Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 153849,21 pages doi:10.1155/2012/153849

Research Article

Pointwise Multipliers of Triebel-Lizorkin Spaces on Carnot-Caratheodory Spaces

Yanchang Han1 and Fang Wang2

1 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

2 School of Geographical Science, Guangzhou University, Guangzhou 510006, China

Correspondence should be addressed to Yanchang Han, hanych@scnu.edu.cn and Fang Wang, wfdili@163.com

Received 19 April 2012; Accepted 7 June 2012

Academic Editor: Yongsheng S. Han

Copyright © 2012 Y. Han and F. Wang. 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.

Let (X,d,f) be a Carnot-Caratheodory space, namely, X is a smooth manifold, d is a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. f is a nonnegative Borel regular measure on X satisfying that there exists constant C0 € [1, to) such that for all x € X and 0 < r < diam X, f(B(x, 2r)) := f({y € X : d{x,y) < 2r}) < C0f(B(x,r)) < to (doubling property). Using the discrete Calderon reproducing formula and the Plancherel-Polya characterization of the inhomogeneous Triebel-Lizorkin spaces developed in Han et al., in press and Han et al., 2008, pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces are obtained.

1. Introduction

The multiplier theory of function spaces has been studied for a long time, and a lot of results have been obtained. As we know, the multiplier theory is one of the important parts in the studies of the Gleason problem, function space properties, and general operator theory. The pointwise multipliers on Rd are studied as a part of the researches of function spaces in several monographs, [1-8]. Pointwise multipliers have been found many important applications in partial differential equations.

However, it was not clear how to generalize the pointwise multipliers on R" to spaces of homogeneous type introduced by Coifman and Weiss (see [9]) because the Fourier transform is no longer available. The main purpose of this paper is to establish pointwise multipliers on inhomogeneous Triebel-Lizorkin spaces in the setting of Carnot-Caratheodory spaces. To be more precisely, we first recall some necessary definitions. In this paper, we always assume that (X, d) is a metric space with a regular Borel measure f such that all balls

defined by d have finite and positive measures. In what follows, set diam(X) = sup{d(x,y) : x,y e X}, and for any x eX and r > 0, set B(x,r) = {y eX: d(x,y) < r}.

Definition 1.1 (see [10]). Let (X,d) be a metric space with a Borel regular measure ¡i such that all balls defined by d have finite and positive measures. The triple (X,d,i) is called a space of homogeneous type if there exists a constant C0 e [1, to) such that for all x eX and r > 0,

¡(B(x,2r)) < C0p(B(x,r)) (doubling property). (1.1)

Remark 1.2. We point out that the doubling condition (1.1) implies that there exist positive constants C and n such that for all x eX and X > 1,

¡¡(B(x,Xr)) < CXni(B(x,r)), (1.2)

where C is independent of x and r. Denote by n the homogeneous "dimension" of X as in [10].

A space of homogeneous type is called an RD space, if there exist constants a0, C0 e (1, to) such that for all x eX and 0 <r < diam(X)/fl0,

C0^(B(x,r)) < ¡(B(x,a0r)) (1.3)

that is, some "reverse" doubling condition holds.

Clearly, any Ahlfors n-regular metric measure space (X, d, ¡i) (which means that there exists some n > 0 such that ¡(B(x,r)) ~ rn for x e X and 0 < r < diam(X)/2) is a (n,n)-space (see [10]), also is an RD space and space of homogeneous type in the sense of Coifman in [10]. In other words, ¡i satisfies the doubling condition which is weaker than Ahlfors n-regular metric measure space and RD-spaces.

Another typical such a space is Carnot-Caratheodory space. One example with unbounded total measure studied in [11] is that X arises as the boundary of an unbounded model polynomial domain in C2. Let ^ = {(z,w) e C2 : Im(w) > P(z)}, where P is a real, subharmonic, non-harmonic polynomial of degree m. Then X = can be identified with C x R = {(z,t) : z e C,t e_R}. The basic (0,1) Levi vector field is then Z = d/dz - i(dP/dz)(d/dt), and we write Z = Xi + iX2. The real vector fields {Xi, X2} and their commutators of order < m span the tangent space to X at each point. See [10, 12] for more details and references therein.

We will also suppose that ¡(X) = to,^({x}) = 0 for all x e X. For any x,y e X and 6 > 0, set Vs(x) = ¡(B(x,S)) and V(x,y) = ¡(B(x,d(x,y))). It follows from (1.1) that V(x,y) ~ V(y,x). The following notion of approximations of the identity on RD spaces was first introduced in [10]. Let Z+ = N u{0}.

We begin with recalling the definition of an approximation to the identity, which plays the same role as the heat kernel H(s,x,y) does in Nagel-Stein's theory [11].

