Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 417341, 8 pages http://dx.doi.org/10.1155/2014/417341

Research Article

Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent

Baohua Dong and Jingshi Xu

Department of Mathematics, Hainan Normal University, Haikou 571158, China Correspondence should be addressed to Jingshi Xu; jingshixu@126.com Received 29 May 2013; Accepted 30 October 2013; Published 22 January 2014 Academic Editor: Kehe Zhu

Copyright © 2014 B. Dong and J. Xu. 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 introduce new Besov and Triebel-Lizorkin spaces with variable integrable exponent, which are different from those introduced by the second author early Then we characterize these spaces by the boundedness of the local Hardy-Littlewood maximal operator on variable exponent Lebesgue space. Finally the completeness and the lifting property of these spaces are also given.

1. Introduction

Variable exponent function spaces have attracted many attentions because of their applications in some aspects, such as partial differential equations with nonstandard growth [1], electrorheological fluids [2], and image restoration [3-5]. In fact, since the variable Lebesgue and Sobolev spaces were systemically studied by Kovacik and Rikosnikin [6], there are many spaces introduced, such as, Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, and Hardy spaces with variable exponent; see [7-20] and references therein. When the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces, many results in classical harmonic analysis and function theory hold for the variable exponent case; see [21-23].

Let p(-) : R" ^ [1,^0 be a measurable function. Denote by Lp(')(Rn ) the space of all measurable functions f on R" such that for some X > 0

dx < rn,

with the norm

\\f\\L*>

dx ^ 1

Then Lp{:)( R" ) is a Banach space with the norm y • H^o

We will use the following notations: p_ := essinf {p(x) : x e R"| and p+ := esssup{p(x) : x e R"|. The set P(R") consists of all p(-) satisfying p_ >1 and p+ < >x>. Moreover, we define P°(R") to be the set of measurable functions p(-) on R" with the range in (0, rn) such that 0 < p_ < p+ < >x>. Given p(-) e P°(R"), one can define the space Lp(:)(Rn) as above. This is equivalent to defining it to be the set of all functions f such that \f\p0 e Lp(')/p0(R"), where 0 < p° < p_ and p(-)/p° e P(R"). We also define a quasinorm on this

space by ||/||Lf,<.)(R„) := W\f\P° W^lpo (r„).

Let f be a locally integrable function on R"; the local variant of the Hardy-Littlewood maximal operator is given by

Mlocf(x):= sup ]l:\\f(y)\dy, Vxe R",

Q3x,|Q|SC M JO

for some constant C. We denote Bioc(R") the set of p(-) e P(Rn) such that Mloc is bounded on Lp(:)(Rn). In 2013 Danelia et al. gave characterizations of Bioc(R"), a vector-estimate for the local Hardy-Littlewood maximal operator if p(-) e Bioc(R"), and a Littlewood-Paley square-function characterization of the variable exponent Lebesgue spaces Lp(°(R") when p(-) belongs to Bloc(R") in [24].

In 2001 Rychkov used the boundedness of the local Har-dy-Littlewood maximal operator to prove a stronger result of the Peetre type for spaces F^(R") and B^q(Rn) and gave the lifting property for these spaces in [25].

Motivated by the previous papers, the goal of this paper is to introduce new Besov and Triebel-Lizorkin spaces with variable exponent. To state our result, we need some notations.

Throughout this paper |S| denotes the Lebesgue measure for a measurable set S c R". N0 denotes the set of all nonnegative integers. Let D := C™(R") and D' be the dual space of D. For se R, := max|s, 0} and [s] is the largest integer less than or equal to s.

Given a function <p on R", let L f e N0 denote the maximal number such that <p has vanishing moments up to order L In other words, JR„ xa<p(x)dx = 0 for all multiindices a with |a| ^ L^. If no moments of <p vanish, then put L^ = -l.Apair of functions (<p0, <p) is called satisfying the Ms condition, if L ^o(^)d^ = 0 and Lf ^ [s].

Take a function <p0 e D satisfying Ms condition, which is possible for any s. (Indeed, the assumption L f ^ [s] is void for s < 0 and is satisfied automatically for 0 ^ s < 1. For s ^ 1, any <p0 e D with Fourier transform <p0(£) = 1 + 0(|£|[s]+1) near the origin will do the job.) The notation S^(R") was introduced by Schott in [26]. More precisely, let it be the set of all f e D' for which the estimate

|(/,y}| ^ A sup ||D"y(x)| exp (N|%|) : % e R",|a| ^ ,

all ye D, (4)

is valid with some constants A = Ay, N = Ny. It is evidently that S,(R") includes temperate distributions S'(R").

