Weighted Hardy inequalities
and Hardy transforms of weights
by
Joan Cerdà and Joaquim Martı́n
1
Introduction
If w is a weight on R+ = [0, ∞), we denote W (t) = 0t w(x) dx, and T :
X −→ Y indicates that T is a bounded operator between X and Y , two
function spaces on R+ . X d will denote the subset of all non–increasing and
nonnegative functions (briefly, decreasing functions) of X.
We recall that Ap , if p > 1, is defined by the condition
R
(Ap )
sup
I
1 Z
|I|
1 Z
w(x) dx
|I|
I
0
p−1
w(x)1−p dx
< ∞,
I
where the suppremum is taken over all intervals I and, if p = 1, by M w ≤
Cw. Here M is the Hardy–Littlewood maximal function and it is well–known
(see [Mu1]) that w ∈ Ap if and only if M : Lp (w) −→ RLp (w) (1 < p < ∞).
In [Mu2], the weights w such that S1 f (t) = (1/t) 0t f (x) dx (the Hardy
operator) is bounded on Lp (w) (1 ≤ p < ∞) are described as the weights of
the class Mp , defined for 1 < p < ∞ by the estimate
(Mp )
sup
∞
Z
t
t>0
w(x) 1/p dx
xp
t
Z
0
1/p0
w(x)−p /p dx
< ∞.
0
The class M1 is defined by S2 w ≤ Cw, where S2 f (t) = t∞ f (x)x−1 dx.
It is also a known fact that S2 : Lp (w) −→ Lp (w) if and only if
R
(M p )
sup
t
Z
1/p Z
0
t>0
0
w(x)−p /p 1/p0
<∞
dx
xp0
∞
w(x) dx
t
when 1 < p < ∞. The class M 1 is defined by S1 w ≤ Cw.
Also (see [ArM]), S1 : Lp (w)d −→ Lp (w) for 1 ≤ p < ∞ if and only if w
satisfies
Z
(Bp )
t
∞
w(x)
C
dx ≤ p
p
x
t
Z
t
w(x) dx,
0
(1)
which defines the class Bp (for any p ∈ (0, ∞)). As seen in [So], it is easily
seen that (1) is equivalent to
Z
∞
t
W (x)
W (t)
dx ' p ,
xp+1
t
(2)
i.e., t∞ W (x)x−(p+1) dx ≤ CW (t)t−p , since W is increasing. As usual, F '
G indicates the existence of a universal constant c > 0 so that c−1 F ≤ G ≤
cF .
For weak type estimates, it was proved in [AnM] that S1 : L1 (w) −→
L1,∞ (w) if and only if w ∈ M1,∞ , the class of weights w such that, for some
α > 0,
R
∞ t α w(x)
Z
(M1,∞ )
x
t
x
dx ≤ C(α) inf w(x)
0≤x≤t
(3)
Only this case p = 1 is interesting, since S1 : Lp (w) −→ Lp,∞ (w) implies
S1 : Lp (w) −→ Lp (w) when p > 1 (see [AnM, Theorem 3]).
The restriction of S1 to decreasing functions was studied in [Ne1]. Again,
for 1 < p < ∞, S1 : Lp (w)d −→ Lp,∞ (w) implies S1 : Lp (w)d −→ Lp (w).
If p = 1, it is proved in [CGS] that S1 : L1 (w)d −→ L1,∞ (w) if and only
if w belongs to the class B1,∞ defined by the condition
1
t
(B1,∞ )
Z
0
t
1
w(x) dx ≤ C
s
s
Z
w(x) dx if s ≤ t.
0
Remark 1.1 Neugebauer also proved that the property S2 : Lp (w)d −→
Lp (w) does not depend on p ∈ [1, ∞) (cf.[Ne2]), and it holds if and only if
(B 1 )
Z
t
(S1 w)(x) dx ≤ C
Z
t
w(x) dx
0
0
i.e., S1 S1 w ≤ CS1 w. This condition defines the class B 1 of weights.
2
Monotone weights
The following facts will be applied to powers W α of W .
Proposition 2.1 If w is decreasing, the following properties are equivalent:
R
R r+2h
w(x) dx (r, h ≥ 0).
(a) w is doubling, i.e., rr+h w(x) dx ' r+h
(b) w ' S1 w.
(c) w ' M w, i.e., w ∈ A1 .
(d) w ∈ Ap for one (or all) p > 1.