Definition 1.3 (see [10,12]). A sequence {Sk}keZ+ of operators is said to be an approximation to the identity (in short, ATI) if there exists constant C1 > 0 such that for all k e Z+ and all x,x',y, and y' e X, Sk (x,y), the kernel of Sk satisfies the following conditions:

S,(x,y) = 0 if p(x,y) > Q2-k, \Sk(x,y) \ <

| Sk(x, y) - Sk(x', y) \ < 2kp(x, x')

V2-k (x) + V2-^y)'

V?-, (x) + V>-,(y) for p(x,x') < max{C1,1}2-k,

- S„(x.y>) \ < 2V(W) V2-,(x) + V2-, (y) (15)

for p(y,y') < max{Ci, 1}2-k,

\ [S,(x,y) - S,(x,y')] - [S,(x',y) - S,(x',y')\\ < 22,p(x,x')p(y,y')

V2-, (x) + V2-,(y)

for p(x,x') < max{Ci, 1}2-k and p(y,y') < max{Ci, 1}2-k,

Sk(x,y) d^(y) ^ = 1. (1.7)

J X J X

The space of test functions plays a key role in this paper; see [10].

Definition 1.4. Fix two exponents 0 < ¡5 < 1 and j > 0. A function f defined on X is said to be a test function of type (¡, j) centered at x0 e X with width r > 0 if there exists a nonnegative constant C such that f satisfies the following conditions:

lf(x)| < C(Vr(xo)+ V(x,xo)) (r + d(x,xo)Y' (1.8)

|f(x) f(x)| < C^r + d(x,x00 (Vr(x0)+V(x,x0))(r + d(x/x0))r (1'9)

for d(x,x') < (1/2)(r + d(x,x0)).

If f is a test function of type (¡, j) centered at x0 with width r > 0, we write f e M(x0,r,p,j), and the norm of f in M(x0,r, p,j) is defined by

Mx0,r,p,r) = inf{C > 0 : (1.8) and (1.9) hold}. (1.10)

We denote by M(ß,J) the class of all f e M(x0,1,ß , y). It is easy to see that M(x1r r , ß , j) = M(ß,Y) with the equivalent norms for all x1 eX and r > 0. Furthermore, it is also easy to check that M(ß, Y) is a Banach space with respect to the norm in M(ß, Y).

In what follows, for given e e (0,1], we let Jl(fi,y) be the completion of the space J(e,e) in J(fi,y) when 0 < ¡, y < e. Obviously Jl(e,e) = J(e,e). Moreover, f e Jl(fi,y) if and only if f e J(fi,y) when 0 < ¡, y < e and there exists {fj }yeN c J(e,e) such that

Wf - fj\\j(5,t ) ^ 0as j ^ to. If f eJi(5,y ^ we then define \\f Wi^Y) = \\f II J(P,J). Obviously Jl(p,y) is a Banach space, and we also have \\f \\y) = limy ^TO\\fj\\j(P,r) for the above chosen

{fj }jeN.

We denote by (Jl(fi, y))' the dual space of Jl(fi, y) consisting of all linear functionals L from Jl(p, y) to C with the property that there exists a constant C such that for all f e Jl(fi, y),

\Af )\< C\\f lU(5ry). (1.11)

We denote by (h,f) the natural pairing of elements h e (Jl(fi,y))' and f e Jl(fi,y). Since Jl(x1,r,p,y) = Jl(p,y) with the equivalent norms for all x1 e X and r > 0. Thus, for all h e (Jl(5,y))', (h,f) is well defined for all f e Jl(x0,r,p,y) with x0 eX and r > 0.

The following constructions, which provide an analogue of the grid of Euclidean dyadic cubes on spaces of homogeneous type, were given by Christ in [13].

Lemma 1.5. Let X be a space of homogeneous type. Then there exist a collection {Qa c X : k e Z+,a e Ik} of open subsets, where Ik is some (possible finite) index set, and constants 6 e (0,1) and C5, C6 > 0 such that

(i) ¡(X \ UaQk) = 0 for each fixed k and Qk n Q^ = 0 if a /

(ii) for any a, ¡5, k, l with l > k, either Ql c Qk or Ql n Qk = 0;

(iii) for each (k, a) and each l < k there is a unique 5 such that Qk c Q^;

(iv) diam(Qk) < C26k;

(v) each Qk contains some ball B(zk, C36k), where zka e X.

In fact, we can think of Qk as being a dyadic cube with a diameter roughly 6k and centered at zka. In what follows, we always suppose 6 = 1/2. See [14] for how to remove this restriction. Also, in the following, for k e Z+,t e Ik, we will denote by Qk'v,v = 1,..., N(k,T, M), the set of all cubes Qk+M c Qk, where M is a fixed large positive integer.

Now, we can introduce the inhomogeneous Triebel-Lizorkin spaces Fpq(X) via the approximation in Definition 1.3. Note that the Triebel-Lizorkin spaces have been already investigated for decades in the study of partial differential equations, interpolation theory, and approximation theory.

Definition 1.6. Suppose that {Sk}keZ+ is an ATI and let D0 = So, and Dk = Sk - Sk-1 for k e N. Let M be a fixed large positive integer, Q{°v be as above. Suppose that -1 < s < 1.

The inhomogeneous Triebel-Lizorkin space F^ (X) formax(n/(n+1),n/(n+1+s)) <p < to and max(n/(n + 1),n/(n + 1 + s)) < q <to is the collection of f e (Jl(fi,y))', for some 5 and y satisfying

max( 0,s,-s + n(-1-r - 1 ) ) < 5 < 1, -'-f—r - n<y < 1 (1.12)

min p, 1 min p, 1

Journal of Function Spaces and Applications such that

Fspq (X)

N(0,T,M)

Teln v=1

(1.13)

LP (X)

where mgov (D0(/)) are averages of D0(/) over Q°'v.

