Scholarly article on topic 'Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences'

Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences Academic research paper on "Mathematics"

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Academic research paper on topic "Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences"

Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 328908,24 pages doi:10.1155/2012/328908

Research Article

Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences

Douadi Drihem

Department of Mathematics, Laboratory of Pure and Applied Mathematics, M'Sila University, P.O. Box 166, M'Sila 28000, Algeria

Correspondence should be addressed to Douadi Drihem, douadidr@yahoo.fr Received 10 December 2010; Accepted 10 February 2011 Academic Editor: Hans Triebel

Copyright © 2012 Douadi Drihem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present characterizations of the Besov-type spaces BpT and the Triebel-Lizorkin-type spaces Fpq by differences. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking t = 0.

1. Introduction

The Bpq spaces and Fp'Jq spaces have been studied extensively in recent years. When t = 0 they coincide with the usual function spaces Bp,q and Fpq, respectively, studied in detail by Triebel in [1-3]. When s e R, t e [0, to) and 1 < p , q < to the Bp'Tq spaces were first introduced by El Baraka in [4, 5]. In these papers, El Baraka investigated embeddings as well as Littlewood-Paley characterizations of Campanato spaces. El Baraka showed that the spaces Bp'q cover certain Campanato spaces, studied in [6, 7]. Later on, Drihem gave in [8] a characterization for BppJq spaces by local means and maximal functions. For a complete treatment of BppJq spaces and Fpq spaces we refer the reader the work of Yuan et al. [9]. Yang and Yuan, in [1012], have introduced the scales of homogeneous Besov-Triebel-Lizorkin-type spaces Bpq and Fp'q (p = to), which generalize the homogeneous Besov-Triebel-Lizorkin spaces Bp'Tq, Fp'Jq and established the relation between Fps'q and Qa spaces. See also [13] for further results.

Our main purpose in this paper is to characterize these function spaces by differences. These results are a generalization of some results given in [17], and [9, Chapter 4, Section 4.3]. All these results generalize the existing classical results on Besov spaces and Triebel-Lizorkin spaces by taking t = 0.

The paper is organized as follows. Section 2.1 collects fundamental notation and concepts and Section 2.2 covers results from the theory of these function spaces. Some

necessary tools are given in Section 3. These results are used in Section 4 to obtain the characterization of Bp'Tq spaces and Fp'Jq spaces by differences.

2. Preliminaries

2.1. Notation and Conventions

As usual, Rn the n-dimensional real Euclidean space, N the collection of all natural numbers, and N0 = N u {0}. The letter Z stands for the set of all integer numbers. For a multi-index a = (ai,...,a„) e N", we write |a| = ai + ••• + an and Da = d^/dx? ■■■dx"n. For v e Z, let Bv be the ball of R" with radius 2-v and v+ = max{v,0}. The Euclidean scalar product of x = (x1,..., xn) and y = (y1,..., yn) is given by x • y = x1y1 + ••• + xnyn. We denote by |Q| the n-dimensional Lebesgue measure of Q c Rn. For any measurable subset Q c Rn the Lebesgue space Lp(Q), 0 <p <g consists of all measurable functions for which

||/ | Lp(Q)|| = (\ \f(x)\pdx)/P< go, 0 <p< go,

VjQ 7 (2.1) ||f | Lg(Q)|| = ess-sup\f (x)\ < g.

By S(Rn) we denote the Schwartz space of all complex-valued, infinitely differentiable, and rapidly decreasing functions on Rn and by S'(Rn) the dual space of all tempered distributions on Rn. We define the Fourier transform of a function f e S(Rn) by

F(f )(Z) = (2n )-n/2\ e-'xf (x)dx. (2.2)

Its inverse is denoted by F 1 f. Both F and F 1 are extended to the dual Schwartz space S'(Rn) in the usual way.

Let t e [0, g) and p e (0, g]. Let LPT(Rn) be the collection of functions f e Lfoc(Rn) such that

\\f | Lp (R") || = sup Bf \f X< (2.3)

where the supremum is taken over all J e Z \ N and all balls Bj of Rn with radius 2 J. Obviously, when t = 0, then LP(Rn) = LP(Rn). Furthermore,

LP(Rn) <^S'(Rn), (2.4)

(see [9, page 46]).

If s e R, 0 < q <œ and J e Z, then j+ is the set of all sequences {fk }k>j+ of complex numbers such that

||{fk }k>j+ I J = ( E 2ksq\fkA q < œ, (2.5)

\k>J+ /

with the obvious modification if q = œ. We recall that for any 0 <d < 1 and any J e Z

te (2.6)

(z\fk\) < E\fk\°,

\k>J+ / k>J+

(y + z)d < max(l,2d-1) (yd + zd), y,z > 0, d> 0. (2.7)

Let / be an arbitrary function on R" and x, h e R". Then

Ahf (x)= f (x + h) - f (x) , AM+1/(x) = Ah(Af^ (x), M e N. (2.8)

These are the well-known differences of functions which play an important role in the theory of function spaces. Using mathematical induction one can show the explicit formula

Aff (x) = X(-iycMf(x + (M - j)h), (2.9)

where CM are the binomial coefficients.

Recall that njrN(x) = 2j"(1 + 2j|x|)-N, for any x e R", j e No and N> 0. By c we denote generic positive constants, which may have different values at different occurrences.

