Accepted 13 August 2006 We study extended Hardy inequalities using Littlewood-Paley theory and nonlinear esti- mates’ method in Besov spaces

JUNYONG ZHANG

Received 2 May 2006; Revised 2 August 2006; Accepted 13 August 2006

We study extended Hardy inequalities using Littlewood-Paley theory and nonlinear esti- mates’ method in Besov spaces. Our results improve and extend the well-known results of Cazenave (2003).

Copyright © 2006 Junyong Zhang. 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.

1. Introduction

A remarkable result of Hardy-type inequality comes from the following proposition, the proof of which is given by Cazenave [2].

Proposition 1.1. Let 1p <∞. Ifq < nis such that 0qp, then |u(·)|^p/| · |^q∈ L¹(Rⁿ) for everyu∈W^1,p(Rⁿ). Furthermore,

Rⁿ

u(·)^p

| · |^q dx p n−q

_q

u_L^p−pq∇u^q_L^p, (1.1) for everyu∈W^1,p(Rⁿ).

It is easy to see that the proposition fails whens >1, wheres=q/ p. In this paper we are trying to find out what happens ifs >1. We show that it does not only become true but obtains better estimates.

The described result is stated and proved inSection 3. The method invoked is different from that by Cazenave in [2]; it relies on some Littlewood-Paley theory and Besov spaces’

theory that are cited inSection 2.

2. Preliminaries

In this section we introduce some equivalent definitions and norms for Besov space needed in this paper. The reader is referred to the well-known books of Runst and Sickel [5], Triebel [6], and Miao [4] for details.

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 69379, Pages1–5 DOI 10.1155/JIA/2006/69379

We first introduce the following equivalent norms for the homogeneous Besov spaces B˙^s_p,m:

u_B^˙s_p,m

|α|=[s]

_+∞

0 t^−mσsup

|y|t

y∂^αu^m_pdt t

1/m

, (2.1)

where

yuτyu−u, τyu(·)=u(·+y),

∂^α=∂^α1¹∂^α2²···∂^α_nⁿ, ∂_i= ∂

∂x_i, i=1, 2,...,n. (2.2) α=(α1,α2,...,αn) ands=[s] +σ with 0< σ <1, namely,σ=s−[s], where [s] denotes the largest integer not larger thans. In the casem= ∞, the normu_B^˙sp,∞ in the above definition should be modified as follows:

u_B^˙s_p,∞

|α|=[s]

sup

t>0t^−σsup

|y|t

y∂^αu_p, s∈R⁺. (2.3)

We now introduce the Paley-Littlewood definition of Besov spaces.

Letϕ0∈C_c^∞(Rⁿ) with

ϕ0(ξ)=

⎧⎨

⎩

1, |ξ|1,

0, |ξ|2, (2.4)

be the real-valued bump function. It is easy to see that

ϕj(ξ)=ϕ0

2^−jξ, j∈Z, ψj(ξ)=ϕ0

2^−jξ−ϕ0

2^−j+1ξ, j∈Z, (2.5) are also real-valued radial bump functions satisfying that

sup

ξ∈Rⁿ

2^j|α|∂^αψj(ξ)<∞, j∈Z, sup

ξ∈Rⁿ

2^j|α|∂^αϕj(ξ)<∞, j∈Z. (2.6) We have the Littlewood-Paley decomposition:

ϕ0(ξ) +

∞ j=0

ψ_j(ξ)=1, ξ∈Rⁿ,

j∈Z

ψj(ξ)=1, ξ∈Rⁿ\{0},

jlim→+∞ϕj(ξ)=1, ξ∈Rⁿ.

(2.7)

For convenience, we introduce the following notations:

jf =Ᏺ⁻¹ψjᏲf =ψj∗f, j∈Z,

S_jf =Ᏺ⁻¹ϕ_jᏲf =ϕ_j∗f, j∈Z. (2.8) Then we have the following Littlewood-Paley definition of Besov spaces and Triebel spaces:

B˙^s_p,m=

⎧⎨

⎩f ∈᏿Rⁿ

| f_B^˙s_p,m=

j∈Z

2^jsmjf^m_p

1/m

=

j∈Z

2^jsmψ_j∗f^m_p

1/m

<∞

⎫⎬

⎭, F˙_p^s,m=

⎧⎨

⎩f ∈᏿Rⁿ

| f_F^˙s_p,m=

j∈Z

2^jsmjf^m

1/m

p

=

j∈Z

2^jsmψj∗f^m

1/m

p

<∞

⎫⎬

⎭, B˙^s_p,_∞=

f ∈᏿Rⁿ

| f_B^˙s_p,∞=sup

j∈Z2^js_jf_p=sup

j∈Z2^jsψ_j∗f_p<∞

, F˙^s_p,_∞=

f ∈᏿Rⁿ

| f_F^˙s_p,∞= sup

j∈Z2^js_jf

p= sup

j∈Z2^jsψ_j∗f

p<∞

. (2.9)

Remark 2.1. We have the identities (equivalent quasinorms)Lp=F⁰_p,2, ˙H^s=F˙2,2^s =B˙2,2^s . 3. Main result

Theorem 3.1. Let 1p <∞. If 0s < n/ p, a constantC exits such that for any u∈ B˙^s_p,1(Rⁿ),

Rⁿ

u(x)^p

|x|^sp dxCuB^p˙^sp,1. (3.1) Remark 3.2. (i) Ifs=0, the result will be more precise replacing ˙B⁰_p,1by ˙F⁰_p,2.

