Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 610649,9 pages doi:10.1155/2012/610649

Research Article

Boundedness of Weighted Hardy Operator and Its Adjoint on Triebel-Lizorkin-Type Spaces

Canqin Tang and Ruohong Zhou

Department of Mathematics, Dalian Maritime University, Liaoning, Dalian, 116026, China Correspondence should be addressed to Canqin Tang, tangcq2000@dlmu.edu.cn Received 26 October 2010; Accepted 18 November 2010 Academic Editor: Hans Triebel

Copyright © 2012 C. Tang and R. Zhou. 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 p € [1, to], q € [1, to), t € (0, to), and a € (0,1) such that t > 1/p - 1/q and a < n(1/p - t), let Uf be the weighted Hardy operator and Vf its adjoint operator with respect to the weight function f. In this paper, the authors establish a sufficient and necessary condition on weight function f to ensure the boundedness of Uf and Vf on the Triebel-Lizorkin-type spaces Fp,',q(Rn) and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results on O-type spaces.

1. Introduction

This paper focuses on the boundedness of the weighted Hardy operator Uf and its adjoint operator Vf on Triebel-Lizorkin-type spaces and their predual spaces. Recall that, for a fixed function f : [0,1] ^ [0, to), the weighted Hardy operator Uf is defined by

Uff (x) = I" f (tx)f(t)dt, x € Rn, (1.1)

see Carton-Lebrun and Fosset [1]. Accordingly, the adjoint operator Vf of Uf, named by the weighted Cesaro average operator, is defined by

f1 /X\ (1.2) Vvf (x) = J f (-j)rnf(t)dt, x € Rn.

This paper is motivated by the following facts. On one hand, Uf is related to the Har-dy-Littlewood maximal operators in harmonic analysis. If f = 1 and n = 1, then Uf goes

back to the classical Hardy-Littlewood average U:

Uf (x) = - f (x)dy, x / 0. (1.3)

Its adjoint operator Vf is the classical Ces&ro average operator:

! f(y)

V/(x) = '

if ^dy, Jx y

x > 0,

r m <L4)

J—^dy, x< 0. y

Xiao in [2] obtained the boundedness of Uv on BMO(Rn), the boundedness of Vf on H1(Rn), and their corresponding operator norms. Recall that Qa(Rn) can be viewed as a generalization of BMO(Rn) (cf. [3]). Recently Tang and Zhai in [4] obtained the boundedness and norm of Uv on Qa(Rn). They even work to more general spaces Qp,q(Rn). The boundedness of Uv and Vf on Q°p'q(Rn) and its dual space are worked out. Recall that Q spaces (Qa spaces) were originally defined by Aulaskari et al. [5] in 1995 as spaces of holomorphic functions on the unit disk, which are geometric in the sense that they transform naturally under conformal mappings, and later, in 2000, were extended to the n-dimensional Euclidean spaces Rn by Essen et al. By [3, Theorem 2.3], Qa(Rn) is always a subspace of BMO(Rn). As a generation of Q spaces, the spaces Qpq(Rn) when a e (0,1) and 2 < q <p < to were first introduced by Cui and Yang [6] and later extended to a e (0,1), p e (0, to], and q e [1, to] in [7]. We also refer to [4, 8-11] for more studies on these spaces.

On the other hand, it is well known that Besov spaces f>aq(Rn) and Triebel-Lizorkin spaces ip^R") allow a unified approach to various types of function spaces, such as Holder-Zygmund spaces, Soblev spaces, Hardy spaces, BMO, and Bessel-potential spaces. Over hundreds of papers and books are focused on these spaces and their applications. Very recently, Yang and Yuan [7, 12] introduced the scales of homogeneous Besov-Triebel-Lizorkin-type spaces Bp,fq(Rn) and tpq(Rn) for all a e R, t e [0, to), and p,q e (0, to]. These spaces unify and generalize the homogeneous Besov spaces Bp^(Rn), Triebel-Lizorkin spaces Jt"aq(Rn), and the spaces Qa(Rn) simultaneously. The following equivalent definition of the spaces Fpaq(Rn) were given in [7, Section 3].

Definition 1.1. Let a e (0,1), p e (0, to), q e (1, to), and t e [0, to]. The space (Rn) is defined to be the space of all measurable functions / on Rn such that

1 i /• / /• wq * p/q 11/p

,,,=sup' {{(J f/q«)p/q.4 <to, (1.5)

,q I11 [Jl\J I x-y | <2/(I) \x - y\ q J j

where the supremum is taken over all cubes I with the edges parallel to the coordinate axes in Rn .

Motivated by the above facts, a natural question is following: under what condition on f the weighted Hardy operator and its adjoint are bounded on Triebel-Lizorkin-type spaces

F^q (Rn)? The main purpose of this paper is to give the sufficient and necessary condition on f such that the operators Uf and Vf are bounded on Triebel-Lizorkin-type spaces Fp,q (Rn). We have the following main results.