The restrictions (1.12) guarantee that the definitions of the inhomogeneous Triebel-Lizorkin space Fp,q(X) for max(n/(n + 1), n/(n + 1 + s)) < p < œ and max(n/(n + 1),n/(n + 1 + s)) < q < œ are independent of the choices of ^ and y satisfying these conditions and Fp'q(X) c (MT(^r))', MT(^r) c MM) c Fp,q(X) in [10].

The classical scale of inhomogeneous Triebel-Lizorkin spaces contains many well-known function spaces. For example, if a> 0,p = q = œ, one recovers the Holder-Zygmund spaces Ca(X), that is,

FT(X) = Ca(X), a > 0. (1.14)

The space Ca(X) is defined as the collection of f such that

e«(X) = llflL + sup lf (1 - 1 < (1.15)

y d(x,y)

If -1 < s < 1 and 1 < p < to, are the Bessel potential spaces

(Lebesgue spaces, Liouville spaces). If m = 0,1,2,... and 1 <p < to, then Wp" (X) = Hpm (X) = Fpm,2(X) are the usual Sobolev spaces. If n/(n + 1) < p < 1, then Fp,2(X) = hp(X) are the inhomogeneous Hardy spaces, which are closely related to the Hardy spaces Hp (X) in [15, 16] (more precisely: the homogeneous spaces Fp/2(X) coincide with the usual Hardy spaces Hp(X)). We will use here the notation Fp,2(X) = Hp(X), s e (-1,1), n/(n + 1) <p <to. Then these spaces will be denoted as the (inhomogeneous) Hardy-Sobolev spaces, which include the above Lebesgue-Sobolev spaces for 1 <p < to.

The inhomogeneous Triebel-Lizorkin spaces have the following Plancherel-Polya characterizations in [10], which will be one of the the basic tools to prove the main results of this paper.

Lemma 1.7. Let {Dk}keZ+ be as in Definition 1.6, -1 < s < 1. Thenifmax(n/(n+1),n/(n+1+s)) < p < to and max(n/(n + 1),n/(n +1 + s)) < q < to, for all f e (Jl(fi,y))' with ¡,y satisfying (1.12), one has

N(0,r,M)

Z Z (Do

teIo v=1

{<x N(r,k,M)

k=1reIk V=1 N(0,T,M)

reI0 v=1

2 f^f)(z)IXQk,v(■)

q> 1/q

Z E I Do

f N(T,k,M)

^ k=1T<Elk V=1

t q^ 1/q

2 sup |Dk(f)(z)IxqP(■)

(1.16)

We now introduce the following definition of the pointwise multiplier.

Definition 1.8. Suppose that g is a given function on X. Then g is called a pointwise multiplier for Fp,q(X) if f ^ gf admits a bounded linear mapping from Fp,'q(X) into itself.

The main result in this paper is the following.

Theorem 1.9. Let -1 < s < 1, max(n/(n + 1),n/(n +1 + s)) <p < to and max(n/(n + 1),n/(n + 1 + s)) < q < to and a > max(s,n/min{p,q, 1} - n - s), then g e Ca(X) with 0 < a < 1 is a multiplier for Fp,'q(X). In other words, f ^ gf yields a bounded linear mapping from Fp,q(X) into itself and there is a positive constant C such that

\\gf\l Fp,q(X) < C\\g\\c«(X)\\f \\Fp-q(X) (L17)

holds for all g e Ca(X) and f e Fp,q(X).

We would like to point out that the study of pointwise multipliers is one of important problems in the theory of function spaces. It has attracted a lot of attention in the decades since starting with [7]. Pointwise multipliers in general spaces Fp,q (Rn), where 0 <p < to, 0 < q < to, s e R have been studied in great detail in [4,5, 8].

Theorem 1.9 was proved in [8] for pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces on Rn based on the Fourier transform. In the present setting, however, we do not have the Fourier transform at our disposal. Since the Fourier transform on Carnot-Caratheodory spaces is not available and hence the idea used in [8] does not work for this more general setting. A new idea to prove Theorem 1.9 is to use the discrete Calderon reproducing formula, which was developed in [10]. Therefore, this scheme easily extends to geometrical settings where the Fourier transform does not exist. The Fourier transform is missing but a version of pointwise multiplier is still present.

Throughout, we also denote by C a positive constant independent of main parameters involved, which may vary at different occurrences. Constants with subscripts do not change through the whole paper. We use f < g and f > g to denote f < Cg and f > Cg, respectively. If f < g < f, we then write f ~ g. For any a,b e R, set a A b = min{a, b},a V b = max{a, b}. If p> 1, set 1/p + 1/p' = 1.

2. Proof of Theorem 1.9

In this section, we will prove Theorem 1.9. Since there is no the Fourier transforms on spaces of homogeneous type, the proof of Theorem 1.9 is quite different from the proof of Theorem 2.8.2 in [8]. The key new ingredient in the proof of Theorem 1.9 is to apply the following discrete Calderon reproducing formulae established in [10,12]. This formula can be stated as follows.