Now, we give the definition of Besov spaces and Triebel-Lizorkin spaces with variable exponent.

Definition 1. Let p(-) e P°(R"), s e R, 0 < q < ot, <p0 as above, <p(x) := <p0(x) - 2-"^0(x/2), and <Pj(x) := 2J"<p(2Jx) for j e N.

(i) The Besov space with variable exponent £^(R") is the set of / with

|/e s: (R"):

R») := ( * /

Ilph(R")

(ii) For _p+ < ot, the Triebel-Lizorkin space with variable exponent F^(R") is the set of / with

|/e S: (R") :

"f;C)(r")

The key point is to prove that different choices of <p0 in Definition 1 do not really change the spaces, leading to equivalent quasinorms. For / e S'(R") that has been proved by the second author in 2008, see [19]. To go on, we recall variant Peetre-type maximal functions which was introduced by Rychkov in [25]. Let

. r, , K' •Pj.A.fi/M := suP -

¿A.B 00

X e R", j e N°, (9)

™j,A,B (y):=(l + 2j |y|)A2WB, A,£^0. (10)

Now it is the position to state our main result.

Theorem 2. Let se R, 0 < q < ot, and p(-) e P0(R") with 0 < _p0 < min|p_,^} such that p(-)/p0 e Bloc(R"). Suppose that <p0, C0 e D and the pairs (<p0, <p := <p0 - 2-"^0(-/2)), and (£0, C := C0 - 2-"C0('/2)) satisfy the Ms condition. Then there arepositive constants A0 := A0(s, B0 := B0(n), and C such that for each A ^ A 0, B > £0/_p_, and a// / e S^(R") one has

f;,(<)(r")'

Ilph(R")

Since <p*AB ^ */| forany <p0, one immediately gets a consequence of Theorem 2.

Corollary 3. The spaces F^R") and F^R") with s e R,

0 < q < ot, and p(-) e P0(R") with 0 < p0 < min|p_,^} such that p(-)/p0 e Bioc(R") are independent of the particular choice of the function <p0 in Definition 1. The quasinorms arising for different <p0 are equivalent.

The proof of Theorem 2 will be given in Section 2. In Section 3 we study the completeness and the lifting property of these spaces by using Theorem 2. We will use the notation a < fc if there exists a constant C > 0 such that a ^ Cfc. If a < fc and b < a we will write a ~ fc. Finally we claim that C is always a positive constant but it may change from line to line. Other notations will be explained when we meet them.

2. Proof of Theorem 2

IIf^r»)

We will use the idea of [25] by Rychkov to prove Theorem 2. First we need some lemmas.

Lemma 4 (see [25, Theorem 1.6]). Let a function <p0 e D have nonzero integral, and let (p(x) = <p0(x) - 2-"<p0(x/2). Thenfor

any N ^ 0 there exist two functions f0, f e D, such that y has vanishing moments up to order N and

f = Yfj * fj *f, Vfe

where fj(x) := 2,n fj(2}x) and fj(x) := 2,n<p(2'x) for j e N.

Before the next lemma we denote a special convolution operator which is given by

KBf(x):=\ \ f(y) (BZ0).

Lemma 5 (see [25, Lemma 2.10]). Let 0 < r < rn, <p0 e D, L„ tp0dx = 0, and (p = <p0 - 2~n<p0(-/2) and A > n]r, B ^ 0. Then there is a constant C depending only on n, r, <p0, A, B such that for all f e S^(Rn) and each x e Rn, j e N0, one has

x |Mloc ( I cpk * f \r) (x) + KBr (\cpk * f \r) (*)}.

Lemma 6 (see [24, Corollary 3.2]). Let p(-) e Bloc(Rn) and 1 < q < rn; then there exists a positive constant C such that for all sequences {fj}'J=0 of locally integrable functions on Rn

l(Mlocfj)q

^ c (I \ u r)

LÍH(R") \j=0 )

Lt^iR")

Lemma 7 (see [23,Lemma 2.1.14]). Let X be a real or complex vector space and q be a semimodular on X. Then ||x||g ^ 1 and q(x) ^ 1 are equivalent. If q is continuous, then also ||x||e < 1 and q(x) < 1 are equivalent, as are ||%||e = 1 and q(x) = 1.