2.2. The BpT Spaces and Fpq Spaces

In this subsection we present the Fourier analytical definition of BpT spaces, Fp'^ spaces and recall their basic properties. We first need the concept of a smooth dyadic resolution of unity.

Definition 2.1. Let ¥ be a function in S(R") satisfying ¥(x) = 1 for |x| < 1 and ¥(x) = 0 for |x| > 3/2. We put ^,(x) = ¥(x), (x) = ¥(x/2) - ¥(x) and

Vj (x) = for j = 2,3,.... (2.10)

Then we have supp Vj c {x e R" : 2j-1 < |x| < 3 ■ 2j-1}, Vj(x) = 1 for 3 ■ 2j-2 < |x| < 2j and ¥(x) + Xj>1 Vj (x) = 1 for all x e R". The system of functions {Vj} is called a smooth dyadic resolution of unity. We define the convolution operators Aj by the following:

Ajf = rV, * f, j e N, Aof = F-1¥ * f, f e S(R").

(2.11)

Thus we obtain the Littlewood-Paley decomposition

f = !>/,

(2.12)

of all f e S'(R") (convergence in S'(Rn)).

The Bpq spaces and Fp'Jq spaces are defined in the following way.

Definition 2.2. (i) Let s e R, t e [0, to) and 0 <p, q <to. The space Bp^ is the collection of all f e S'(Rn) such that

||f | Bs/J _ sup-

Z2isclUAjflLp(BJ )ir

(2.13)

where the supremum is taken over all J e Z and all balls Bj of R" with radius 2 J.

(ii) Let s e R, t e [0, to), 0 <p < to and 0 < q < to. The space Fp'Jq is the collection of all f eS(R") suchthat

if l || = sup

BJ \BJ[

/ \ 1/q

X2jsq\Ajf\q ) | U{Bj)

■J>J+ /

(2.14)

where the supremum is taken over all J e Z and all balls Bj of Rn with radius 2 J.

Remark 2.3. The spaces Bp^ and Fpq are independent of the particular choice of the smooth dyadic resolution of unity {^j} appearing in their definitions. They are quasi-Banach spaces (Banach spaces if p > 1, q > 1). In particular,

bS,0 _ BS Bp,q Bp,q,

fS,0 _ fS

rp,q _ гp,q,

(2.15)

where Bs q and Fp q are the Besov spaces and Triebel-Lizorkin spaces respectively. If we replace the balls Bj by dyadic cubes P (with side length 2-J) we obtain equivalent norms.

The full treatment of both scales of spaces can be found in [9]. Let Ajf (j e N0) be the functions introduced in Definition 2.1. For any a > 0, any x e Rn and any J e Z we denote (Peetre's maximal functions)

\ Ajf (y) \

A J (x) _ sup 1 j1 j e N0, f e S

j'J yeBj i1 + 2j\x - y\)

(2.16)

We now present a fundamental characterization of Bp'Z spaces and Fpq spaces.

Theorem 2.4. Let s e M, t e [0, to), 0 <p, q <to and a> n/p. Then

||/ | BSfqir = sup2;sq||A;;/ I Lp(Bj) ||q) , (2.17)

is an equivalent quasinorm in BppJq.

Theorem 2.5. Let s e M, t e [0, to), 0 < p < to, 0 < q < to and a > n/min(p,q). Then ||/ | Fp;q II* = supB (1/|Bj |T )y(2y>J+ 2jsq| Aj |q )1/q | Lp (Bj )||, is an equivalent quasinorm in Fpq.

Remark 2.6. Theorem 2.4 for 0 < p, q < to is given in [8, Theorem 4.5]. For Theorem 2.5 see [12, Theorem 1.1]. In addition if a > nmax(1 /p,T), then in Theorem 2.4 A*/ can be replaced by A*,a/, where

I A; f(v)l

A*,a/(x) = sup 1 jfKyn j e N0, / e S(Mn). (2.18)

' + 2j 1 X - y^

3. Some Technical Lemmas

To prove our results, we need some technical lemmas. The following lemma for Ajf, in place of A*,af, is given in [14, pages 87-89] (for the Bppq spaces and 1 < p < to). Further results, can be found in [12, Lemma 2.4].

Lemma 3.1. Let Ajf be as in Definition 2.1 and let s e R, a > 0, t e [0, to) and 0 < p, q < to(0 <p < to for the space Fpq). Then there is a constant c > 0, independent of j, such that for any f e S(Rn)

||A;'af||TO< c2j(n/p-s-nT)||f | Ap;q\\, j e N0. (3.1)

Here one uses ApJq to denote either Bp'Jq or Fpq.

Proof. Let f,f0 e S(Rn) be two functions such that Ff = 1 and Ffo = 1 on supp y1 and supp y0 respectively. Then

| Ajf (y)| = |f * Ajf (y)|, y e Rn, (3.2)

with fj = 2(j-1)nf (2j-1) if j e N. Since y e S(Rn), the right-hand side is bounded by cnj-1/N * | Af |(y), for any N> 0. Hence we get for all f e S'(Rn) and any x e Rn

A*'a/(x) < c sup * I Aj/Ky).

Using the same method given in [9, Proposition 2.6] we obtain for any y e R"

Hi-in * |A,/1( y) < c2i(»^-s-»T )\/ | A^\\.

