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

Authors: Canqin Tang; Ruohong Zhou
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

Letp∈[1,∞], q∈[1,∞), τ∈(0,∞), and α∈(0,1) such that τ>1/p-1/q and α≤n(1/p-τ), let Uψ be the weighted Hardy operator and Vψ its adjoint operator with respect to the weight function ψ. In this paper, the authors establish a sufficient and necessary condition on weight function ψ to ensure the boundedness of Uψ and Vψ on the Triebel-Lizorkin-type spaces Ḟp,qα,τ(ℝn) and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results on Q-type spaces.


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## Body

## 1. Introduction

This paper focuses on the boundedness of the weighted Hardy operatorUψ and its adjoint operator Vψ on Triebel-Lizorkin-type spaces and their predual spaces. Recall that, for a fixed function ψ:[0,1]→[0,∞), the weighted Hardy operator Uψ is defined by
(1.1)Uψf(x)=∫01f(tx)ψ(t)dt,x∈Rn,
see Carton-Lebrun and Fosset [1]. Accordingly, the adjoint operator Vψ of Uψ, named by the weighted Cesàro average operator, is defined by
(1.2)Vψf(x)=∫01f(xt)t-nψ(t)dt,x∈Rn.This paper is motivated by the following facts. On one hand,Uψ is related to the Hardy-Littlewood maximal operators in harmonic analysis. If ψ≡1 and n=1, then Uψ goes back to the classical Hardy-Littlewood average U: (1.3)Uf(x)=1x∫0xf(x)dy,x≠0.
Its adjoint operator Vψ is the classical Cesàro average operator:
(1.4)Vf(x)={∫x∞f(y)ydy,x>0,-∫-∞xf(y)ydy,x<0.Xiao in [2] obtained the boundedness of Uψ on BMO(ℝn), the boundedness of Vψ on H1(ℝn), and their corresponding operator norms. Recall that Qα(ℝn) can be viewed as a generalization of BMO(ℝn) (cf. [3]). Recently Tang and Zhai in [4] obtained the boundedness and norm of Uψ on Qα(ℝn). They even work to more general spaces Qpα,q(ℝn). The boundedness of Uψ and Vψ on Qpα,q(ℝn) and its dual space are worked out. Recall that Q spaces (Qα 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 Essén et al. By [3, Theorem 2.3], Qα(ℝn) is always a subspace of BMO(ℝn). As a generation of Q spaces, the spaces Qpα,q(ℝn) when α∈(0,1) and 2≤q<p<∞ were first introduced by Cui and Yang [6] and later extended to α∈(0,1), p∈(0,∞], and q∈[1,∞] 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 spacesḂp,qα(ℝn) and Triebel-Lizorkin spaces Ḟp,qα(ℝn) allow a unified approach to various types of function spaces, such as Hölder-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 Ḃp,qα,τ(ℝn) and Ḟp,qα,τ(ℝn) for all α∈ℝ, τ∈[0,∞), and p,q∈(0,∞]. These spaces unify and generalize the homogeneous Besov spaces Ḃp,qα(ℝn), Triebel-Lizorkin spaces Ḟp,qα(ℝn), and the spaces Qα(ℝn) simultaneously. The following equivalent definition of the spaces Ḟp,qα,τ(ℝn) were given in [7, Section 3].Definition 1.1.
Letα∈(0,1), p∈(0,∞), q∈(1,∞), and τ∈[0,∞]. The space Ḟp,qα,τ(ℝn) is defined to be the space of all measurable functions f on ℝn such that
(1.5)‖f‖Ḟp,qα,τ(Rn)=sup⁡⁡1|I|τ{∫I(∫|x-y|<2l(I)|f(x)-f(y)|q|x-y|n+qαdy)p/qdx}1/p<∞,
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 onψ the weighted Hardy operator and its adjoint are bounded on Triebel-Lizorkin-type spaces Ḟp,qα,τ(ℝn)? The main purpose of this paper is to give the sufficient and necessary condition on ψ such that the operators Uψ and Vψ are bounded on Triebel-Lizorkin-type spaces Ḟp,qα,τ(ℝn). We have the following main results.Theorem 1.2.