Lemma 8. Let p(-) e Bioc(Rn) and 1 < q < >x>. Then there exists a positive constant B0 = B0(n) > 0 such that for B ^ Bo/P-

m x1/q

I\f,\q

LP« (R")

where C is a positive constant and [f¡}'¡=0 are locally integrable functions on R".

Proof. By homogeneity, it suffices to consider the case

rn s1/q

l\ft t

LÍ*->(R")

Let Rn = UQe1 Q,where 11 is the set of all unit dyadic cubes in Rn. Then it is easy to get

kb/(x)=\ I f(y) \2-B^dy

\ I f(y) I dy.

< ^ 2~B dist(x,Q) I1 3Q JQ

Since p- > 1, by Minkowski's inequality and Holder's inequality,

lKBft(x)q

-B dist(x,Q)

¡=0\ 11BQ jQ

B dist(x,Q)

\ I ft (y) I dy

I (\ I f:¡ (r) |dy

I 2-B dist(x,Q)p(x)/2 11BQ

I (\ I f:¡ (r) |dy

¡=0 Q

p(x)/q

1/p(x)

1/p'(x)

*[ I 2

B dist(x,Q)pf (x)/2

Since B > 0 and p- > 1, the latter factor is uniformly bounded in x. We take the p(x)th power of the above inequality and integrate it. We get

\ (ikbUx)

JR" V ;-n

p( x)/ q

q ) dx

< I \ 2-B dist(x,Q)p(x)/2dx ~ 11bq' r"

I (\ I f¡ (y) |dy

¡=0 Q

p( x)/ q

It is easy to know that

2-B dist(xQ)p(x)/2dx< \ dx Q

for p- >1 and also that

\ I f¡ (y)\ d^ Miocf¡ (x), VxeQ. (23)

By these two observations and (21), we have

I (I KBf,(x)i

Jr» \ j=o

I(Mloc/, (*))*

Applying Lemmas 6 and 7 we obtain

, / m \

I (I KBf,(x)i

Jr" \<=o

Using Lemma 7 again we obtain

Ife/,) '

dx < 1.

Thus we have

IPH(R")

I I /> I ')

IPH(R")

This finishes the proof.

We give a notation of norm in (lj) which will be used in the following context:

IP^R")

Lemma 9 (see [27, Lemma 2]). Let 0 < q < œ>, 5 > 0. For

Ji=0|L?()(£,)

; [27, Lemn

any sequence 1^,1: of nonnegative numbers denote

Gj = ^^

{"] : Il Il r T:|

.. G,}° L <cl{^j}0

Jo ii iK^Jo ii

1 v(j 1 "lq

holds, where C is constant and only depends on q, S.

Lemma 10 (see [19, Lemma 3]). Let 0 < q < >x>, S > 0, and p(-) e P0(R"). For any sequence of nonnegative measurable functions on R", denote

Gj (*) = l2-|fc-j|SÄ (*), xí R".

{Gj}:"iK-)(ig) <Ci"{^j}

o "ip()(i„)'

{g,.}:|| <c2|| {«,.}

o lk(i10)

hold with some constants C1 = C1(^, 5) and C2 = C2(p(-),^, 5).

Proof of Theorem 2. By Lemma 4, take ^ e D, with large enough so that (13) is true. It follows that

I C,- I

: /•

M I Cj (z) |-| % I dz

k=o JR"1 1

Ijjk sup

I <?k * /(*-y-z) I

zeR" mk,A,B (z)

Jjfc := I |Cj (z) I mk,A,B (z) dz.

Jr» ' 1

Because of the elementary inequality

mj,A,B (z) ^ mj,A,B M mj,A,B (z -

m,',A,B M ^

mk,A,B M > j ^

2(j-k)A-k,A,B (>0> i<fc

we have the following fact:

^•pílB/m,

if j ^ fc

I/jk2(k-j)A<A,B/M, if i<fc.

To estimate 7jfc, note that

K» * wtL <

2(k-j)(If+1)2k", if j ^ fc, 2(j-k)(I^+1)2j", if j < fc,

which follows easily from the moment conditions on £ and y. Furthermore, Cj * is supported in the ball ||z| < max (2-j,2-fc)|, inwhich

mk,A,B (z) < *

1, if j ^ fc

2(k-j)A, if j<k.