The proof is completed.

Lemma 3.2. Let M e N, J e Z \ N, A > 0, t e [0, to) and 1 < p < to. Then there is a constant c> 0, independent of J and A, such that

|Af/ 0|dh | Lp (Bj )

J\h\<A1 1

< cA"1 Bj|TII/ | L

for any M Bj of Rn with radius 2 J and any function f such that ||f | Lp|| < to. Proof. Since 1 < p <to, the left-hand side is bounded by

f IIAM/ | Lp(Bj)\\dh. (3.6)

J\h\<An 11

From the definition of AM/ we have

|AM/(*)|<XCM|/(X + (M - m)h)|. (3.7)

Take the Lp (Bj)-norm to estimate (3.6) form above by

M f u /

EcMl \\/|Lp( m=0 J\h\<AU V

Bj)\\dhr

where if x0 the centre of Bj then x0 + (M-m)h is the centre of Bj. Using the fact that |Bj| = |Bj| to estimate (3.8) from above by cA^Bj-^||f | Lp|.

The lemma is proved. □

Remark 3.3. Let M, A, t, and p be as in Lemma 3.2. Let J e N0. By the embedding Lp(B0) ^ Lp(Bj) there is a constant c > 0, independent of J and A, such that

|AM/(■) | dh | Lp(Bj)

for any ball Bj of Rn with radius 2 J and any function f such that ||f | LVp || < to.

For s> 0, M e N, t e [0, to), 1 < p < to and 0 < q <to, we set

P,q\\m

\\/ | LP(R")

sup T^TT

Bj \BJ \

t-sqsup\AM/ofy ) | Lp{Bj) \h\<t1 1 1

(3.10)

Here the supremum is taken over all J e Z and all balls Bj of Mn with radius 2 J.

Lemma 3.4. Let s > 0, M e N, J e Z, t e [0, to), 1 < p < to and 0 < q < to. Then there is a constant c> 0, independent of J, such that

E 2isq(\ \

j>j+ vMS1'

AMv/(')|dv

E 2isq(\ I

j>j+ vm^

AM/(>(v)\dH ) I lp(bj)

I Lp(Bj)

< c\bj\T\\/ | Ff,T

< c\bj\T\\/ | f}s,t

PiïWM'

(3.11)

(3.12)

for any ball Bj of Rn with radius 2 J, any w e S(Rn) and any function f such that ||f | F}

p,q "m

Proof. Let P be a dyadic cube with side length 2-J. This result, for P in place of Bj, is already known, see [9, Lemmas 4.3, 4.4]. By simple modifications of their arguments we will give another proof of (3.12). The proof is given only when 0 < q < œ. The case q = œ is similar. Before proving this result we note that for any x € R" and any i e Z

£2(s+n)vq( f \aM/(x)\dh

v>i V|h|<2"vl 1

K JT "1 ^(LK/m^t)

q\ 1/q

qd^1/q

(3.13)

Here we will prove that the left-hand side of (3.12) is bounded by

\\/ | LP(R;

)\ + sup " Bj \BJ \

m ---\qdt

rsqsup\AM/(-)f t

JO |h|<t 1 1

| Lp Bj

= \\/1F

p,q\\m-

Obviously, ||f | Fpq ||M < ||f | Fpq||M. We write

2js f \ A^J(x)w(v)\dv _ f 2js f \ A^J(x)u(v)\dv

J\v\>V 1 k_0 J 2k<\v\<2k+1 1

< cf 2(s+n)j-Nkf \Aff (x)\dh,

I-n .>2k-J<lhl<2k-J+1 1

where N > 0 is at our disposal and we have used the properties of the function w,

|w(x)|< c(1 + |x|)-N, (3.16)

for any x e Rn and any N > 0. Now the right-hand side of (3.15) in ^ -+-norm is bounded by (with a = min(1,q))

cf f 2-N°k( f 2(s+")jqff \Aff (x)\dh

k_0 \j >J+ 2k-j<\h\<2k-j+1 1

/J+-1 V"

_ X '+X ' _ (Jj(x)+ Hj(x))

\k_0 k>J+ /

< 2^-1((ij(x))1/CT + (Uj(x))1/CT),

/ (3.17)

by (2.7). Here we put = 0 if J + = 0. Take the Lp(Bj)-norm we obtain that the left-hand

side of (3.12) is bounded by

,i/a-i ) 1/a | lp (Bj) || + 21/a-11| (IIj)1/a | Lp (Bj) ||. (3.18)

Let us estimate (Ij)1/a in Lp(Bj)-norm. After a change of variable j - k -1 = v, we get for any x e Rn (here J + = J and k< J)

J+-1 / /r \q\a/q

Ij(x) < 2(s+n-N)ak( £ 2(s+n)vq( |AMf(x)ldh) )

k=0 \v>J+-k-1 \J |h|<2-^ / /

J+-2 J+-1 (3.19)

= £••• + £ •••

k=0 k=J+-1

= M1(x) + M2(x).