(e) w ∈ M p for one (or all) p ≥ 1.
(f ) w ∈ B p for one (or all) p ≥ 1.
(g) inf x>0 w(rx)/w(x) > 1/r for some r > 1.
2
Proof If w is doubling, then w(r) ≤ (1/r) 0r w(x) dx ≤ (C/r) r2r w(x) dx ≤
Cw(r), i.e., w ' S1 w. Since we are assuming that w is decreasing, S1 w '
M w and then w ' M w. Obviously, (d) follows from (c), and it is known
that Ap –weights are doubling. It was proved in [CM] that w ∈ M p if and
only if w ∈ M 1 , that is S1 w ' w, as well as the equivalence of (e), (f) and
(g).
R
R
Corollary 2.1 If w ∈ Ap on Rn (1 ≤ p < ∞), then w∗ ∈ A1 . Here
w∗ (t) = inf{λ > 0; |{w > λ}| ≤ t}, the non–increasing rearrangement of w.
Proof Since w satisfies [(1/|Q|) Q ws ]1/s ≤ (C/|Q|) Q w (Q any cube) for
some s > 1 and (M ws )1/s ∈ A1 (see [GR]). Thus M w ∈ A1 , since M (M w) ≤
M ((M ws )1/s ) ≤ c(M ws )1/s ≤ cCM w.
But (M w)∗ ' S1 w∗ , thus S1 S1 w∗ ' (M M w)∗ ≤ C(M w)∗ ' S1 w∗ , and
∗
w ∈ B 1 . It follows from Proposition 2.1 that w∗ ∈ A1 .
With the same proof, if w ∈ Ap on a cube Q ⊂ Rn , then w∗ ∈ A1 [0, |Q|]
(cf. [Wi]).
Obviously, any decreasing weight w belongs to Mp , and hence to Bp , for
all p > 1. This is not true if p = 1, as the simple example w ≡ 1 shows. If
p = 1, a decreasing
w belongs to B1 if and only if w ∈ M1 ,
R doubling weight
C Rx
0
since, if w ∈ B1 , x∞ w(s) ds
≤
s
x 0 w(s) ds ≤ C w(x) by Proposition 2.1–
(b), and then w ∈ M1 .
R
R
Proposition 2.2 If w is decreasing, w ∈ M1 if and only if supx>0 w(rx)/w(x) <
1 for some r > 1, and then wα ∈ M1 for any α > 0.
Proof If
R∞
x
w(s)
s
ds ≤ Cw(x) and r > eC , then
(ln r)w(rx) ≤
Z
rx
x
w(s)
ds ≤
s
Z
∞
x
w(s)
ds ≤ Cw(x)
s
and w(rx)/w(x) ≤ C/ln r < 1. Conversely, if δ := supx>0
w ∈ M1 , since
Z
∞
x
∞
X
w(s)
ds =
s
n=0
Z
xr n+1
xr n
w(rx)
w(x)
< 1, then
∞
X
w(s)
ln r
δn =
ds ≤ (ln r)w(x)
w(x).
s
1−δ
n=0
Proposition 2.3 For an increasing weight w and 1 < p < ∞, the following
properties are equivalent:
(a) w ∈ Ap ,
(b) w ∈ Mp ,
R
(c) w ∈ Bp and t∞ w(x)x−p dx ' w(t)t1−p , and
(d) inf x>0 w(rx)/w(x) > rp−1 for some r < 1.
3
Proof By [CU; Corollary 6.3], (c) implies (a) and, since S1 f ≤ M f for any
f ≥ 0, (a) implies (b). Also (b) implies (c), since if w ∈ Mp then w ∈ Bp
and
Z ∞
Z
w(x)
w(t)
1 w(t)
1 t
w(x) dx ≤ C p−1 .
≤
dx
≤
C
p−1
p
p
pt
x
t 0
t
t
If (d) holds and 0 < a < 1 is such that w(rx)/w(x) ≥ ap−1 > rp−1 , then
Z
∞
x
∞ Z x/r n+1
X
w(s)
w(s)
ds =
ds
p
n
s
sp
n=0 x/r
≤
∞
X
n=0
w
x rn+1
x1−p 1 1
−
.
1−p
r
rn(1−p) 1 − p
1
Since w(x) ≥ ap−1 w(x/r), also w(x/rn+1 ) ≤ w(x)a(n+1)(1−p) and it follows
that
Z ∞
1
1 − rp−1
w(s)
ds
≤
w(x)x1−p ,
p
p−1 − r p−1
s
1
−
p
a
x
0
and w satisfies property (c). Conversely, if w ∈ Ap , then w1−p ∈ Ap0 .