Lemma 2.1. Suppose that {Sk}keZ+ is an approximation to the identity as in Definition 1.3. Set Dk = Sk - Sk-1 for k e N and D0 = S0. Then for any fixed M e N large enough, there exists a family

of functions {D k (x,y)}keZ+ and {D k (x,y)}keZ+ such that for any fixed yk'v e Qk'v, k e N,t e Ik and v e {1,...,N(k,r,M)} andall f e (M(p,j))' with 0 < < e and x e X

N(0,T,M) *

f(x)^E E \ 0vD0(x'y)dKy)mQ°T-v(D0(f))

TeI0 v=1 •'qt'v N(k,T,M)

+ EE E v(QkT'v)Dk(x'ykT'v)Dk(f)(ykT'v)

keNT elk v=1 N(0'T'M) *

= E E Tv Do (x' y) d^y) mQT'v(Do (f)

T^Tn v= 1 J QTv V

N(k,T,M)

keNT eIk v=1

EE E v(Qkk'v)Dk(x,ykk'v)Dk(f)(ykk'v),

where the series converges in the norm of Fp'q(X) with max(n/(n + 1), n/(n + 1 + s)) < p < to and max(n/(n + 1),n/(n + 1 + s)) <q < to,-1 < s < 1, and M(p', j') for f e M(p,j) with ¡5' < p and Y < j, and (M(p', Y))' for f e (M(p,j))' with e > p' > p and e > y' >Y. Moreover, Dk(x,y)

and Dk (x,y), the kernels of Dk and Dk, satisfy the similar estimates but with x and y interchanged in (2.3): for 0 < e< 1,

I ~ I 1 2-ke

ID^x'y) 1 < CV2-k(x) + V2-k(y) + V{x,y) (2-k + d(x,y)Y' (2.2)

I ~ ~ f I / d(x, x ) \ 1 2

\Dk(x,y) - Dk(x',y) | < C^2-k + d{x,y)) V2-k(x) + V2-k (y) + V(x,y) (:2-k + d(x,y))e'

for d(x,x') < (2-k + d(x,y))/2,

D k(x,y)d^(y) = D k(x,y)dp(x) = 0, (2.4)

J X J X

when k e N; J^ Dk(x,y)dp(y) = J^ Dk(x,y)dp(x) = 1 when k = 0. To prove Theorem 1.9, we first show the following lemma.

Lemma 2.2. Let {Sk(x,y)}keZ++ and {Gk(x,y)}keZ++ be two approximations to the identity as in Lemma 2.1 above and Dk = Sk - Sk-1,Ek = Gk - Gk-1 for k e N and D0 = S0,E0 = G0. For any given e e (0,1) and g e Ca(X) with 0 < a < e, then

I ~ I , 1 2-(kAk')e

IEkgDkix,y) | < \\g\\ca(x)T]kk]aV2-kk>Mx)+ V^k*) (y) + V(x,y) (2-(k^)+ d(x,y))e'

where k, k' e Z+.

Proof. We first show the inequality (2.5) of the case k = k' = 0 above. In fact, in this case, it follows that

0gD0(x,y)I < \\g\\ca(X)j E0(x,z)D 0 (z,y)d^(z)

1 1 (Z6) < \\g\L 1 1

1 Ca(X) V1 (x) + V1 (y) + V(x,y) (1 + d(x,y))e' We next consider the case k' > k for k> 0 and k' > 0 or k > 0 and k' > 0. We write lEkgDk'(x,y) | = j[Ek(x,z)g(z) - Ek(x,y)g(y)]DDk-(z,y)d^(z)

< J" | Ek(x, z) - Ek (x, y) 11 g(z) 11Dk (z, y) | dp(z) (2.7)

+ j I Ek (x,y)^Ig(z) - g (y) 11D k(z, y) | dMz)

:= E + F. For E, we can obtain

1 2-ke

E < \\g\\ 2-(k'-k)e_1__2_ (28)

< Ilgllca(X) V2-(k,k>)(x) + V2-(k,k>) (y) + V(x,y) (2-k + d(x,y))e, (2.8)

where k, k' e Z.

Journal of Function Spaces and Applications We may rewrite this integral as

1 2-ke

F <||g|| 1 2

Ca(X)V2-k(x) + V2-k(y) + V(x,y) (2-k + d{x,y)Y

( a 1 2-k'e Jwi d(-z'y V2-k-(z) + V^ (y) + V(z,y) (2-k' + d(z,y)yd^z)

( a 1 2-k'e JW2 V2-k' (z) + V2-* (y) + V(z,y) (2-k + d(z,y))ed^(z\

1 2-k^ := ||g||ca(X) V2-k(x) + V2-k (y) + V(x,y) (2-k + d(x,y))e [Fl + f2]-

where W1 = {z : d(z,y) > 2-k} and W2 = {z : d(z,y) < 2-k}.