Letψ:[0,1]→[0,∞) be a function, and let p∈[1,∞], q∈[1,∞), τ∈(0,∞), and α∈(0,1) such that τ>1/p-1/q and α≤n(1/p-τ). Then Uψ:Ḟp,qα,τ(ℝn)→Ḟp,qα,τ(ℝn) exists as a bounded operator if
(1.6)∫01t-n(1/p-τ)+αψ(t)dt<∞.
Moreover, when (1.6) holds, the norm of Uψ on Ḟp,qα,τ(ℝn) is given by
(1.7)‖Uψ‖Ḟp,qα,τ(Rn)→Ḟp,qα,τ(Rn)=∫01t-n(1/p-τ)+αψ(t)dt.We give the proof of Theorem1.2 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 λI, λ>0, is the cube with the same center as I and sidelength λl(I). For all q∈(0,∞], 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 Theorem1.2.Proof of Theorem1.2.
Suppose (1.6) holds. If f∈Ḟp,qα,τ(ℝn), then for any cube I⊂ℝn, applying Minkowski's inequality, we have
(2.1){∫I(∫|x-y|<2l(I)|Uψf(x)-Uψf(y)|q|x-y|n+qαdy)p/qdx}1/p={∫I(∫|x-y|<2l(I)|∫01[f(tx)-f(ty)]ψ(t)dt|q|x-y|n+qαdy)p/qdx}1/p≤{∫I[∫01(∫|x-y|<2l(I)|f(tx)-f(ty)|q|x-y|n+qαdy)1/qψ(t)dt]pdx}1/p≤∫01{∫I(∫|x-y|<2l(I)|f(tx)-f(ty)|q|x-y|n+qαdy)p/qdx}1/pψ(t)dt.
Notice that
(2.2)∫I(∫|x-y|<2l(I)|f(tx)-f(ty)|q|x-y|n+qαdy)p/qdx=∫tI(∫|u-v|<2l(tI)|f(u)-f(v)|q|u-v|n+qαdv)p/qdu⋅tpα-n.
Then, by (1.6), we have
(2.3)1|I|τ{∫I(∫|x-y|<2l(I)|Uψf(x)-Uψf(y)|q|x-y|n+qαdy)p/qdx}1/p≤1|tI|τ∫01{∫tI(∫|u-v|<2l(tI)|f(u)-f(v)|q|u-v|n+qαdv)p/qdu}1/ptα-n/p+nτψ(t)dt≤‖f‖Ḟp,qα,τ(Rn)∫01tα-n(1/p-τ)ψ(t)dt.
Thus, the operator Uψ is bounded on Ḟp,qα,τ(ℝn) with operator norm no more than
(2.4)∫01tα-n(1/p-τ)ψ(t)dt.
Conversely, ifUψ is bounded on Ḟp,qα,τ(ℝn), then we can choose the function
(2.5)f0(x)={-|x|-n(1/p-τ)+α,x∈Rln,|x|-n(1/p-τ)+α,x∈Rrn,
where ℝln and ℝrn denote, respectively, the left and right halves of ℝn, separated by the hyperplane x1=0 (x1 is the first coordinate of x∈ℝn).
We now show that0<∥f0∥Ḟp,qα,τ(ℝn)<∞. Notice that the assumption on α implies that τ<1/p. For any cube I in ℝn, we consider two cases.
Case 1.