0
0
By Proposition 2.1, S1 (w1−p ) ' w1−p and there exists s > 1 such that
inf x>0
3
0
w(sx) 1−p
w(x)
> 1s , and inf x>0 w(rx)/w(x) > rp−1 with r = 1/s.
Bp –weights as derivatives of Ap+1 –weights
The main result of this section states that w ∈ Bp if and only if W ∈ Ap+1 .
Theorem 3.1 Let w be a weight on R+ and 0 < p < ∞. Then w ∈ Bp if
and only if W α ∈ Apα+1 for one (or for all) α > 0.
Proof Since w ∈ Bp is equivalent to (2), it follows from Proposition 2.3
applied to W , that (2) holds if and only if W ∈ Ap+1 . Since W ∈ Ap+1 if
and only if inf x>0 W (rx)α /W (x)α > rpα for some r < 1, i.e. W α ∈ Apα+1 .
Remark 3.1 The above results can be used to see that w ∈ Bp if and only
R
if W α−1 w ∈ Bpα (cf. [Ne1; Theorem 6.5]), since W α (t) = α 0t W α−1 w.
Neugebauer ([Ne1]) presented some properties of Bp suggested by the
analogous properties of Ap , and gave short proofs of facts such as Bp implies
Bp−ε (see also [Ma]). Here we give a very easy proof from the corresponding
result for Ap .
4
Corollary 3.1 (a) If w ∈ Bp (0 < p < ∞), then w ∈ Bp−ε for some
ε ∈ (0, p).
(b) w ∈ B∞ = ∪p>0 Bp if and only if W ∈ ∆2 , i.e., W (2t) ≤ CW (t).
R
(c) w ∈ Bp (p ∈ (0, ∞)) if and only if 0t W (x)−1/p dx ' tW (t)−1/p (cf.
[So]).
Proof (a) From Theorem 3.1, W ∈ Ap+1−ε and w ∈ Bp−ε .
(b) If w ∈ Bp then W ∈ Ap+1 and W ∈ ∆2 . If W ∈ ∆2 , W (t/2)/W (t) ≥
1/C > (1/2)q (q > 1), so W ∈ Aq+1 (cf. Proposition 2.3) and w ∈ Bq
(Theorem 3.1).
0
(c) Since, for 1 < q < ∞, w ∈ Aq if and only if w1−q ∈ Aq0 (see [GR]),
0
we have W ∈ Ap+1 if and only if W 1−(p+1) ∈ A(p+1)0 , which means that
W −1/p ∈ A1+1/p and W −1/p is a doubling and decreasing weight, and Proposition 2.1 applies.
4
Hardy transforms of Bp –weights
Let us see that w ∈ Bp if and only if S1 w ∈ Mp .
Theorem 4.1 If 1 ≤ p < ∞,
S1 : Lp (w)d −→ Lp (w)
S1 : Lp (S1 w) −→ Lp (S1 w).
if and only if
Proof First assume S1 w ∈ Mp , i.e. S1 : Lp (S1 w) −→ Lp (S1 w). If 1 < p <
R
R
0
0
∞, w1 (t) := (S1 w)(t) = W (t)/t satisfies ( t∞ w1x(x)
dx)1/p ( 0t w1 (x)−p /p dx)1/p ≤
p
Rt
C and, W being increasing,
0
−p0 /p
w1
p0
(x) dx =
p0 /p dx
0 (x/W (x))
Rt
0
≥
1
tp
p0 W (t)p0 /p .
0
0
(x)
t
1/p ≤ p0 1/p C and w ∈ B follows from (2) .
dx)1/p ( W (t)
Thus ( t∞ W
p
p0 /p )
xp+1
In the case p = 1, S1 w ∈ M1 means that S2 S1 w ≤ CS1 w and then
S1 S2 w ≤ CS1 w. Since S2 w is decreasing, S2 w ≤ S1 S2 w ≤ CS1 w and hence
w ∈ B1 .