Journal of Function Spaces and Applications Here we put £k+=02 ••• = 0 if J + < 1. We have

j+-2 /c 2-J+k+2 /c \qdt \a/q

M1(x) < c g 2(s+n-N)ak M f-(s+n)^J | AMf (x^dhjj

AMf (x)|dh ) - ) . (3.20)

Since Lp/a(Bj) is a normed spaces and Bj c Bj-k-2, the right-hand side in Lp/a(Bj)-norm can be estimated from above by

c ^ 2(s+n-N)ak k=0

/ ^2-J+k+2 0

i-sqsup|AMf (-)| - | Lp (Bj-k-2)

(3.21)

J+-2 a

< c|Bj r^ 2(s+n-N+nT w ||f | Fspqq hm) k=0

We choose N > 2(s + n)+ Tn. This yields that the last expression is bounded by

(3.22)

where c > 0 is independent of J. Now (M2)1/a in Lp(Bj)-norm is bounded by

q \ 1/q

c2(s+n-N)J+

| Lp{Bj)

q\ 1/q

< c2(s+n-N)J

. c2(s+n-N)J+

(£2(s+n)vq( f |AMf(■) |dh

\v>0 V |h|<2-v 1

2(s+n)vq^^J AMf q | Lp (Bj )

(3.23)

| AMf (■)|dh | Lp(Bj)

where we have used (2.7). Using the embedding Lp(B0) ^ Lp(Bj) and Remark 3.3, we obtain

M2 | Lp/a(Bj)|| = ||(M2)1/a | Lp(BJ)||°

< c2(s+n-N)aj+ (||f |Fp;q||M)°

(3.24)

= c2(s+n-N+nT)aJ+|Bj|Ta(||f | F%H'M)°

< c|BJГ(||f|Fsaм )a,

because of N > s + n + nT. Therefore,

||(ij ) 1/(7 | LP (Bj )f =||IJ|Lp/° (Bj )||

IM1 | Lp/7 (BJ) || + || M2 | Lp/7 (BJ) || (3.25)

< c|Bj|T7(Uf^s-M .

Now let us estimate (IIj)1/7 in LP (Bj)-norm. We write

/k+J++1 TO \a/q

IIJ(•) = c£2-N7k( £ ••• + £ ... )

k>J+ \ j=J+ j=k+J++2 /

//k+J+ + 1 \7/q / to \7/qs

<cE2-N7k( ( £ .. ) + (£•••)

k>J+ y \ j=J+ / \j=k+J++2 /

= c £ 2-N7k(Sk (•) + Sk (•)).

After a change of variable j - k - 1 = c, we get

S(x))1/7 < c2(s+n^kf £ 2(s+n)№Yf |AMf(x)|

AMf (x)|dh

q \ 1/q

/r2k-J++2 /r \qw. \ 1/q

< c2(s+n)kM + f-(s+n)^J ^f (x)|dhj d

r / r2k-J++2 1/q

< c2(s+n)k |AMf (x)|dh( t-(s+n)q^

J|h|<2k-J++^ 1 \ J 2-J+ t

< c2(s+n)J++(s+n)k f | AMf(x)|dh.

J \h\<2k-J++2 1

Therefore there exists a constant c > 0 independent of J and k such that

S ) 1/7 | Lp (Bj) || < c2(s+n)J++(s+n)k j + AMf ()dh | Lp (Bj) < c2(s+Tn)J++(s+2n)k|BJ|T||f | lp||

< c2(2s+2n+Tn)k |Bj |t ||f | F;q ||M,

(3.26)

(3.27)

by Lemma 3.2 (combined with Remark 3.3) and the fact that k > J+. Now let us estimate

(Sfc2)1/ain Lp (Bj)-norm. We have

q\ 1/q

. Mf(x)\dh '

V=J++1 V|h|:

(3.29)

2-J+ 1/q

(Sfc2 (x))1/CT < c2(n+s')k f ]T 2(s+n)vq( f \aM/(x)\

\v=J++1 \J| h | <2-vl

/ 2-J+

< c2(n+s)ki f r^supUff(x)\q-

\Jo |h|<t ' t

Therefore,

||(Sk)l/o | LP (Bj)|| < c2(n+s)k

rsq sup \ AM/(■)

|h|<t h

| lp(bj)

(3.3o)

< c2(n+s)k\Bj\Tf | F,

S,T 11 '

pwIIm-

Consequently,

||(IIj )1/a|Lp(Bj )||° = ||ПJ|Lp/a (Bj )||

< £ 2-Nak(|| (Sk )1/a | Lp (Bj )||° + ||(S> )1/a | Lp(Bj )||°)

° (3.31)

< c^ 2(2s+2n+Tn-N)ak|Bj|T^||f | FpquM)

< c^r^^^U'M)°,

since N > 2(s + n)+ rn. This finishes the proof of Lemma 3.4. □

For s> 0, M e N, t e [0, to), 1 < p <to, and 0 < q <to, we set

1 ic 2-J++1 dt \1/q

||f | B^Um = ||f | Lp(Rn)|| + sup_( i-sqsup||AMf | Lp(Bj)||qd ) . (3.32)

bj |Bj| \^0 |h|<f" 11 1 /

Here the supremum is taken over all J e Z and all balls Bj of R" with radius 2 J. Similar arguments yield.

Lemma 3.5. Let s > 0, M e N, J e Z, t e [0, to), 1 < p < to and 0 < q < to. Then there is a constant c> 0, independent of J, such that

^W J JM/ | Lp(Bj)|M ) < CIBjItII/ | BtIIm,

j>J+ |v|<1

(3.33)

M c I rp/nII II\ \ /„InJ^l ns,Tl

{MJ|AMv/ | Lp(Bj)||«(v)^)q^ < c|Bj|T||/ | Bj

for any ball Bj of Rn with radius 2-J, any w e S(Rn) and any function f such that ||f | Bpqhm < to.