Theorem 1.2. Let f : [0,1] ^ [0, to) be a function, and let p e [1, to], q e [1, to), t e (0, to), and a e (0,1) such that t > 1/p - 1/q and a < n(1/p - t). Then Uf : Fpq(Rn) ^ Fpq (Rn) exists as a bounded operator if

C rn(1/p-T)+af(t)dt< to. (1.6)

Moreover, when (1.6) holds, the norm of Uf on Fpq (Rn) is given by

llUf llfpq(Rn) ^(Rn) = t n(1/p t)+V (t)dt. J 0

We give the proof of Theorem 1.2 in Section 2.

Finally, we make some conventions on notations. Throughout the paper, Rn denotes the n-dimensional Euclidean space, with Euclidean norm | x|, and Lebesgue measure dx. A cube I will always mean a cube in Rn with the edges parallel to the coordinate axes with sidelength l(I) and volume | I|. The dilated cube XI, X > 0, is the cube with the same center as I and sidelength Xl(I).Forall q e (0, to], q' denotes the adjoint index of q, namely, 1/q+1/q' = 1. C will often be used to denote a positive constant, but it may vary from line to line.

2. Proof of Main Results

We begin with the proof of Theorem 1.2.

Proof of Theorem 1.2. Suppose (1.6) holds. If f e Fpq (Rn), then for any cube I c Rn, applying Minkowski's inequality, we have

jii CL-yl

\Uf (x) - Uf (y)\q d ^p/q 1

I \ n+qa dy

<2l(I) \x - y\ q

)p/q ^

i |x-y|<2l(i)

m, /:{L a

\ 1 \q \ p/q ^ 1/p

u:f(tx) -f(ty)]y(t)dt\ \

-\-n+a-dy I

\ x - y\ n qa

l f (tx) - f (ty)|

0 \J| x-yl <2l(I) \x - y\

\f (tx) - f (ty)\q

l\J|x-yl<2l(I) \x - y\

n+qa tyJ 9(t)dt dx f

P ^ 1/P dx

f(t) dt.

Notice that

f ( f ffn dy

J^J|x-y|<2/(i) |x - y I ^

q \ p/q

■ UJ,

Then, by (1.6), we have

tl \J | u-v| <2l(tI)

|q \ p^ |u - v|n+a

lf (u) " f (V)|qdvl du.tP-"

|I| [Jl \J|x-y|<2/(I) 1 x - y1 q J j

< ' J1 |f(f \ff^dv)'/qdu}"''^f (t)dt

R | Jo [ J tl \J u-v^tl) u - v| q J j

< WfWtzcrO £ ta-n(1/p-T)^(t)dt.

Thus, the operator Uv is bounded on F^(Rn) with operator norm no more than

Conversely, if Uf is bounded on F^ (Rn), then we can choose the function

|-|xp(1/p-T)^r x e R-

r ta-n(1/p-r(t)dt. (2.4)

/c(x) = i (1/ ) l (2.5)

¡Jxp(1/p-T)+a, x e Rn,

where RJ1 and Rn denote, respectively, the left and right halves of Rn, separated by the hyperplane xi = 0 (xi is the first coordinate of x e Rn).

We now show that 0 < II/oII^t(R„) < to. Notice that the assumption on a implies that t < 1/p. For any cube I in Rn, we consider two cases.

Case 1. If p < q, then p/q < 1. Since t > 1/p - 1/q, then a> 0 > n(1/p - 1/q) - nT, and hence

f |/o(z)|qdz J I+h

<f |z|[-n(1/p-T)+a]qdz + f |z|[-n(1/p-T)+a]qdz

JlnilzKlhl) JI nilzMhl)

'In{|z|<|h|} ^ In{|z|>|h|]

< |h|[-"(i/p-T)+"]q+n + \h\[-n(i/p-T )+a]q |i1

which together with Holder's inequality, t > 1/p - 1/q, yields that

If (x + "^dTdx

h | <2l(I) | h | n+qa

< tt (U

h|<2i(I) J I+h | h| n+qa J | 1 (2.7)

|h|L-n(1/p-T)+«]q+n + |h|[-n(1/p-T)+a]q |i | ^p/q ^ ^ (r Ihl[-n(1/p-T)+a]q+n + |h|[-n(1/p-T)+a]qm ^ p/q

< h-¡r+h-1 dh\ |I|1-p/q

[J|h|<2i(i) |h| q

< Cl(I)nTp.

L |f0w|'"fe hbdh\'m""q

Case 2. If p > q, then p/q > 1. Using Minkowski's inequality, similar to the computation of Case 1, we also obtain

I h <2l(I)

|f](x + h)|q

| h | n+qa

)p/q "J dx

J |h|<2l(I) JI

< Cl(I)nT.

If0(x + h) |q |h|n+qa

q\ p/q

Combining Cases 1 and 2 yields that

IT {/i(U ^^ -P* }

^ (|w<2Wi^fqdh)p/qdx},/" ,29>

namely, f e Fp,q(Rn) and 0 / ||f0 H^t(rn) < to.

Uff0(x) = f0(x) I" tn(1/p-T)+af(t)dt, 0

(2.10)

we then have

n\taq (Rn) ^ paq («»)

= i"1 r(n/p-T)+> (t)dt, :

(2.11)

which completes the proof of Theorem 1.2.