Observe further, a simple argument yields the same estimate for k, k' e Z+ as follows:

œ ç 1 2-ke

Fl < ,?kJ iz:2Kd(z,v)<2i+M dz' y ^k^yT^kTyy^VTzy) (2-k' + d(z,y))e ^

if-kJ{z:2i<d(z/y)<2i+1 ^^y V2-k-(z) + V2-k (y) + V(z,y) (2-k' + d(z,y)Y

(2.10)

< 2'+ )) 2(i+1)a2-k'e2-ie < 2-(k'-k)a

< .¿J ,,/R/,, niW < < '

KB(y, 20)

where we used 0 < a < e. For F2, we have

F2 < 2-k'a < 2-(k'-k)a. (2.11)

Combining the estimate for F, we conclude that if k' > k, then (2.5) holds. The case k > k' is similar to above. This finishes the proof of Lemma 2.2. □

Now we show the following technical version of Theorem 1.9. Proposition 2.3. For any g e Ca(X), f e J(5, y) with ¡5 and y satisfying (1.12), then

||fg|| fs/(x) < c|g|ea(X)|f |f;,(x)' (2.12)

where -1 < s < 1, max(n/(n+1) ,n/(n+1+s)) <p < œ and max(n/(n+1),n/(n+1+s)) <q <œ and a > max(s,n/min{p,q, 1} - n - s).

Proof. Using the Calderon reproducing formula, for any g e Ca(X),f e jfl(p, Y), We write

\\gf\\F;q(x)

N(0,T,M)

< i E E

^t elo V=1

N(0,T',M)

E E mQ0v

t'gIo V'=1

E0gDQ0f ) ^(Q0/)

mQ0f (dq(f))

N(0,T,M)

. E E KqQ'v)

^t eiq v=1

» N(k',T',M)

EE E KQk'v')mQTv

k'=1T'eIk, v'=1

x( |EogD k'(

;ykJ'v') I) Dk' fly:

(f )( I

^ 2ksq

N(0,T',M)

^ e KQ°Z) EkgDQqf (•)

T'eIQ v'=1

mQfi |Do( f I)

P A 1/P

& N(k',T',M)

EE E KQk^)|EkgDDk<-,y^)ID(f)(yTfc/v')|

LP (X) q A 1/q

k'=1T'eIy v'=1

:= L1 + L2 + L3 + L4.

Applying the Holder inequality for p > 1 and

|) <Z\aklp

LP (X)

(2.13)

(2.14)

for all ak e C and p < 1, for L1, it follows that

{N(0,T,M) N(q,T',M)

E E E ^Q°Q'')mQQf (Dq (f))

t eIQ v=1 t' eIQ v'=1 T

inf inf

xeQT'v yeQ0'f

V1(x)+ V^y) + V(x,y) (1 + d(x,y))e_

{.N(q,T ',M)

e e kqqo

T'eIQ v'=1

mQ,,'(DQ( f))

kqq'tk yQV)]

\\g\\c a<X)\\f \

(2.15)

where we used the fact that for any y0/ e Q0/, Vl(yQ'v')^(Q0/).

For L2, in fact for all k' e N,r' e Ik' and v' = 1,2,..., N(k',r', M), and all x e X,z e

^Qk''^ ~ V2-k'(z), Vi (yky) + v(x,ykT'y) ~ Vi(z) + V(x,z), (2.16) Vi(z) < 2k'n^Qkk/v'), V2+1 (z) < 2in V1 (z), 1 + d(x,yk,vV) ~ 1 + d(x,z). (2.17)

From the inequality (2.5), the Holder inequality for q > 1 and (2.14) for q < 1, the Fefferman-Stein vector-valued maximal function inequality in [17] and Lemma 1.7, it follows that

L2 < lull*w|E E ^Q°,v)

N(0'T'M)

^t elo v=1

N(k',r ',M)

^2-k'(s+a^ E 2k'S\Dkif)(yk;'v')\

T'elk v'=1

xeQ0vV1(x) + V^') + ^x,yTfc^) (1 + d(x,yk;v))\

N(0,T,M)

e E KqO,v)

TeI0 v=1

N(kl,r' ,M)

E^M E E ^QkTy) r\2k'SDkif)(yky)\

k'=1 \ t 'eIk' v'=1

(*( *?") + + V(x0,v,yTfc>'))

(1+d^y; ,*))£

c»(x)

< |g||c«(X)

N(0,T,M)

e E KQ0,V)

TeI0 v=1

V"12-k'(s+a)

¿1 x?,v)

x0,v ) Jd(x?,v,z)<1 T'eik;

N(k',t',M) Vi x

2k'sDi

' (f)(yT'/V'^) \ \ Xq^v (z) dH(z)

Vi(z)1-r V2-1 (z)

l=0 V2^(4'V) J2><d(4'\z)<2>+1 pQy} 1-r Vl(z)'-r

N(k ,T ,M)

^ E \2k'SDkif)(yk;'v')\ Xgk'Z (zMz)

r'ely v = 1

< II^Mc(X)

2-k'(n+s+a-n/r)

/ N(k' ,T, M) \

m( £ X 2ksr\Dk'(f)(/k'v)\xQk'/ )

\r'<Elk' v'=1 T J _

< IIgM

ca (X)

y1 2~k' (n+s+a-n/r)(qf\ 1) k'=1

/ N(k',r',M) \

m( £ X 2ksr \ Dk' (f )(yk''V)\lQk''X )

\T '^Ik' V'=1 T J

q/r ~~

MgMc a Mf Mi

(2.18)

where we choose r satisfying max(n/(n + e),n/(n + s + a)) <r < min{1,p,q}. By the inequality (2.5), similarly,

N(0,T',M)

E E vQV) mQV (\D0(f)\)

T' I V'=1 QT'

X T'eI0 v'=1

(2.19)

(V!(x) + V^yOV) + V(x,y0?))P1 (1 + d{x,yr))

e(pA 1)

Ca(X)\\ f\Fi'q(X)

where s can be any number in (-e,a) c (-1,1).

