EXTENSIONS OF HARDY INEQUALITY

advertisement
EXTENSIONS OF HARDY INEQUALITY
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 estimates’ 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 1 p < ∞. If q < n is such that 0 q p, then |u(·)| p / | · |q ∈
L1 (Rn ) for every u ∈ W 1,p (Rn ). Furthermore,
u(·) p
Rn
| · |q
dx p
n−q
q
p −q
q
uL p ∇uL p ,
(1.1)
for every u ∈ W 1,p (Rn ).
It is easy to see that the proposition fails when s > 1, where s = q/ p. In this paper we
are trying to find out what happens if s > 1. We show that it does not only become true
but obtains better estimates.
The described result is stated and proved in Section 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 in Section 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, Pages 1–5
DOI 10.1155/JIA/2006/69379
2
Extensions of Hardy inequality
We first introduce the following equivalent norms for the homogeneous Besov spaces
Ḃ sp,m :
u Ḃ sp,m
+∞
t
0
|α|=[s]
m dt
sup y ∂α u
−mσ
p
| y |t
1/m
t
,
(2.1)
where
y u τ y u − u,
τ y u(·) = u(· + y),
∂α = ∂α1 1 ∂α2 2 · · · ∂αnn ,
∂i =
∂
,
∂xi
i = 1,2,...,n.
(2.2)
α = (α1 ,α2 ,...,αn ) and s = [s] + σ with 0 < σ < 1, namely, σ = s − [s], where [s] denotes
the largest integer not larger than s. In the case m = ∞, the norm uḂsp,∞ in the above
definition should be modified as follows:
uḂsp,∞ sup t −σ sup y ∂α u p ,
s ∈ R+ .
| y |t
|α|=[s] t>0
(2.3)
We now introduce the Paley-Littlewood definition of Besov spaces.
Let ϕ
0 ∈ Cc∞ (Rn ) with
⎧
⎨1,
|ξ | 1,
ϕ
0 (ξ) = ⎩
0,
(2.4)
|ξ | 2,
be the real-valued bump function. It is easy to see that
ϕ
j (ξ) = ϕ
0 2− j ξ ,
j ∈ Z,
ψ
j (ξ) = ϕ
0 2− j ξ − ϕ
0 2− j+1 ξ ,
j ∈ Z,
(2.5)
are also real-valued radial bump functions satisfying that
sup 2 j |α| ∂α ψ
j (ξ) < ∞,
ξ ∈Rn
sup 2 j |α| ∂α ϕ
j (ξ) < ∞,
ξ ∈Rn
j ∈ Z,
j ∈ Z.
(2.6)
We have the Littlewood-Paley decomposition:
ϕ
0 (ξ) +
∞
ψ
j (ξ) = 1,
ξ ∈ Rn ,
j =0
ψ
j (ξ) = 1,
ξ ∈ Rn \{0},
j ∈Z
lim ϕ
j (ξ) = 1,
j →+∞
ξ ∈ Rn .
(2.7)
Junyong Zhang 3
For convenience, we introduce the following notations:
j f = Ᏺ−1 ψ
j Ᏺ f = ψ j ∗ f ,
j ∈ Z,
S j f = Ᏺ−1 ϕ
j Ᏺ f = ϕ j ∗ f ,
j ∈ Z.
(2.8)
Then we have the following Littlewood-Paley definition of Besov spaces and Triebel
spaces:
Ḃsp,m
=
⎧
⎨
⎩
f ∈ ᏿ Rn | f Ḃsp,m =
j ∈Z
=
s
Ḟ p,m
2
jsm m
j f p
m
2 jsm ψ j ∗ f ⎫
⎬
1/m
< ∞⎭ ,
p
j ∈Z
⎧
⎨
1/m
1/m m
n
jsm
s
j f
= f ∈ ᏿ R | f Ḟ p,m = 2
⎩
j ∈Z
p
(2.9)
⎫
1/m ⎬
m
jsm <∞ ,
=
2
ψ
∗ f
j
⎭
Ḃ sp,∞
j ∈Z
n
js
js
f ∈ ᏿ R | f Ḃsp,∞ = sup 2 j f p = sup 2 ψ j ∗ f p < ∞ ,
=
j ∈Z
s
Ḟ p,
∞=
p
f ∈ ᏿
Rn | f Ḟ p,s ∞
j ∈Z
js js = sup 2 j f = sup 2 ψ j ∗ f < ∞ .
j ∈Z
j ∈Z
p
p
s
s
0
Remark 2.1. We have the identities (equivalent quasinorms) L p = F p,2
, Ḣ s = Ḟ2,2
= Ḃ2,2
.