By the last two estimates,

2(k-j)(Ic+1), if j^fc 2(j-k)(Ir+1-A), if j < fc.

We put this estimate in (36) and see that if we choose > 2A - [s] and take into account L^ ^ [s], then we arrive at

2jSC;,A,B/(^)sl2-£|fc-,|2fcs^fc'A,B/(x) (40) fc=0

with some e > 0. It is easy to see that, in the right side of (40), we have essentially the convolution with the sequence |2-efc|,

which is of course a bounded operator on any l?, 0 < q < to. Now by Lemmas 9 and 10 for ) we easily obtain

12^, B/(0f

l2fc>fc*,A, ,/(•)? fc=0

Y^llf* fll?

llSj>A>-^ Hlp«(R»)

Li(->(R") 1/?

LPH(R")

Il^'kR")

In other words, we reduce matters to prove (11) and (12) with = <Po, C = <p. Below we do it only for (11); the argument for (12) is similar.

Let 0 < r < q and A > A0 := n/r + maxj-s, 0}.By Lemma 5 and a discrete version of the Hardy inequality

TO /TO

X2^e(X|fefc|) <X2^>fcr (fl.T>0), (42) j=0 \fc=j / j=0

which we apply with 0 := ^(A + s - n/r) and r = ^/r, we have

+ (|<Pj */|r)(^)

Proof. We only give the proof for F^(R") and for it

canbeprovedbythe similarway. Weuse thesimilar argument in [25]. Let f e S^(R") and ye D with supp y c B(0,1). We set C0 = y and j = 0 in the left side of (11). Analyzing the proof of Theorem 2 shows that only finite numbers of derivatives of the kernels are involved in the estimates, and therefore we know

I / *y(--y) I

y6R" (1 + |y|r2№

LPH(R")

llFsf. (R") p() jajSL

sup I D"y|, (44)

where L is a constant and depends on _p_, s, n, but not on / and y.

It is easy to know

I / * y(") |

sup J-t-L = sup - „

y6R» (1 + |y|)A2l^ M6R" (1 + |x - w|)A2j*-"jB

sup . .

I«-X|S1 (1 + |w-x|)A2j*-"jB

| / * y(") |

Wetake || • H^em») on both sides of the last inequality and

I /*y(--y) I

y6R" (1 + |y|r2№

L?()(R")

I / * y(") |

sup-1-J-1-

M6R» (1 + |--M|)A2j-"jB

L?()(R")

I / * y(")

j.-«j<1 (1 + |--M|)A2j-"jB

LPH(R")

^ I /*y(") |

s(1 + |v|)a2iv|b ^

LPH(R")

> | /*y(w) |. By (44) and (46) we have

Note that Br • p_/r ^ ß0 and 1 < ^/r < to. Let r := p0; by

Lemmas 6 and 8, the operators f ^ jMioc(|/|r)}1/r and f ^ (l/|r)}1/r are all bounded on Li( )(l?). Hence the desired estimate (11) with £0 = <p0, £ = <p, follows.

This finishes the proof. □

3. Some Applications

In this section, we will consider the completeness, the lifting property, and the related quasinorms of these spaces introduced in previous section.

Theorem 11. Let se R, 0 < q < to, and p(-) e P0(R") with 0 < _p0 < minjp_,^} such that e Bloc(R"). Iften the

quasinormed spaces B^^R") and F*(?) (R") are quasi-Banach spaces.

I / * y(^) I < ||/||f« (R») ( sup |-D"y| ) exp (N|x|), (47)

where N, L are constants and depend on _p_, s, n, but not on / and y.

Then we know that the following estimate

I / * yM |

< H/ll^ (Rn) sup ||D"y (x)| exp (N |*|) : * e R", |«| ^ L|

is valid for all / e S'e (R") and ye D with some constants N, L which may depend on _p_, s, n, but not on / and y. Thus we obtain that F^^R") is continuously embedded in S^(R").

Now we conclude the proof of the theorem in a normal way. If a sequence of distributions is Cauchy sequence in

F^?) (R"), then by (48) it converges "pointwise." By the completeness of D', the sequence has a limit f in D'. Again by (48), we have f e S^(R"), since Cauchy sequences are bounded. Finally, by Lebesgue's theorem on dominated convergence it is easily seen that fj ^ f in Fs^)(Rn).