We now consider the estimate of L4, Similar to the estimate of L2, using the equality (2.5), the Holder inequality if q > 1 and (2.14) if q < 1, the Fefferman-Stein vector-valued maximal function inequality in [17], we obtain

l4 £ 1ы1е«ш

£ 1Ы1е«ш

Ё Ё [2(fc-fc )s 2-|fc'-fc|a 2~k'n 2<-клк")п 2[fc'-(fcAfc')]™/rj

k=1k'=1

/ N<k',T',M) ! \

m( £ E 2ks Dk (f )( y^') IQk;y I)

V'elk, v;=1 V .

<2.20)

/ N<k',T',M) \

M( X E 2ksIDk; (f )(yT-;V)I -Xq^v )

\T;eIk; v;=1 T /

LP <X)

Lp <X)

£ 1Ы1е«<х)Н/llf« <X),

where the first and the second inequalities follow from the estimates

supj^2(k-k')s 2_|k/-k|a 2~k'n 2<kлk')n 2[k'-<^k')]n/r < £ k k;=0

sUpV i2<k-k')s 2-\k-k\a 2-kkn 2<kлk')n 2[k'-<^k')]n/r 1 qЛl < £ k; k=0L J

<2.21)

with max(n/(n + 1),n/(n + s + a)) < r < min(1,p,^}, s can be any number in (-e,a), which verifies Proposition 2.3. □

To introduce our definition of pointwise multiplication, the interesting estimate is

needed.

Lemma 2.4. Let {Sk(x, y) }keZ+ be я approximation to the identity as in Lemma 2.1 above and Dk = Sk - Sfc-i for k e N and D0 = So. For any g e Ca(X) with 0 < a < 1, h e Л(в,у) with в and у satisfying (1.12). Then

КDk(.yg,H)I <c||g||w,PIUM2-k^'Vi<M)tv(„,„> (ГГHyXtf ■ <122)

wfere k e Z+.

Proof. We first prove inequality (2.22) with k = 0. In fact, since DQ(z,y) = 0 if d(z,y) > 2C1, then

k do( •ygh \ < M gM

Ca X^Mßj)

d(y,z)<2C1 V\(z) + Vx(y) V1 (xo) + V(xo,z) (1 + d(z,xo))Y d^(z)

= MgMCa(X)MMß„. ,

J d(y,z)<2C1,d(xo,z)>d(xo,y)/2 J d(y,z)<2Ci,d(xo,z)<d(xo,y)/2

:= MgMca{X)\\h\\Mß,r)№ + ß2]-

(2.23)

For B1, since V(xQ,y) = ¡(B(xQ,d(xQ,y))) < ¡(B(xQ,2d(xQ,z)))^(B(xQ,d(xQ,z))) = V(xQ, z), it follows that

Vt(xo)+ V(xo,y) (1 + d(y,xo))

(2.24)

For B2, the fact d(y,z) < 2C1 ,d(xQ,z) < d(xQ,y)/2 implies that 2C1 > d(y,z) > d(xQ,y)/2, then

B2 <-<

Vl(xo) + V(xo,y) ~ V1(xo) + V(xo,y) (1 + d(y,xo))Y

(2.25)

That is, (2.22) with k = 0 holds.

To prove (2.22) with k e N, we write

\iDk (^,y)g,h)\ = Dk (x,y)[g(x)h(x) - g (y)h(y)]d^(x)

< \Dk (x,y)\\g(x) - g (y)\\h(x)\d^(x)

+ \Dk(x,y)\\g(y)\\h(x) - h(y)\d^(x)

:= I + II.

(2.26)

For I, since Dk(x, y) = 0 if d(x, y) > 2C12-k, we obtain

I < \\g\\

d(x,y)<2C12-k V2-k (x) + V2-k (y) V[(xq) + V(xo,x) (1 + d(x,xo))Y

I d(x,y)<2C12-k,d(x,y)<(1/2)(1+d(y,x0)) J 2C12-k>d(x,y)>(1/2)(1+d(y,x0))

= \\g\\ca{X)MMMrkaI + I2 ].

(2.27)

For I1, by d(x,y) < (1/2)(1 + d(y,xQ)) implies that d(y,xQ) < 2(d(x,xQ) + 1) and

v1(xo) + v(xq,x) ~ V1(xq) + ¡(B(xQ,d(xQ,y))) V1(xq) + V{xo,y)'

(2.28)

V[(xo) + V(xo,y) (1 + d(y,xo))Y Jd(x,y)<2C12-k V2-k (x) + V2-k (y) 1 1 Vl(xq) + V(xo,y) (1 + d(y,xo)y.