Let now w ∈ Bp . If p = 1, S2 S1 w = S2 w + S1 w ≤ (C + 1)S1 w and
S1 w ∈ M1 . In the case 1 < p < ∞, to prove that w1 = S1 w satisfies
R
Z
t
∞
w1 (x) 1/p dx
xp
Z
t
1/p0
0
w1 (x)−p /p dx
≤C
(4)
0
1/p
(x)
observe that (2) implies I1 := ( t∞ W
dx)1/p ≤ C W (t)t .
xp+1
α−1
e := W
On the other hand w
w ∈ Bp0 for α = p0 /p (cf. Remark 3.1)
R
which means that
0
I2p
Z
:=
0
t
0
0
xp −1
0 W
e (x)
Rt
xp −1
dx '
W (x)α
Z
0
t
p0
dx ≤ C 0 et (see [So; Theorem 2.5 (ii)]). Then
W (t)
0
xp −1
Rx
dx =
e
ds
0 w(s)
5
Z
0
t
0
xp −1
0
C 0 tp
0
C 0 tp
dx ≤
= α .
f (x)
f (t)
W (t)
W
W
Thus, I1 · I2 ≤ CC 0 gives (4).
A similar result holds for the weak type Hardy inequalities:
Theorem 4.2 S1 : L1 (w)d −→ L1,∞ (w) if and only if S1 : L1 (S1 w) −→
L1,∞ (S1 w).
Proof If w ∈ B1,∞ , from (S1 w)(x) ≤ C(S1 w)(t) (t < x) we see that S1 w
satisfies (3) with α = 1, since
∞
Z
t
t
dx
(S1 w)(x)
≤ C(S1 w)(t)
x
x
Z
∞
t
dx = C(S1 w)(t).
x2
t
Assume now S1 w ∈ M1,∞ . Then, as in [AnM; proof of Theorem 2],
1
y
Z
y
(S1 w)(x) dx ≤ C inf (S1 w)(s)
(0 < t < y),
0≤s≤t
t
1 2t 1 t
1 2t
and, for y = 2t, 2t
t (S1 w)(x) dx ≥ 2t t ( x 0 w(s) ds) dx =
Hence (S1 w)(t) ≤ C inf 0≤s≤t (S1 w)(s) and w ∈ B1,∞ .
R
R
R
ln 2
2 (S1 w)(t).
Theorem 4.3 (a) S2 : L1 (w)d −→ L1 (w) if and only if S2 : L1 (S1 w) −→
L1 (S1 w).
(b) If S2 : Lp0 (w)d −→ Lp0 (w) with p0 ∈ [1, ∞), then S2 : Lp (S1 w) −→
Lp (S1 w) for all p ∈ [1, ∞).
(c) If W ∈ ∆2 and S2 : Lp (S1 w) −→ Lp (S1 w) for some p ∈ [1, ∞), then
S2 : Lq (w)d −→ Lq (w) for any q ∈ [1, ∞).
Proof (a) If w ∈ B 1 ,
S1 w ∈ M 1 ,
Z
Rt
0
∞
(S1 w)(x) dx ≤ C
Z
0
w(x) dx, i.e. S1 w ∈ M 1 . If
∞
f (x)(S1 S1 w)(x) dx =
0
Rt
(S2 f )(x)(S1 w)(x) dx ≤ C
0
Z
∞
f (x)(S1 w)(x) dx
0
when f ≥ 0, and then S1 S1 w ≤ CS1 w, i.e., w ∈ B 1 .
(b) Since S2 : L1 (w)d −→ L1 (w) (see Remark 1.1), S1 w ∈ M 1 and also
S2 : Lp (S1 w) −→ Lp (S1 w), since M p ⊂ M q if q > p (cf. [BMR]).
(c) Let 1 < p < ∞ and S2 : Lp (S1 w) −→ Lp (S1 w) (if p = 1, see Theorem
3.3). Then
Z
t
2t
R
1/p R
0
t
∞ S1 w(x)−p /p
S
w(x)
dx
0
1
0
t
xp
0
S1 w(x)−p /p
dx =
xp0
Z
2t
t
1/p0
dx
≤ C and in our case
0
W (x)−p /p
0
dx ≥ (ln 2)W (2t)−p /p .
x
−p0 /p
0
0
Thus ( 0 (S1 w)(x) dx)1/p ≤ C( t W (x)x
dx)−1/p ≤ C(ln 2)−1/p W (2t)−1/p .
Rt
Since W ∈ ∆2 , 0 S1 w(x) dx ≤ C 0 W (t) and now we apply Remark 1.1.
Rt
R 2t
6
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Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, E-08071 Barcelona, Spain
E-Mail: [email protected], [email protected]
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