This finishes the proof. □

In next context, we study the action of the Bessel potential operators in our Besov and Triebel-Lizorkin spaces with variable exponent. More precisely, we consider the following t-dilated version:

T = (id - t2A) a/2, ae R, t >0,

where id denotes the identity operator.

For f e S(R") this operator acts by the rule Tatf = (Ka)t *f, where

Ka = F-1 {(\ + 4n2\x\2)-a/2}

formally j е2"'м(1 + 4n2\x\2)-a/2dx e S' (R»).

It is well known that if a > 0, then Ka e L1 representation

n) and has the

(x) ~ \

a-n -n\x\2/u2 -u2/An

и e 1 1 ' e '

(see Stein's book [28] for these matters), from which it follows rather easily that Ka with a > 0 is Cm away from the origin and

\DaKa (x)| < e

\x\ Z 1

with an absolute constant E > 0. By the identity Ka = (id - A)NKa+2N, N e N,

we see that for a ^ 0 the distribution Ka agrees in R" \ {0} with a Cm function, which again satisfies (52).

By thesameargumentinpage170 of[25]weknowthatthe convolution K" * f can be defined as an for any f e B^(Rn) u F^R"), provided that t ^ t0(n) ■ p_. The next theorem states explicitly where it acts.

Theorem 12 (the lifting property). Let p(-) e P0(R") with 0 < p0 < min{p_,q} such that p(^)/p0 e Bloc(Rn). Then there is a constant t0 = t0(n) > 0 so that for all 0 < q < >x>, s e R, and every positive t < t0 ■ p_ one has

T (RB)

T : F% (R")

ß^q (r") isomorphically, F^"'4 (R") isomorphically.

Proof. The idea of the proof comes from [25]. We use again (13) with f and f having vanishing moments up to large order L. By an argument similar to that one used above to define Tat

on Besov and Triebel-Lizorkin spaces with variable exponent, one can establish the identity

* (K *f) = TK *<Pi * Vj * ((P} *f)> le N0.

From [25] by choosing L sufficiently large, we have

2г("я) \fl * (Kat * f) (x)\ $ (x),

£>0, t <

2B ln 2

Now by using Theorem 2 with A = A0 and B = B0/p_, it follows easily that if f belongs to B^ (R") or F^ (R"),then Kat *

f is in B^'q(Rn) or Fs+'q(Rn), respectively. Then the condition on t becomes t < t0 ■ p_ with t0 = E/2B0 ln 2.

The fact that the maps in (54) are actually onto follows

from the identity TtaTat = id. This finishes the proof.

It follows from Theorem 12 that \\TtsfWp«*(R„) is an equivalent quasinorm on Fp^R") for small t > 0 and analogously for Bpl^R"). The next theorem gives a version of this result for s e N involving "pure" derivatives.

Theorem 13. Let s e N, 0 < q < rn, and p() e P0(Rn) with 0 < p0 < min{p_,q} such that p(^)/p0 e Bloc(Rn). Then for any \a\ ^ s

I\\Daf\\P0*, Vfe S'e,

I\\Daf\\B7), *fe S,

Proof. For brevity, we only give the outline of the proof for Triebel-Lizorkin space with variable exponent; for Besov space with variable exponent it can be proved by similar way. The 'V inequality follows immediately from Definition 1 by partial integration and invoking Theorem 2.

The 'V inequality for s even follows from Theorem 12. To obtain it for s odd, it suffices to consider the case s = l.In view of Theorem 12, it is sufficient to prove the estimate

\\K1 *

Y \\Daf\\P«.4 for small t.

MS1 pH

From [25] again, we have

Vj *f = 2-ï(<PV)j * 477■

v=1 UXV

All this leads to the following counterpart of (56): [8

\fi * * f) Ml ^ 2-£VCU.B/ (*)

12-^'K/ yjAB ijf jw,

j= 1 ]=1 V UXv '

£ > 0, (60)

Acknowledgments

References

where, for v = 1,..., n, <pv e D has vanishing moments up to order L - 1 and satisfy

-1 (£)'■ (6« [12

From (58), it is easily deduced by virtue of Theorem 12. [13 This finishes the proof. □

Conflict of Interests [14

The authors declare that there is no conflict of interests

regarding the publication of this paper. [15

The authors would like to thank the referee for his carefully reading which made the presentation more readable. Jingshi Xu is supported by the National Natural Science Foundation [17 of China (Grant nos. 11071064, 11361020, and 11226167) and the Natural Science Foundation of Hainan Province (no. 113004). [18

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