Ifp<q, then p/q<1. Since τ>1/p-1/q, then α>0>n(1/p-1/q)-nτ, and hence
(2.6)∫I+h|f0(z)|qdz≤∫I∩{|z|≤|h|}|z|[-n(1/p-τ)+α]qdz+∫I∩{|z|>|h|}|z|[-n(1/p-τ)+α]qdz≤|h|[-n(1/p-τ)+α]q+n+|h|[-n(1/p-τ)+α]q|I|,
which together with Hölder's inequality, τ>1/p-1/q, yields that
(2.7)∫I(∫|h|<2l(I)|f0(x+h)|q|h|n+qαdh)p/qdx≤{∫I(∫|h|<2l(I)|f0(x+h)|q|h|n+qαdh)dx}p/q|I|1-p/q={∫|h|<2l(I)∫I+h|f0(z)|qdz1|h|n+qαdh}p/q|I|1-p/q≤{∫|h|<2l(I)|h|[-n(1/p-τ)+α]q+n+|h|[-n(1/p-τ)+α]q|I||h|n+qαdh}p/q|I|1-p/q≤{∫|h|<2l(I)|h|[-n(1/p-τ)+α]q+n+|h|[-n(1/p-τ)+α]q|I||h|n+qαdh}p/q|I|1-p/q≤Cl(I)nτp.Case 2.
Ifp≥q, then p/q≥1. Using Minkowski's inequality, similar to the computation of Case 1, we also obtain
(2.8){∫I(∫|h|<2l(I)|f0(x+h)|q|h|n+qαdh)p/qdx}1/p≤{∫|h|<2l(I)[∫I(|f0(x+h)|q|h|n+qα)p/qdx]q/pdh}1/q≤Cl(I)nτ.
Combining Cases 1 and 2 yields that
(2.9)1|I|τ{∫I(∫|h|<2l(I)|f0(x+h)-f0(x)|q|h|n+qαdh)p/qdx}1/p≤1|I|τ{∫I(∫|h|<2l(I)|f0(x+h)|q+|f0(x)|q|h|n+qαdh)p/qdx}1/p≤C,
namely, f0∈Ḟp,qα,τ(ℝn) and 0≠∥f0∥Ḟp,qα,τ(ℝn)<∞.
Since(2.10)Uψf0(x)=f0(x)∫01tn(1/p-τ)+αψ(t)dt,
we then have
(2.11)‖Uψ‖Ḟp,qα,τ(Rn)→Ḟp,qα,τ(Rn)=∫01t-(n/p-τ)+αψ(t)dt,
which completes the proof of Theorem 1.2.Remark 2.1.
In fact, if we choosep=q and τ=1/p-1/r, then
(2.12)∫01tα-n(1/p-τ)ψ(t)dt=∫01tα-n/rψ(t)dt.
Therefore, Theorem 1.2 is consistent with [4, Theorem 4.1] since Qrα,p(ℝn)=Ḟp,pα,1/p-1/r(ℝn).The weighted Hardy operatorUψ and the weighted Cesàro average operator Vψ are adjoint mutually, namely,
(2.13)∫Rng(x)Uψf(x)dx=∫Rnf(x)Vψg(x)dx.
Recall that
(2.14)Vψf(x)=∫01f(xt)t-nψ(t)dt,x∈Rn,
then we have (2.15){∫I(∫|x-y|<2l(I)|Vψf(x)-Vψf(y)|q|x-y|n+qαdy)p/qdx}1/p={∫I(∫|x-y|<2l(I)|∫01[f(x/t)-f(y/t)]t-nψ(t)dt|q|x-y|n+qαdy)p/qdx}1/p≤{∫I[∫01(∫|x-y|<2l(I)|f(x/t)-f(y/t)|q|x-y|n+qαdy)1/qt-nψ(t)dt]pdx}1/p≤∫01{∫I(∫|x-y|<2l(I)|f(x/t)-f(y/t)|q|x-y|n+qαdy)p/qdx}1/pt-nψ(t)dt,
where
(2.16)∫I(∫|x-y|<2l(I)|f(x/t)-f(y/t)|q|x-y|n+qαdy)p/qdx=∫(1/t)I(∫|u-v|<2l((1/t)I)|f(u)-f(v)|q|u-v|n+qαdv)p/qdu⋅t-pα+n.Similar to the proof of Theorem1.2, we have next theorem.Theorem 2.2.