Remark 2.1. In fact, if we choose p = q and t = 1/p - 1/r, then

i"1 ta-n(1/p-TV(t)dt = i"1 ta-n/rf (t)dt ::

(2.12)

Therefore, Theorem 1.2 is consistent with [4, Theorem 4.1] since

The weighted Hardy operator Uf and the weighted Cesaro average operator Vf are adjoint mutually, namely,

g(x)Uvf (x)dx = f (x)Vvg(x)dx.

J Rn J Rn

(2.13)

Recall that

Vf (x) = J" ^x)t-n9(t)dt, x e Rn

(2.14)

then we have

{Ii (I

\Vf (x) - Vf (y)\ d

n+qa dy

x-y| <2l(I) \x - y\

'IV Jlx-yl <2l(I)

f / f \lo1[f(x/t) - f (y/t)]t-n9(t)dt\q \

-\-in+qa-dy \ dx

Jl V Jl x-y l <2l(I) \x - y\ J

I0(i..........

01 { I x-y

l x-y l <2l(I) \x - y\

\f(x/t) - f (y/t)\q

I x-y <2l(I) \ x - y\

T> (t)dt,

(2.15)

Journal of Function Spaces and Applications where

f (f f 7

JI\J\x-y\<2l(I) \x - y\ /

^^ ^^ (2.16)

\f (u) - f (v)\q

J(1/t)I \J\u

v\<2i((1/t)I) \u - v|

_ vn+qa

dv) du- t-pa+n.

Similar to the proof of Theorem 1.2, we have next theorem.

Theorem 2.2. Let f : [0,1] ^ [0, to) be a function, and let p e [1, to], q e [1, to), t e (0, to), and a e (0,1) such that t > 1/p - 1/q and a < n(1/p - t). Then Vf : Fpq(Rn) ^ F^(Rn) exists as a bounded operator if

C t-n(1-1/p+T)->(t)dt < to. (2.17)

Moreover, when (2.17) holds, the norm of Vf on F^T (Rn) is given by

9 HPS:

( rn(1-1/P+T)-«9(t)dt.

(2.18)

Recall that the Triebel-Lizorkin-Hausdorff spaces FH-,^(Rn) were originally introduced in [7,12] and proved therein to be the predual spaces of Triebel-Lizorkin-type spaces F^q(Rn). Triebel-Lizorkin-Hausdorff spaces unify and generalize Triebel-Lizorkin spaces (see, e.g., [13]) and Hardy-Hausdorff spaces HH-a in [14]. These spaces were further studied in [15].

By dual argument, we have the following theorems.

Theorem 2.3. Let f : [0,1] ^ [0, to) be a function, and let p e [1, to], q e (1, to], t e (0, 1/(max{p,q})'], and a e (0,1) such that t > 1/p' - 1/q' and a < n(1/p' - t). Then Uf : FH%T(Rn) ^ FHpaT(Rn) exists as a bounded operator if

C rn(1/p,-T)->(t)dt < to. (2.19)

Moreover, when (2.19) holds, the norm of Uf on FH-CqT (Rn) is given by

«llpic

n) ^ PH-

= I" t-n(1/p'-T)-af(t)dt.

(2.2o)

Theorem 2.4. Let f : [0,1] ^ [0, to) be a function, and let p e [1, to], q e (1, to], t e (0, 1/(max{p,q})'], and a e (0,1) such that t > 1/p - 1/q' and a < n(1/p' - t). Then Vf : FHpOq^(Rn) ^ FHP0qT(Rn) exists as a bounded operator if

C rn(1-1/p,+T)->(t)dt < to. (2.21)

Moreover, when (2.21) holds, the norm of Vf on FHpaq,T(Rn) is given by

\Vf\ I(R") ^ (R")

= I" t-n(1-1/p,+T)-y(t)dt. (2.22)

Remark 2.5. When p = q' and t = 1/q' - 1/r, then

( t-n(i-i/p>+T)-af (t)dt = ( t-n(1-1/r)-af(t)dt, (2.23)

and Theorem 2.4 is consistent with [4, Theorem 5.8] since FHq,J/q-1/r (Rn) is predual space of Q^R").

Our Theorems actually induce the following results:

Il Vf II Flyq(Rn) ^(Rn) II II Ffia;T(Rn) ^Ffiaq(R")'

ll^llFiyq (Rn) ^ Ffi-aT, (Rn)= II ^Wfh^ (Rn) ^ Ffiaq (Rn).

(2.24)

In particular,

11 U<f\ I FlT,^/,-1/p (R") ^ FJÏ-,aq1/,-1/p(Rn) " Il Vf II Qaq (R") ^ Qa'q (R")' IlV II FÏÇ'1 /,-1/p(R") ^FYT*1/q-1/p(R") = Il II Q^R") ^Qaq(R").

(2.25)

Acknowledgments

The authors cordially thank Professor D. C. Yang and Dr. W. Yuan for their valuable comments. This work was supported by the Fundamental Research Funds for the Central Universities (20 1 1QN058) and (20 1 1 QN150).

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