Now we recall the following lemma which is useful for us.

Lemma 3.6. Let 0 < a < 1, J e Z and 0 < q <to. Let (ek} be a sequences of positive real numbers, such that

||{£k}k>j+ | J = I< to. (3.34)

The sequences {Sk : Sk = lj=j+ ak-j£j }k>j+ and {^k : nk = ^TO=kaj-kej }k>j+, are in ¿q j+ with

.. .. .. J || <

c depends only on a and q.

| W1 ej1 +1 {n^fc>j+1 eJ || < ci, (3.35)

4. Characterizations with Differences

We are able to state the main results of this paper.

Theorem 4.1. Let 1 < p < to, 0 <q <to, t e [0, to) and M e N. Assume

min p, q p

< s < M, 0 < t < — (4.1)

n n 1 ..

---- < s < M - nT + -, t >-. (4.2)

min p, q p p

Then ||f | FpJq ||M is an equivalent quasinorm in FpJq.

Theorem 4.2. Let 1 < p <to, 0 <q <to, t e [0, to) and M e N. Assume

0 < s < M, 0 < t< 1 (4.3)

0 <s<M - nT + -, t >-. (4.4)

s,T|| ____________________T?s,T

Then ||/ | Bp'rqIIM is an equivalent quasinorm in Bp'q

Remark 4.3. Theorems 4.1 and 4.2 for 0 < t < s/n + 1/p and 0 < t < s/n + 1/p -max(1/ min(p,q) - 1,0), respectively, are given in [9] Theorems 4.6 and 4.7, respectively.

Proof of Theorem 4.1. Let Bj be any ball centered at x0 e Rn and of radius 2-J, J e Z. We will do the proof in three steps.

Step 1. We have with s> 0,

XIV| = X2-js2js| Ajf | < csup2js| Ajf |

j>0 j>0

jeNo 1/q

< c(£2jsq| Ajf | \j>o

Let f e F?;!. Then,

| L? (R")|| <

X|Ajf| I L?(R")

^^2jsq|Ajf|q^ | L?(R")

< c||f IF

?,q H •

Step 2. For any x e Bj we put

Hj (x) = i-sqsup|Aff (x)|q^

jo |h|<t' 1 t

After a change of variable t = 2 y, we get

Hj (x) = ln 2 2ysq sup | Af f (x) | dy

J J+-1 |fc|<2-y'

Hj (x) < c^ 2ksq sup | Af

k>J+ |h|<2-k+^

Let f, f0 e S(Rn) be two functions such that Ff = 1 and Ff0 = 1 on supp^i and supp¥ respectively. Using the mean value theorem we obtain for any x e Bj, j e N0, and \h\ < 2-fc+!

|AhAjf (x)| = |Ah(^j * A

< 2-k sup £|D>; * Ajf (y) |,

|x-y|<c2-k |a|=1

with some positive constant c, independent of j and k, and fj(•) = 2(j-1)nf (2j-1-) for j = 1,2,____By induction on M, we show that

|AMAjf (x)|< 2-kM sup ZDyj * Ajf(y)|. (4.11)

|x-y|<c2-k |a|=M

We see that if |a| = M and a> 0

|D>; * Ajf (y)| = 2(j-1)n f Da(f(2j-1(y - z)))Ajf(z)dz

J R» v v ' '

< 2(j-1)(n+M) JR | (D>) j (y - z)) 11 Ajf (z^dz.

Suppose that 0 < j < J + - 1. The right-hand side in (4.12) may be estimated as follows: c2j(n+M)A*,af (y) JR | (Daf) (2j-1(y - z))|(1 + 2j|y - z|)adz < c2jMAjaf (y).

Then we obtain for any x e Bj, |h|< 2-k+1 and any k > J+

(4.12)

(4.13)

f (x)| < c2(j-k)M sup A*,af (y)

|x-y|<c2-k

< c2<j-k'M(1 + 2'-k)a sup (iA-:f(y),)a <«4)

V 7 |x-y|<c2-k C1 + 21 ^ - yO

< c2(j-k)MA*'af (x),

if 0 < j < j+ - 1.

Suppose now that j+ < j < k. By our assumption on x and k we have

|y - X0| < |x - X0| + |x - y| < 2-J + c2-k < (c + 1)2-J, (4.15)

which implies that y is located in some ball Bj, where Bj = {y e Rn : |y - xo| < (c + 1)2-J}. Writing the integral in (4.12) as follows

f •••dz + W _ •••dz = Ijj (y) + Elljj-i(y). (4.16)

jBj-1 i>0 j Bj-i-2\Bj-i-1 i>0

We recall that

A*af (y) = sue |Ajf (z)|u a, (4.17)

hlfky J a+2i |y - z|)a

Journal of Function Spaces and Applications for any j e N0, f e S(Rn) and any l e Z. We have

Ij,j (y) < Ajj-f (y) f |(D»(2j-1(y - z))|(1 + 2j|y - Z|) d

J i?J-1

< c2-jnAjj-1f (y) < c2-jnA*j2f (y).