3. Main result
Theorem 3.1. Let 1 p < ∞. If 0 s < n/ p, a constant C exits such that for any u ∈
Ḃ sp,1 (Rn ),
u(x) p
Rn
|x|sp
p
dx C uḂsp,1 .
(3.1)
0
Remark 3.2. (i) If s = 0, the result will be more precise replacing Ḃ0p,1 by Ḟ p,2
.
(ii) Noting interpolation inequality in [1] by Bergh and Löfström between Ḣ 0,p and
Ḣ 1,p , the theorem implies the proposition when 0 < s < 1.
1
= Ḣ 1,p .
(iii) If s = 1, the result will be more precise replacing Ḃ1p,1 by Ḟ p,2
s
s
s
for Ḃ2,1
.
(iv) If p = 2, we have more precise proposition substituting Ḟ2,2 = Ḃ2,2
s
s
(v) The Hardy-type inequality will be excellent substituting Ḟ p,2 for Ḃ p,1 , but it fails
using this method, in fact we obtain this estimate:
u(x) p
Rn
s
where Ḟ p,2
is a Triebel space.
|x|sp
p −1
dx C uḞ p,2
s uḂ s ,
p,1
(3.2)
4
Extensions of Hardy inequality
In order to prove the theorem, we need the following two lemmas, the first of which
was easily proved using Littlewood-Paley theory in Lemarié-Rieusset [3] and the other
will be proved here.
n/ p−s
Lemma 3.3. Let s be in ]0,n[. Then for any p in [1, ∞], | · |−s ∈ Ḃ p,∞ .
Lemma 3.4. Let 1 p < ∞. If 0 s < n/ p, then u p ∈ Ḃq0
,1 for every u ∈ Ḃ sp,1 , where q
=
q/(q − 1) and q = n/sp.
Proof of Lemma 3.4. By equivalent definition and norms for Besov space, it is sufficient
to establish that
p
u Ḃq0
,1
p
uḂs .
(3.3)
p,1
Hence
F Ḃq0
,1 +∞
0
dt
sup y F q
t
| y |≤t
.
(3.4)
Let F(u) = |u(x)| p . Using Newton-Leibniz formula and inequality (|a| + |b|) p 2 p (|a| p +
|b| p ), we deduce that
1 τ y F(u) − F(u) = dF θτ y |u| + (1 − θ)|u| C τ y u p−1 + |u| p−1 τ y u − u,
0
(3.5)
where C is a constant.
By definition of y and thanks to the Hölder inequality, we have that
y F C u p −1 τ y u − u ,
(p−1)χ1
q
χ2
(3.6)
where 1/χ1 = (p − 1)(1/ p − s/n) and 1/χ2 = 1/ p − s/n.
Note that
Ḃsp,1 Rn
Ḃ sp,1 Rn
L(p−1)χ1 Rn ,
(3.7)
Ḃχ02 ,1 Rn .
Thus we infer that
p
u Ḃq0
,1
p −1
p
u(p−1)χ1 uḂχ02 ,1 C uḂs
p,1
(3.8)
implying the lemma.
Proof of Theorem 3.1. Let us define
Is,p (u) Rn
u(x) p
|x|sp
dx = | · |−sp , |u| p .
(3.9)
Junyong Zhang 5
Using Littlewood-Paley decomposition, we can write
Is,p (u) =
j | · |−sp , j |u| p
| j − j |2
j |u| p C sup j | · |−sp q
j
j ∈Z
q
(3.10)
C | · |−sp Ḃq,0 ∞ u p Ḃ0
,
q ,1
where q = n/sp > 1. Lemma 3.3 claims that | · |−sp belongs to Ḃq,0 ∞ and Lemma 3.4 claims
p
p
in particular that u p Ḃq0
,1 uḂsp,1 . Thus Is,p (u) C uḂsp,1 , which implies the theorem.
Acknowledgment
The author is grateful to the referees for their valuable suggestions.
References
[1] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.
[2] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10,
American Mathematical Society, Rhode Island, 2003.
[3] P. G. Lemarié-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 Mathematical 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]
Download