(2.29)

For I2, by 2C12-k > d(x,y) > (1/2)(1 + d(y,xo)) implies that

1 + d(y,xQ) < C,

V2-k (x) + V2-k.(y) ~ V(y,x) ~ V1(xQ) + V(xQ,y)

(2.30)

V1(xo) + V(xo,y) (1 + d(xo,y)y Jx V1(xo) + V(xo,x) (1 + d(x,xo))Y

(2.31)

Vl(xq) + V(xo,y) (1 + d(xo,y))

To obtain the estimate of the II, we write

II< IUII

d(x, y)

\Dk (x,y)\

V1(xq) + V(x0,y) (1 + d{y,xo))M 11

V1(xo) + V(xq,x) (1 + d(x,xo))r

1 1 Vx(xo) + V(xo,y) (1 + d(y,xo)))

c«(x) WWUm [II1 + II2 + II3],

(2.32)

where m1 = {x e X : d(x,y) < (1/2)(1 + d(y,xQ))} and m2 = {x e X : d(x,y) > (1/2)(1 +

d(y,xo))}.

For II1, we have

Vl(xq) + V(xo,y) (1 + d(xo,y))'

\Dk(x,y) \ dy(x)

Vl(xq) + V(xo,y) (1 + d(xo,y)y

(2.33)

For II2, similar to I2, we have

\Dk(x,y) \

V1(xo) + V(xo,y) (1 + d(xo,y)y

(2.34)

V1(xo) + V(xo,y) (1 + d(xQ,y)y J. Vl(xq) + V(xo,x) (1 + d(x,xo))r 1 2-kp

Vl(xq) + V(xo,y) (1 + d(xo,y))r

(2.35)

Journal of Function Spaces and Applications For II3, obviously

Vi(xo)+ V(xo,y) (1 + d(y,x0))r Jx 2-kß 1 Vi(xo)+ V(xo,y) (1 + d(y,xo))r

\Dk(x,y) \ dy(x)

(2.36)

Combining the estimate of I and II, we obtain that for k e N

KDk(;yS.H)\ < C\\S\\„„»MiM(ßr)2-k'ß"'vi(xo) + v(xo, y (1 + d.(y,xo))r- (237)

Thus, (2.22) also holds. This finishes the proof of Lemma 2.4. □

If g e Ca(X) and f e Fp,q (X) for s e (-1,1), max(n/(n + l),d/(d + e + s)) <p< to, max(n/(n + 1),d/(d + e + s)) < q < to, it is not clear in general what is meant by gf (pointwise multiplication). Our approach is the following.

Lemma 2.5. For any f e Fp,q (X) with max(n/(n + 1),n/(n + 1 + s)) <p < to and max(n/(n + 1),n/(n+l+s)) < q < to,-1 < s < 1,and g e Ca(X) with 1 > a> max(s, (n/min{p,q, 1})-n-s). There exist a constant C and a sequence {fj }jeN such that fj e Jt(e, e), \\fj\\Fspq(X) < C\\f \\pspq(X) and limy ^TO(gfj,h) converges for any h e J(p,j) with ¡5 and y satisfying (1.12).

Proof. For any f e Fp,q(X) with max(n/(n + 1),n/(n + 1 + s)) < p < to and max(n/(n + 1),n/(n +1 + s)) < q < to,-1 < s < 1, and g e Ca(X) with a > max(s, (n/min{p,q, 1}) - n - s), denote

j N(o,T,M) .

fj = Z E ^^KH^f) )Dq?v(x)

T=1 V=1 ^ '

(2.38)

j j N(k,T,M)

+ EE E y(QkT'v)Dk(x,ykT'v)D k (f)(ykr'v),

k=1T=1 V=1

where DQo,v(x) = (1/y(Q°'V)) jQo,v Do(x,y)dy(y).

It is easy to see that fj e Jl(e, e). Applying a similar proof as in Proposition 2.3 with g = 1 and f = fj gives ||fj < C||f ||F^(X). We write

Kfj - Ugh)| <

j N(0,t,M) -

E E viQ^y^Dofj^^gh

T=m+1 v=1 ^ '

j j N(k,T,M) x , , x ,

E E E yT,v)'gh)Dk(f)(yk'v)

k=m+lT=1 v=1

j j N(k,T,M)

k=lT=m+1 v=1

:= R + T + Y.

E E E ^Qkr'V){Dk(•,y^), gh)Dk(f)(ykT'v)

(2.39)

The fact (2.22) implies

j N(0,t,M)

E E KQo'OIDQ^(gh|

T=m+1 v=1

^T>logm ' {y:2T<d(xo,y)<2T+1}

Vi(xo) + V(xo,y)

(1 + d{y,xo))Jp

< Mc«WllhlU(ß,r)| E 2-TYP' 1

7 dKy)

LT>iogm

(2.40)

j N(0,t,M)

E E KQ0'O|DQ?,4gh|F

T=m+1 v=1

Ff (X)

,m<d(x0 ,y°/)<jV1 (Xo) + V(xo, y°V) (1 + d(yT'V, Xo))

||g|c(X)|f HfT (X)HhIU(ßr)m

: p> 1

: p < 1,

(2.41)

Fps (X)

this finishes the proof of R ^ 0 as j,m ^ to.