Let us estimate IIj,j-i. Since f e S(Rn), we have

|Daf(x)|< c(1 + \x\)-2N, (4.19)

for any x e Rn and any N > 0. Then for any N large enough, IIjj-i(y) does not exceed

T / \ -2N+a

cAjj-i-2f y I _ (1 + 2j-1 |y - z|) dz

J BJ-i-2 \ BJ-i-1

< c2-iNAjj-i-2f (y) f (1 + 2j-1 |y - z|\-N+adz

' J Rn /

< c2-iN-jnAjj-i-2f (y),

where we have used 2j !\y - z\ > (c + 1)2j J+i 1 > (c + 1)2i Therefore,

Dafj * Ajf (y) | < c2jM£ 2-iN Ajj-fy). (4.21)

Then we obtain for any x e Bj any \h\< 2 k+1 and any J + < j < k

| AM Ajf (x)|< c2(j-k)M£ 2-iN sup Ajj-i-2fy)

i>0 \x-y\<c2-k

A*, a f (y) (4.22)

< c2j-k)M( 1 + 2j-k)^2-iN sup (1 J- ,)..

v 7 £0 \x-y\<c2-k C1 + 2 |x - yv

Consequently, for any J + < j < k there is a constant c > 0 independent of J, j, and k such that

| AM Ajf (x)| < c2(j-k)M £2-iN a;^ (x). (4.23)

Finally for j > k + 1 we have for x e Bj and |h| < 2-k+1

|AM Ajf (x)| < X CM|Ajf (x + (M - m)h) | sup

|x-y|<C2-k

h j v*! | <

< 2M sup |Ajf(y) |

(4.24)

m — (1 + 2j|x - y|)a.

2j x y )a

|x-y|<C2-k i1 + 21 |x - y\)

We remark also that by our assumption on x and k we have

|y - x0| <|x - x0| + |x - y| < 2-J + C2-k < (C + 1)2-J, (4.25)

and this implies that y is located in some ball Bj, where Bj = {y e Rn : |y - x0| < (C + 1)2 J}. Then,

|aMAjf (x)|< c2(j-k)aAjf(x), (4.26)

if j > k+1, where Aj'^f is given in (4.17) (with Bj a ball centered at x0 and of radius (C+1)2 J). We write,

AMf (x) = E AM Ajf (x)

J+-1 k (4.27)

= £••• + £••• + E •••.

j=0 j=J + j>k+1

J+ 1 + s

Here we put ^J=0 ••• = 0 if J + = 0. Let us estimate each term in ¿s j+ -norm. We have by (4.14)

and Lemma 3.1

J+-i , J+-1

X|aM Ajf (x ^ 2(j-k)MA*'af (x) j=0 j=0

< c^ 2(j-k)M+j(n/p-s-nT)|f | FsTq||

j=0 ' (4.28)

< c2-kM ^ 2j(M+n/p-s-nT) ||f | Fs/q ||

j=0 p,q

< c2J(M+n/p-s-nT)-kM ||f | F^W

where the last inequality can be obtained by our assumption on s and t. The last expression in esn T+ -norm does not exceed

c2jn(1/p-T)

j2(k-j+)(s-M)}k>j+1 ej ||/1 Fsqi < c2Jn(1/p-T) 1/1 Fsq|, (4.29)

since s < M. Therefore,

(Hj(x))1/q < c2Jn(1/p-T)|/ | F,

q\ 1/q

+ c( E(EE 2(j-fc)(M-s)+sj-iN|A*j-i-2/(x)|

V>J+\ i>0 j=J+

+ c (E (E2(j-k)(a-s)+sj | A*J?/(x) | ^ \ .

(4.30)

(4.31)

(4.32)

The second term can be estimated by (with a = min(1, q))

c I E 2-iNa (El E 2(j-k)(M-s)+sj | A*jt-2/(x) | i>0 \fc>j+ \;=J+

q\ a/qN

(4.33)

Since again s < M, then we can apply Lemma 3.6 to estimate the last expression by

E2-Na (E 2jsq|A;j-t-2/(x)|q \

i>0 \j>J+ J

(4.34)

Since Lp/a (Bj) is a normed space, so (4.31) in Lp (Bj)-norm is dominated by

cl E2-

z?™;/ \ | Lp/a{Bj)

>]2jsq|A*j-i-2/|q J | Lp(Bj)

a \ 1/a

Using the embedding Lp(Bj-i-2) c LP(Bj) and the fact that J + > (J - i - 2)+ to estimate this expression from above by

( X 2jiAJ-i-2fr ^ |L^BJ-i-2)

\j>(J-i-2)+ /

< c|Bj|^£2i("T-N)^ ||f | F

< c|Bj|T||f |F?;q||,

(4.36)

where the first inequality is obtained by Theorem 2.5 and the second inequality follows by taking N > nT. Taking a e (n/ min(p,q),s), then using again Lemma 3.6 to estimate (4.32) by

c( £2jsq|A

(4.37)

This expression, by Theorem 2.5, in Lp(Bj)-norm is bounded by c\Bj\T||f \ Fpq||. Hence we have for any J e Z and any ball Bj of Rn with radius 2-J

