

0, n p1

+ n p2−n

< s <min n

p1

,r

, 1

p ⁼ 1 p1+ 1

p2−r n, 1

q2≥ 1 p2−r

n, r < n p2

resp.,r= n p2

.

(2.1)

Then it holds

F^s_p1,q·B^r_p2,q2 F^s_p,q resp.,F^s_p1,q·

B^{n/ p}p2,_∞²∩L_∞ F^s_p1,q. (2.2) Corollary 2.2. Under the hypotheses ofTheorem 2.1. Ifr < n/ p2(resp.,r=n/ p2) then it holds

F_p^s_1,q·F^r_p2,q2 F_p,q^s resp.,F_p^s_1,q·F^{n/ p}p2,q²2 F_p^s_1,qforp2≤1. (2.3) Furthermore, in particular, if 1< p1<∞andr > n/ p1+n/ p2−n, can be takens=0 in (2.3).

Proof. SinceF^r_p2,q2B^r_p2,t witht=(1/ p2−r/n)⁻¹, we obtain the first embedding. How- ever, the second embedding follows fromF^{n/ p}p2,q²2B^{n/ p}p2,∞²∩L_∞. Remark 2.3. InCorollary 2.2, whenr < n/ p2(resp.,r=n/ p2), we obtain [10, Theorems 4.4.3/2(21) and 4.4.4/2(16) (resp., Theorems 4.4.3/2(22) and 4.4.4/2(17))]. The particu- lar cases=0 presents a complement of [10, Theorem 4.4.4/4(i)].

To proveTheorem 2.1, we need the following lemma.

Lemma 2.4. Let 0< p <∞anda > n/ p. Then there exists a constantc >0 such that Q^∗_j^,ag_j∈N|Lp

_∞≤cg|F_p,2⁰ , (2.4) for anyg∈F⁰_p,2.

Proof. First, we define the maximal function ofQjg, of Hardy-Littelewood type, by the formula

MQjg(x)=sup

r>0

B(x,r)1

B(x,r)

Qjg(y)d y, (2.5)

whereB(x,r) is the ball centered atxof radiusrand|B(x,r)|denotes its measure. Next, lett >0 satisfyn/a < t < p. From [13, Theorem 1.3.1], we have

Q^∗_j^,ag(x)≤Q^∗_j^,n/tg(x)≤cMQ_jg^t(x)^1/t ∀x∈Rⁿ

. (2.6)

Then we obtain

sup

j∈NQ^∗_j^,ag|L_p≤c

sup

j∈NMQ_jg^t 1/t

|L_p=csup

j∈NMQ_jg^t|L_p/t

1/t

≤csup

j∈N

Q_jg^t|L_p/t

1/t

.

(2.7)

A proof of the last inequality may be found in [13, Theorem 2.2.2, page 89]. Now, it is easy to see that the last member of (2.7) is bounded by

sup

j∈N

Qjg|Lp

≤cg|F⁰_p,2. (2.8)

Inequality (2.8) follows from the equality between the local Hardy spaceshpandF⁰_p,2, (cf.

see [12, Section 2.2, page 37, and Theorem 2.5.8/1]).

Proof ofTheorem 2.1

Case 1 (r < n/ p2). (i) Estimate of{Πk,1(f,g)}k∈N. Since Πk,1(f,g)(x)=

Rⁿ

Ᏺ⁻¹ϕ(y)Q_k+1g·Δkfx−2^−kyd y

≤cQ^∗_k+1^,a1g(x)Δ^∗_k^,a2f(x) ∀x∈Rⁿ ,

(2.9)

whereQ_k^∗,a1andΔ^∗_k^,a2are defined as inRemark 1.3, we obtain 2^ksΠk,1(f,g)_k∈N|q≤csup

j∈N

Q^∗_j^,a1g 2^ksΔ^∗_k^,a2f_k∈N|q, (2.10)

wherea1anda2are real numbers at our disposal. We set 1/b=1/ p2−r/n. The left-hand side of (2.10), inLp-norm, is bounded by

csup

j∈NQ^∗_j^,a1g|L_b 2^ksΔ^∗_k^,a2f_k∈N|L_p1q. (2.11) Choosea1> n/banda2> n/min(p1,q), then bothLemma 2.4and the embeddingB^r_p2,q2 F_b,2⁰ yield that (2.11) is estimated as desired.