Letψ:[0,1]→[0,∞) be a function, and let p∈[1,∞], q∈[1,∞), τ∈(0,∞), and α∈(0,1) such that τ>1/p-1/q and α≤n(1/p-τ). Then Vψ:Ḟp,qα,τ(ℝn)→Ḟp,qα,τ(ℝn) exists as a bounded operator if
(2.17)∫01t-n(1-1/p+τ)-αψ(t)dt<∞.
Moreover, when (2.17) holds, the norm of Vψ on Ḟp,qα,τ(ℝn) is given by
(2.18)‖Vψ‖Ḟp,qα,τ(Rn)→Ḟp,qα,τ(Rn)=∫01t-n(1-1/p+τ)-αψ(t)dt.Recall that the Triebel-Lizorkin-Hausdorff spacesFḢp′,q′-α,τ(ℝn) were originally introduced in [7, 12] and proved therein to be the predual spaces of Triebel-Lizorkin-type spaces Ḟp,qα,τ(ℝn). Triebel-Lizorkin-Hausdorff spaces unify and generalize Triebel-Lizorkin spaces (see, e.g., [13]) and Hardy-Hausdorff spaces HH-α1 in [14]. These spaces were further studied in [15].By dual argument, we have the following theorems.Theorem 2.3.
Letψ:[0,1]→[0,∞) be a function, and let p∈[1,∞], q∈(1,∞], τ∈(0,1/(max{p,q})′], and α∈(0,1) such that τ>1/p′-1/q′ and α≤n(1/p′-τ). Then Uψ:FḢp,q-α,τ(ℝn)→FḢp,q-α,τ(ℝn) exists as a bounded operator if
(2.19)∫01t-n(1/p′-τ)-αψ(t)dt<∞.
Moreover, when (2.19) holds, the norm of Uψ on FḢp,q-α,τ(ℝn) is given by
(2.20)‖Uψ‖FḢp,q-α,τ(Rn)→FḢp,q-α,τ(Rn)=∫01t-n(1/p′-τ)-αψ(t)dt.Theorem 2.4.
Letψ:[0,1]→[0,∞) be a function, and let p∈[1,∞], q∈(1,∞], τ∈(0,1/(max{p,q})′], and α∈(0,1) such that τ>1/p′-1/q′ and α≤n(1/p′-τ). Then Vψ:FḢp,q-α,τ(ℝn)→FḢp,q-α,τ(ℝn) exists as a bounded operator if
(2.21)∫01t-n(1-1/p′+τ)-αψ(t)dt<∞.
Moreover, when (2.21) holds, the norm of Vψ on FḢp,q-α,τ(ℝn) is given by
(2.22)‖Vψ‖FḢp,q-α,τ(Rn)→FḢp,q-α,τ(Rn)=∫01t-n(1-1/p′+τ)-αψ(t)dt.Remark 2.5.
Whenp′=q′ and τ=1/q′-1/r, then
(2.23)∫01t-n(1-1/p′+τ)-αψ(t)dt=∫01t-n(1-1/r)-αψ(t)dt,
and Theorem 2.4 is consistent with [4, Theorem 5.8] since FḢq′,q′-α,1/q-1/r(ℝn) is predual space of Qrα,q(ℝn).Our Theorems actually induce the following results:(2.24)‖Vψ‖FḢp′,q′-α,τ(Rn)→FḢp′,q′-α,τ(Rn)=‖Uψ‖FḢp,qα,τ(Rn)→FḢp,qα,τ(Rn),‖Uψ‖FḢp′,q′-α,τ(Rn)→FḢp′,q′-α,τ(Rn)=‖Vψ‖FḢp,qα,τ(Rn)→FḢp,qα,τ(Rn).In particular,(2.25)‖Uψ‖FḢq′,q′-α,1/q-1/p(Rn)→FḢq′,q′-α,1/q-1/p(Rn)=‖Vψ‖Qpα,q(Rn)→Qpα,q(Rn),‖Vψ‖FḢq′,q′-α,1/q-1/p(Rn)→FḢq′,q′-α,1/q-1/p(Rn)=‖Uψ‖Qpα,q(Rn)→Qpα,q(Rn).



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