-||(Hj)1/q | L?(Bj)|| < c||f |

(4.38)

with some positive constant c independent of J. From this it follows that

iif ftHm < c\\f\fpq\\,

(4.39)

for any f e F

Step 3. Let ¥ be the function introduced in Definition 2.1 and in addition radial symmetric. We make use of an observation made by Nikol'skij[15] (see also [16]). We put

f(x) = (-1)M+1 2 (-1)iCf¥(x(M - i))

(4.40)

The function f satisfies f (x) = 1 for \ x\ < 1/M and f(x) = 0 for \ x\ > 3/2. Then, taking y0(x) = f (x),y1 (x) = f(x/2) - f(x) and yj (x) = (2-j+!x) for j = 2,3,..., we obtain that [tyj} is a smooth dyadic resolution of unity. This yields that

Bj |BJ |

z \ 1/q

X 2jsq|Ajf|q ) |L? (Bj)

(4.41)

is a norm equivalent in Fpq (see Remark 2.3). Let us prove that

(E2jsq|Aj/|q^ | Lp{Bj)

\j-J+ /

<c|/1F

s,T 11

p,q M,

(4.42)

for any ball Bj of Rn with radius 2 J. First the left-hand side contains A0f only when J + = 0. Then

/ | Lp (Bj )|| < jj|/ | Lp(Bj)|||F-V (y) | dy, (4.43)

where Bj is a ball centered at xo + y and of radius 2 J. Hence

||Ac/ | Lp(Bj)|| < cB^I/ | Lp||

(4.44)

< c|Bj|Ti/^

where we have used the fact that |Bj | = |Bj |. Moreover, it holds for x e Rn and j = 1,2,

Aj/(x) = (-1)

W AMMv/(x)F-1

l¥ (v)dv,

(4.45)

with ¥(■) = F-1¥(-) - 2-nF-1¥(-/2) (see [17, Theorem 3.1]). Now, for j e N we write f |Amv/(x)||F-1¥(v)|dv

J Rn1 11 1

J | AMv/(x) 11F-1¥(v) |dv + (-1M+1 £ ^ | AMv/(x) 11F-1¥(v)|dv.

(4.46)

Then the estimate (4.42) is an obvious consequence of (4.44) and Lemma 3.4. Therefore,

II/FS| < c||/|F,

s,T 11

p,q| M,

(4.47)

which completes the proof of Theorem 4.1. □

Proof of Theorem 4.2. The first two steps closely follow the argument in [17, Theorem 3.1]. Step 1. Let f e Bpq Since s > 0, then we have

| Lp(Rn)|| <£2-js2js|| Aj/ | Lp(Rn)|| < csup2js|| Aj/ | Lp(Rn)|

j>0 jeN0

< c||/1 Bpqi.

Step 2. As in the proof of Theorem 4.1 we have

f.2-J++1 dt \1/q

jo rsqsupl\AMf\lp(bj)(dt)

(4.49)

< c( £ 2ksq sup \\AMf \ Lp(Bj)T

\k>J+ ihi<2-k+1

Let us estimate AM Ajf. If j < k, then as in the proof Theorem 4.1, we have

| AM Ajf (x) | < c2(j-k)M A*'af (x), (4.50)

for any x e Bj, \h\ < 2-k+1 and any a> 0. If j > k we have for x e Rn and \h\ < 2-k+1

| AM Ajf (x)| <£ CM | Ajf (x + (M - m)h)|. (4.51)

Hence we obtain for any j > k and any a> 0

hj \Lp(Bj)H <£ CM\\ Ajf \ Lp(j3j

(4.52)

M \i /~

cM\\A*'af | u(b}

where if x0 the centre of Bj then x0 + (M - m)h is the centre of Bj. We remark also that by our assumption on h and k we have

|x - xo| < |x - (xo + (M - m)h))| + (M - m)|h| < 2-J + M2~k+1 < (2M + 1)2-J, (4.53)

for any x e Bj. We denote Bj the ball in Rn centred at x0 and of radius (2M + 1)2-J. Since Lp(Bj) c Lp(Bj) and Lp(Bj) c Lp(Bj), we get

J+-1 k

|AMf | Lp(Bj)\\ < c£ 2(j-k)M\\ A*,af | Lp (Bj)\\ + c£ 2(j-k)M\\ A*,af | Lp(B,

1=0 j=J+

+cE\\A;-af | Lp(Bj

(4.54)

Here we put ^ j=01 ••• = 0 if J + = 0. Lemma 3.1 gives

j+-1 j+-1

^2(j-k)M\\A*'af \ LP(B^ \\ < c^ 2(j-k)M+j(n/P-s-nT)-Jn/P\f \ BpqH

j=0 11 j 11 j=0 ' (4.55)

< C2J(M-s-nr)-kM\\f | Bpq \

The second inequality follows by our assumption on s and t. Since 0 < s < M, then we can apply Lemma 3.6 to estimate (4.49) by

c\Bj \T 11/1B

{2(fc-J+)(s-M^kJ | <j+ + c( £2jsq\\ A*/ | Lp(Bj

(4.56)

< c\Bj\T\\/ | B

where we have used Theorem 2.4, combined with Remark 2.6, and the equation \Bj| = (2M -

1)n\Bj\.

Step 3. First this step in [17, Theorem 3.1] contains a gap, but using the same arguments given in Step 3 in the proof of Theorem 4.1 (with Lemma 3.5 in place of Lemma 3.4), we can prove that

\\/ I Bp'TqW < C\\/ | B

'pq\\M-

(4.57)

This ends the proof of Theorem 4.2.