(ii) Estimate of{Πk,2(f,g)}k∈N. Letu∈Rsuch that max

0, 1

p1−r n

<1 u<min

1 p1, 1

p1− s n

. (2.12)

We set

1 v⁼

1 p2+1

u, σ=s−n p+n

v, β=s− n p1+n

u. (2.13)

We have

^σ_pL^γv

L^γ_p^s_q, F^s_p1,q B^βu,p1. (2.14) For the first embedding of (2.14), we can see [4, Section 2.3, Theorem 3]. On the other hand, the H¨older inequality yields

2^kσΠk,2(f,g)|Lv≤c2^krΔkg|Lp2 2^kβ

k+1

j=0

2^−jβ·2^jβΔjf |Lu

. (2.15)

We set 1/q2=1/ p−1/ p1. Applying, successively, the H¨older inequality again in_p-norm andLemma 1.5, we obtain the boundcg|B^r_p2,q2f |B^βu,p1. So (2.14) andB^r_p2,q2B^r_p2,q2 give

2^ksΠk,2(f,g)_k∈N|Lp

q≤cg|B^r_p2,q2f |F^s_p1,q. (2.16) (iii) Estimate of{Πk,3(f,g)_k∈N. We first consider 1/ p1+ 1/ p2≤1. Letu∈Rsuch that

max

0, 1 p1−r

n, 1 p1−r+s

n

< 1 u< 1

p1

. (2.17)

We use the notationsv,σ, andβfrom (2.13).Lemma 1.6provides 2^kσΠk,3(f,g)|Lv≤c2^k(β+r)

∞ j=k

2^−j(β+r)·2^j(β+r)Δjg|Lp2Δjf |Lu. (2.18) A similar argument as above yields

2^kσΠk,3(f,g)_k∈N|p

Lv≤c 2^j(β+r)Δjg|Lp2Δjf |Lu

j∈N|p. (2.19) We set 1/q2=1/ p−1/ p1. By the H¨older inequality inp-norm, the right-hand side of (2.19) is bounded bycg|B^r_p2,q2f |B^βu,p1. Then we conclude the desired estimate by (2.14).

We now study case 1/ p1+ 1/ p2>1. Letu∈Rsuch that max

0, 1− 1

p2, 1 p1−r

n

< 1 u< 1

p1

. (2.20)

We employ the notationsυandσfrom (2.13). ByLemma 1.6, we obtain 2^kσΠk,3(f,g)|Lv≤c2^kμ

∞ j=k

2^−jμ·2^j(r+ρ)Δjg|Lp2Δjf |Lu, (2.21) whereρ=s−n/ p1+n/uandμ=s+r−n/ p1−n/ p2+n >0, therefore,

2^kσΠk,3(f,g)_k∈N|p

Lv≤c 2^j(r+ρ)Δjg|Lp2Δjf |Lu

j∈N|p. (2.22)

On the right-hand side of (2.22), we employ the H¨older inequality in p-norm (with 1/ p=1/ p1+ 1/q2),F^s_p1,qBu,p^ρ 1, andB^r_p2,q2B^r_p2,q2successively. Sinceσ > sandv < p, we can finish the proof of this case by applying, in the left-hand side of (2.22), embeddings (2.14).

Case 2 (r=n/ p2). We only estimate{Πk,1(f,g)}k∈N. It is sufficient to see that

2^ksΠk,1(f,g)≤cg|L_∞2^ksΔ^∗_k^,af (2.23)

witha > n/min(p1,q) and to take theLp1(q)-norm.

Theorem 2.5. Let 0< p,p1<∞, 0< p2,q≤ ∞,−∞< s <∞, andr >0 be such that

−r+ max

0, n p1+ n

p2−n

< s < r. (2.24)

If either of the following assertions is satisfied:

(i) 1/ p=1/ p1+ 1/ p2,

(ii) max(1/ p1,s/n) + max(0, 1/ p2−r/n)<1/ p <1/ p1+ 1/ p2, then it holds

F_p^s_1,q·B^r_p2,∞ F^s_p,q. (2.25) Corollary 2.6. Let p, p1,q,r,sbe as inTheorem 2.5and 0< p2<∞. If (i) or (ii) of Theorem 2.5is satisfied, then the embeddingF^s_p1,q·F^r_p2,∞F^s_p,qholds.

The proof ofCorollary 2.6is immediate becauseF^r_p2,∞B^r_p2,∞.

Remark 2.7. We note thatTheorem 2.5(i) whenp2= ∞is given in [4, Section 3.2, The- orem 1]. Also, we note thatCorollary 2.6is given in both [5, Theorem 6.1 withr=pin formula (6.6)] and [10, Theorems 4.4.3/1(7) and 4.4.4/1(7)].

Proof ofTheorem 2.5(i). NotingRemark 2.7, we only need to treat the part 0< p2<∞. (i) Estimate of{Πk,1(f,g)}k∈N. From (2.9) andLemma 2.4, we have

2^ksΠk,1(f,g)_k∈N|Lp

q≤cg|F⁰_p2,2f |F^s_p1,q. (2.26)

By embeddingsB^r_p2,∞B⁰_p2,min(p2,2)F⁰_p2,2, we obtain that the last term of (2.26) is bounded by the desired quantity.

(ii) Estimate of{Πk,2(f,g)}k∈N. The H¨older inequality provides Πk,2(f,g)|Lp≤c

2^−kr

k+1

j=0

2^−j(s−r)·2^−jr

g|B^r_p2,∞f |B^s_p1,∞. (2.27)

The hypothesiss < ryields

2^ksΠk,2(f,g)_k∈N|_min(p,q)L_p≤cg|B^r_p2,∞f |B^s_p1,∞. (2.28)

Using embeddings

^s_min(p,q)L^γp

L^γp

^s_q, F^s_p1,q B^s_p1,∞, (2.29) we obtain the desired result.