(ii) Noting interpolation inequality in [1] by Bergh and L¨ofstr¨om between ˙H^0,p and H˙^1,p, the theorem implies the proposition when 0< s <1.

(iii) Ifs=1, the result will be more precise replacing ˙B¹_p,1by ˙F¹_p,2=H˙^1,p. (iv) Ifp=2, we have more precise proposition substituting ˙F2,2^s =B˙2,2^s for ˙B^s2,1. (v) The Hardy-type inequality will be excellent substituting ˙F^s_p,2for ˙B^s_p,1, but it fails using this method, in fact we obtain this estimate:

Rⁿ

u(x)^p

|x|^sp dxCuF^p˙^−sp,2¹u_B^˙s_p,1, (3.2) where ˙F_p^s,2is a Triebel space.

In order to prove the theorem, we need the following two lemmas, the first of which was easily proved using Littlewood-Paley theory in Lemari´e-Rieusset [3] and the other will be proved here.

Lemma 3.3. Letsbe in ]0,n[. Then for anypin [1,∞],| · |^−s∈B˙^{n/ p}p,∞^−s.

Lemma 3.4. Let 1p <∞. If 0s < n/ p, thenu^p∈B˙⁰_q,1for everyu∈B˙^s_p,1, whereq= q/(q−1) andq=n/sp.

Proof ofLemma 3.4. By equivalent definition and norms for Besov space, it is sufficient to establish that

u^pB˙⁰q,1uB^p˙^sp,1. (3.3)

Hence

F_B^˙0_q,1 +_∞

0 sup

|y|≤t

_yF_qdt

t . (3.4)

LetF(u)= |u(x)|^p. Using Newton-Leibniz formula and inequality (|a|+|b|)^p2^p(|a|^p+

|b|^p), we deduce that τ_yF(u)−F(u)=

1

0dFθτ_y|u|+ (1−θ)|u|Cτ_yu^p−1+|u|^p−1τ_yu−u, (3.5) whereCis a constant.

By definition of_yand thanks to the H¨older inequality, we have that

yF_qCu(^pp⁻−¹1)χ1τyu−u_χ2, (3.6) where 1/χ1=(p−1)(1/ p−s/n) and 1/χ2=1/ p−s/n.

Note that

B˙^s_p,1

Rⁿ

L^(p−1)χ1Rⁿ , B˙^s_p,1

Rⁿ B˙_χ⁰₂,1

Rⁿ

. (3.7)

Thus we infer that

u^pB˙⁰_q,1u(^pp⁻−¹1)χ1u_B^˙0_{χ2 ,1}CuB^p˙^sp,1 (3.8)

implying the lemma.

Proof ofTheorem 3.1. Let us define

Is,p(u)

Rⁿ

u(x)^p

|x|^sp dx=

| · |^−sp,|u|^p

. (3.9)

Using Littlewood-Paley decomposition, we can write Is,p(u)=

|j−j|2

j| · |^−sp,j|u|^p Csup

j

_j| · |^−sp

q

j∈Z

_j|u|^p

q

C| · |^−sp_˙

B⁰q,∞u^pB˙⁰_q,1,

(3.10)

whereq=n/sp >1.Lemma 3.3claims that| · |^−spbelongs to ˙B⁰_q,_∞andLemma 3.4claims in particular thatu^p_B^˙0_q,1uB^p˙^sp,1. ThusIs,p(u)CuB^p˙^sp,1, which implies the theorem.

Acknowledgment

The author is grateful to the referees for their valuable suggestions.

References

[1] J. Bergh and J. L¨ofstr¨om, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.

[2] T. Cazenave, Semilinear Schr¨odinger Equations, Courant Lecture Notes in Mathematics, vol. 10, American Mathematical Society, Rhode Island, 2003.

[3] P. G. Lemari´e-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman &

Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Florida, 2002.

[4] C. Miao, Harmonic Analysis and Application to Differential Equations, 2nd ed., Science Press, Beijing, 2004.

[5] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter, Berlin, 1996.

[6] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathe- matical Library, vol. 18, North-Holland, Amsterdam, 1978.

Junyong Zhang: The Graduate School of China Academy of Engineering Physics, P.O. Box 2101, Beijing 100088, China

E-mail address:[email protected]