Finally we study, in addition, the case t e [1/p, to). Under this condition we can restrict supB in the definition of Bpq and Fp'^ to a supremum taken with respect to balls Bj of Mn with radius 2-J and J e N0.

Lemma 4.4. Let s e M, t e [1/p, to) and 0 <q <to.

Let 0 <p <to. A tempered distribution f belongs to BpT if and only if

11/ | B'p'q||* = sup

■ E2;sq\A;/ | Lp(Bj)\\q . bj \BJ\ \j>J /

(4.58)

where the supremum is taken over all J e N0 and all balls Bj of Mn with radius 2 J. Furthermore, the quasinorms ||f \ Bp'^W* and ||f \ BppJq\\ are equivalent.

Let 0 <p < to. A tempered distribution f belongs to Fp'^ if and only if

\ 1/q 'q » \ U(Bj)

\\/ | Fm\\# = sup Br

Bj \BJ\

E2jSq\A;/\q j>J

< <X>,

(4.59)

where the supremum is taken over all J e N0 and all balls Bj of Mn with radius 2 J. Furthermore, the quasinorms ||f \ Fp'Jq ||# and ||f \ Fp'Jq W are equivalent.

Proof. For each J e Z and m = (mi,..., mn) e Zn, set

Qj,m = {x = (x1r..., xn) e Rn : mj < 2JXt < (mj + 1), i = 1,2,nj.

(4.60)

This lemma for Qjm in place of Bj is given in [9, Lemma 2.2]. By the properties of the dyadic cubes, there exists v,k e N not depending on J such that

Qjm c U^, Bj c U^Qj.

k r\l 1=

(4.61)

Here BJ is a ball of center 2 J and QJ is a dyadic cube of side length 2 J. The proof of this result

is an obvious consequence of the previous embeddings, Lemma 2.2 of [9] and Remark 2.3.

Defining for 1 < p <to

Mp(f) = sup^f \f (x)\pdx) ,

Bo \ J Bo /

(4.62)

where the supremum is taken over all balls B0 of Rn with radius 1.

Theorem 4.5. Let 1 < p < to, 0 < q < to, t e [1 /p, to), M e N and n/ min(p,q) < s < M - "t + n/p. Then

If I IlM = Mpf + sup 1

Bj \BJ \

i-sqsup\AMfOfd^ ) I LP(Bj) 0 |fc|<i' 1 r /

(4.63)

is an equivalent quasinorm in Fp'/q. Here the supremum is taken over all J e N0 and all balls Bj of Rn with radius 2-J.

Theorem 4.6. Let 1 < p <to, 0 <q <to, t e [1 /p, to), M e N and 0 < s < M - "t + n/p. Then

2-J+1 1/q

llf I Bpq 11M = Mp(f) + sup -L, C f j f-sqsup|| AMf | LP(Bj)

Bj \Bj\ \Jo ihi<t" 11 1

(4.64)

is an equivalent quasinorm in Bp'q. Here the supremum is taken over all J e N0 and all balls Bj of Rn with radius 2-J.

Proof. We will prove only Theorem 4.5. The proof of Theorem 4.6 is similar. We employ the same notations given in the proof of Theorem 4.1.

Step 1. Let f e Fp'fq. Since s > 0, then we have

MP(f) 2-jS2jSMp(Af < c sup 2js|| Ajf | LP(Bo)||

j>0 Bo,jeNo

< c||f |Fp;q||#.

(4.65)

Step 2. As in the proof of Theorem 4.1, there is a constant c> 0 independent of J such that

U|(Hj)1/q|Lp(Bj)|| < c||f | Fp'fqW

Jl (4.66)

< cUflF^ ||#,

by Lemma 4.4.

Step 3. The left-hand side in (4.42) (with J+ = J) contains A0f only when J = 0. Then

,/ | Lp(Be)|| < jj|/ | Lp(Bo)\\\?My)\dy < cMpf ).

We recall that for x e R" and j = 1,2,... Aj/(x) = (-1)Mo1 f AM/(x)F-1¥ (v)dv

= (-1)'

A'v/(x)F-1lP (v)dv + (-1)'

J |v|>1

A'iv/(x)F-1¥ (v)dv.

As in the proof of Lemma 3.4, we can prove that

E^J \ A'jv/(■) \dv

I lp(bj)

< c\bj\T ||/if-\M,

J >J \j|v|>11

A"-,v/(-Mv)\dv ) | Lp{Bj)

< c\bj\T /IF?;\M,

(4.67)

(4.68)

(4.69)

for any J e N any ball Bj of R" with radius 2 J and any w e S(R"). The proof is completed.

Remark 4.7. Recently, Yang and Yuan [18, Theorem 2] proved that

= FSo+"(T -1/p) ; e <p< œ, s e R, Bpq = BS+;jT-1/p), e <p <œ,s e R,

(4.7e)

if t > 1/p, 0 < q < œ or if t = 1/p and q = œ. Under these conditions the study of the Triebel-Lizorkin-type space Fp,'q and the Besov-type space Bp'Jq is not interest.

Acknowledgments

The author would like to thank the referee for his very carefully reading and also his many careful and valuable remarks, which improve, the results and the presentation of this paper.

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