We now consider the estimate of T. ByHolder inequality if p,q > 1, and (2.14) if p,q < 1 and (2.22), it follows that

E \2-ksDl(gh) \q

Lp (X) 1-1/P.

Z sup2k[-s+^]^Qk,v) " P\D*k(gh(x)\

k=m+1 xeX

sup \2-ksDk(gh)\

m+1<k<i

sup sup 2-ksp(Qkv) 11/P\Dk(gh)(x)\

xeX m+1<k<i

E \2-kSDk(gh\

Lp (X)

\\g Wc-X) Wf Wf? wWhWßY )2~ms-™(1-1/r>>*W ,

2-m[s+aAß]

2-m [s+n(1-1/p)+aAß]

: p> 1, q > 1,

: p < 1, q > 1, : p> 1, q < 1, : p < 1, q < 1,

p> 1, q > 1,

p < 1, q > 1, p> 1, q < 1, p < 1,q < 1,

(2.42)

where let e be a positive number with s - e + n(1 - 1/p)+aAfi>0 when p < 1 and s + a Aft > 0 if p > 1, and the second inequality can be obtained by using the fact that

FS'q(X) c Bp,max(p,q) (X) c Bsi£'œ(X),

(2.43)

when -e < s - e, see [10, Proposition 5.31].

By the Proposition 2.3, then gh e FpS'q (X) if s e (-1,1) ,p,q> 1. Thus by the Calderon

reproducing that the series converges in the norm of F-Sq (X) for gh e F^(X) with s e (-1,1),p,q> 1 in Lemma 2.1, which proves T ^ 0 asj,m ^ œ whenmax(n/(n + 1),n/(n + 1 + s)) <p < œ and max(n/(n + 1),n/(n + 1 + s)) < q <œ, -1 < s < 1, and g e Ca(X) with a > max(s, (n/min{p,q, 1}) - d - s).

For Y, applying the Holder inequality for p,q > 1 and (2.14) for p,q < 1 and (2.22), we also have

Fp'q(X) \ \{y\m<d(xo,y)<i} j

Bp £,m

Z\2-ksD*k(gh)(y)\q\ dy(y) k=1

1-1/p\

^ p'/q ^

J dKy) f

1 11/ J -œ(X)sup sup 2-k[s-]y(Qk'v)l~Vv\Dk(gh)(y) \

keN ye{y\m<d(x0,y)<i}

Fp,q(X)î \{y\m<d(x0,y)<i}

sup\2-ksDKgh)(y) \[ dy(y)

sup sup 2-ksy(Qk,v)

ye{y\m<d(xo,y)<i)1<k<i

* I \Dk(gh(y)\

: p> 1, q > 1, : p < 1, q > 1,

: p> 1, q < 1, : p < 1,q < 1,

Fps(X)

Fp (X)

c«( x )

llgHc«cx)H/II

ll*llc(X)ll/ll

ll*llc«(x)llf II

< ll* llca(X)llf ""

y-k(s+aAß)q'

Fp-q(X)\\n\\Mß,ry

sUp2-k[s-£+n(i-i/p)+(«Aß)]q'

Fp-q (X)

WM^m-r sup2-kts+aAß]

Fp-q(X)\\h\U(ßr)

m-Y sUp2-k[s+™(1-1/p)+aAß]

Fp-q ex)

\\h\U(ßr)

: p > 1- q> 1: p < 1- q > 1: p > 1- q < 1: p < 1- q < 1-

(2.44)

where let e be a positive number with s - e+n(1 - 1/p) +a A^ > 0 when p < 1 and s+a A^ > 0 if p> 1, and we also used the fact that Fsq(X) c Bp'max(p'q) (X) c Bp-e,TO (X), when -e < s-e. This finishes the proof of Y ^ 0as j,m ^ to, and hence the proof of Lemma 2.5 is concluded. □

The above estimate shows limy^TO(gfj,h) exists and the limit is independent of the choice of fj. Therefore, for g e Ca (X), f € Fp,q (X) with max(n/(n + 1), n/(n + 1 + s)) <p < to and max(n/(n +1), n/(n +1 + s)) < q <to s e (-1,1), a > max(s, (n/min{p,q, 1}) - n - s). We define

(gf-h) = lim (gfj-h)

(2.45)

for any h e Jl(p,y) with ^ and y satisfying (1.12), /y is fundamental sequence defined in Lemma 2.5.

We now prove Theorem 1.9.

Proof of Theorem 1.9. By Proposition 2.3 and Lemma 2.5, for any g e Ca(X),/ e Fp,q (X), Fatou's lemma implies that

llgfly

{N(O-T-M)

E E Kß0-v)

t elo v=1

E 2ka lim I ^^ Cfjg) I

^^ j ^œ

lim £o( fyg)

LP (X)

< lim inf Ufy

lg llca(X) lf If

(2.46)

We complete the proof of Theorem 1.9.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant no. 4o8o1o34).

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