(iii) Estimate of{Πk,3(f,g)}k∈N. We setρ=s+r−max(0,n/ p−n). UsingLemma 1.6, we obtain

2^ksΠk,3(f,g)|Lp≤c2^−kr·2^kρ ∞ j=k

2^−jρ2^jrΔjg|Lp22^jsΔjf |Lp1

≤c2^−krg|B^r_p2,∞f |B^s_p1,∞.

(2.30) In this inequality, we take _min(p,q)-norm and we conclude the desired estimate us- ing (2.29).

Proof ofTheorem 2.5(ii). (1) Estimate of {Πk,1(f,g)}k∈N. We set 1/u=1/ p−1/ p1. As in (2.26), we have the boundcg|F_u,2⁰ f |F_p^s_1,qwhich, by the embeddingsB^r_p2,_∞ B^{n/ p}p2,u^2−n/uF_u,2⁰ , is estimated as desired.

(2) Estimate of {Πk,2(f,g)}k∈N. In part, for technical reasons, we prove this in three separate cases:

Case 1 (s <0). ByLemma 1.6, the H¨older inequality, andLemma 1.8, we have

Πk,2(f,g)|L_p≤c2^{(n/ p1+n/ p2−n/ p−r−s)k}g|B^r_p2,∞f |B^s_p1,∞. (2.31)

Since n/ p1+n/ p2−n/ p−r <0, we obtain an inequality of type (2.28) and finish the proof of this case using (2.29).

Case 2 (0≤s < n/ p1). We set 1/b=1/ p2+ 1/ p1−s/n. We continue with the following subcases.

Subcase 2.1 (r≤n/ p2andp≤b(ors≤n/ p2< randp≤b)). As inCase 1, we have Πk,2(f,g)|Lp≤cγk2^−krg|B^r_p2,∞f |B^s_p1,∞, (2.32)

where

γk=

⎧⎨

⎩

k+ 2 ifp=b,

1−2^{n/b−(n/ p)−s−1} ifp < b. (2.33)

Now since{2^k(s−r)γk}k∈N∈min(p,q), we conclude the desired conclusion using (2.28) and (2.29).

Subcase 2.2 (r≤n/ p2andp > b(ors≤n/ p2< randp > b)). Letu >0 satisfy max

0,1

p⁻ 1 p2

<1 u< 1

p1−s

n. (2.34)

We employ the notationsv,σ, andβfrom (2.13). We have

Πk,2(f,g)|Lv≤cg|B^r_p2,∞f |Bu,^β∞2^−k(β+r). (2.35) Since{2^{−k(β+r−σ)}}k∈N∈p, we can finish the proof of this case using (2.14).

Subcase 2.3 (n/ p2< s < r). We have only casep < bneeds to be verified. As in (2.32), we immediately obtain the result.

Case 3 (s≥n/ p1). We have the following subcases.

Subcase 3.1 (p < p2). We set 1/v=1/ p−1/ p2. Observe that 2^ksΠk,2(f,g)|Lp≤c2^krΔkg|Lp2

2^k(s−r)

k+1

j=0

2^{j(n/ p1−n/v)}Δjf |Lp1

≤cg|B^r_p2,∞f |B^{n/ p}_p1,∞¹ 2^k(s−r)

k+1

j=0

2^−jn/v

.

(2.36)

Then, we calculatemin(p,q)-norm and conclude the desired estimate by the fact that

2^k(s−r)

k+1

j=0

2^−jn/v

k∈N

∈_min(p,q). (2.37)

Subcase 3.2 (s > n/ p1andp_≥p2). It suffices to apply both embeddingB^s_p1,∞B^{n/ p}_p1,1¹and (2.29) to

Πk,2(f,g)|Lp≤cΔkg|LpQk+1f |L_∞

≤c2^{k(n/ p2−r−n/ p)}g|B^r_p2,_∞f |B^{n/ p}_p1,1¹.

(2.38) Subcase 3.3 (s=n/ p1and p≥p2). We chooseα >0 such thatε=α−n/ p+n/ p1+n/ p2− r <0, then it suffices to apply (2.29) to

2^{kn/ p1}Πk,2(f,g)|L_p≤c2^kεg|B^r_p2,∞f |B^{n/ p}p1,∞^1−α. (2.39) (3) Estimate of{Πk,3(f,g)}k∈N. The proof of this case is obtained similarly to the proof ofTheorem 2.1just by replacing (2.17) and (2.20) with

max

0,1 p⁻

1 p2, 1

p1−r+s n

< 1 u< 1

p1, max

0, 1− 1

p2,1 p⁻

1 p2

< 1 u< 1

p1,

(2.40)

respectively.

3. Multiplication of typesF·BandB·B

The next theorem presents a continuation of [3], [5, Theorem 6.1] ,[6], and [7, Section 5].