An. Şt. Univ. Ovidius Constanţa
Vol. 17(2), 2009, 27–38
ON THE BOUUNDEDNESS OF
FRACTIONAL B-MAXIMAL OPERATORS
IN THE LORENTZ SPACES Lp,q,γ (Rn+ ) ∗
Canay Aykol and Ayhan Serbetci
Abstract
In this study, sharp rearrangement inequalities for the fractional Bmaximal function Mα,γ f are obtained in the Lorentz spaces Lp,q,γ and
by using these inequalities the boundedness conditions of the operator
Mα,γ are found. Then, the conditions for the boundedness of the Bmaximal operator Mγ are obtained in Lp,q,γ .
1
Introduction
Let Rn+ = {x = (x1 , ..., xn ) ∈ Rn , xn > 0}. Denote by Lp,γ ≡ Lp,γ (Rn+ ), γ > 0,
the set of all classes of measurable functions with finite norm
!1/p
Z
kf kLp,γ ≡
Rn
+
|f (x)|p xγn dx
< ∞,
If p = ∞, we assume
L∞,γ (Rn+ ) = L∞ (Rn+ ) = {f : kf kL∞ = ess sup |f (x)| < ∞}.
x∈Rn
+
Key Words: Laplace-Bessel differential operator, generalized shift operator, γ-rearrangement, Lorentz spaces, B-maximal function, fractional B-maximal function.
Mathematics Subject Classification: 42B20, 42B25, 42B35, 47G10
Received: March, 2009
Accepted: September, 2009
∗ This research was supported by The Turkish Scientific and Technological Research
Council (TUBITAK, programme 2211.)
27
28
Canay Aykol and Ayhan Serbetci
The generalized translation operator T y , y ∈ Rn+ , is defined (see [8, 9, 10]) for
smooth functions on Rn+ by
Z π
p
f (x′ − y ′ , x2n + yn2 − 2xn yn cos θ )sinγ−1 θdθ,
T y f (x) = Cγ
0
√
where x′ , y ′ ∈ Rn−1 and Cγ = Γ((γ + 1)/2)[ π Γ(γ/2)]−1 .
It is well-known that the generalized shift operator T y is closely related to
the Laplace-Bessel differential operator
∆B =
n
X
γ ∂
∂2
+
.
∂x2k
xn ∂xn
k=1
Further, if f belongs to Lp,γ , 1 ≤ p ≤ ∞, then for all y ∈ Rn+ , the function
T y f belongs to Lp,γ , and
kT y f kLp,γ ≤ kf kLp,γ .
For 0 ≤ α < n + γ, the fractional maximal function associated with ∆B
n
(fractional B-maximal function) is defined at f ∈ Lloc
1,γ (R+ ) by
Z
α
−1
(Mα,γ f )(x) = sup |B(0, r)|γn+γ
T y |f (x)| ynγ dy, x ∈ Rn+ ,
r>0
B(0,r)
where B(0, r) = {y ∈ Rn+ : |y| < r}. For α = 0 we get the maximal function Mγ f associated with the Laplace-Bessel differential operator (B-maximal
function, see [6]).
The aim of this paper is to obtain sharp rearrangement estimates for the
fractional B-maximal function Mα,γ f . We give the necessary and sufficient
conditions for the boundedness of the operator Mα,γ in the Lorentz spaces
Lp,q,γ (Rn+ ) by using the obtained sharp rearrangement estimates. As consequence of these results we find the boundedness conditions for the B-maximal
operator Mγ in the Lorentz spaces Lp,q,γ (Rn+ ).
It is well known that for the classical Hardy-Littlewood maximal operator
the rearrangement inequality
cf ∗∗ (t) ≤ (M f )∗ (t) ≤ Cf ∗∗ (t), t ∈ (0, ∞)
holds, where f ∗ (t) is the nonincreasing rearrangement of f and f ∗∗ (t) =
R
1 t ∗
t 0 f (t)dt.
n
For f ∈ Lloc
1 (R ), similar sharp rearrangement estimates are obtained for
the maximal function of f in [1] and for the fractional maximal function of
ON THE BOUUNDEDNESS OF FRACTIONAL B-MAXIMAL OPERATORS IN THE
LORENTZ SPACES Lp,q,γ (Rn
29
+)
f in [3], and for the fractional maximal function of f ∈ Lp,γ (Rn+ ), γ > 0,
associated with the Laplace-Bessel differential operator in [7]. These estimates
are of great importance in the study of operators on rearrangement-invariant
spaces as well as in interpolation theory.
Throughout the paper, we denote M + (0, ∞) the set of all non-negative
measurable functions on (0, ∞) with respect to the measure xγn dx, and
M + (0, ∞; ↓) the set of all non-increasing functions from M + (0, ∞). We use
the letter C for a positive constant, independent of appropriate parameters
and not necessary the same at each occurrence.
2
Preliminaries
Let f :R Rn+ → R be a measurable function and for any measurable set E,
|E|γ = E xγn dx. We define γ-rearrangement of f in decreasing order by
fγ∗ (t) = inf {s > 0 : f∗,γ (s) ≤ t}, ∀t ∈ (0, ∞),
wheref∗,γ (s) denotes the γ-distribution function of f given by
f∗,γ (s) = {x ∈ (Rn+ ) : |f (x)| > s}γ .
Some properties of γ-rearrangement of functions are given as follows (see
[2, 4, 12]):
1) if 0 < p < ∞, then
Z
Rn
+
|f (x)|p xγn dx =
Z
∞
p
fγ∗ (t) dt;
0
2) for any t > 0,
sup
|E|γ =t
3)
Z
Rn
+
Z
E
|f (x)| xγn dx
|f (x)g(x)| xγn dx
It is well known that
4)
≤
=
Z
t
0
Z
0
fγ∗ (s)ds;
(1)
∞
fγ∗ (t)gγ∗ (t)dt;
t
t
(f + g)∗γ (t) ≤ fγ∗ ( ) + gγ∗ ( )
2
2
(2)
30
Canay Aykol and Ayhan Serbetci
holds.
We denote by W Lp,γ (0, ∞) the weak Lp,γ space of all measurable functions
f with finite norm
kf kW Lp,γ = sup t1/p fγ∗ (t),
t>0
1 ≤ p < ∞.
The function fγ∗∗ : (0, ∞) → [0, ∞] is defined as
fγ∗∗ (t)
1
=
t
Z
t
fγ∗ (s)ds.
0
Definition 1. The Lorentz space Lp,q,γ (Rn+ ) is the collection of all measurable
functions f on Rn+ such that the quantity
 q 1/q

 R ∞ t p1 f ∗ (t) dt
, 0 < p < ∞, 0 < q < ∞
γ
t
0
kf kp,q,γ =
1
∗

sup t p fγ (t),
0 < p ≤ ∞, q = ∞

t>0
is finite.
If 0 < p ≤ ∞, q = ∞, then Lp,∞,γ (Rn+ ) = W Lp,γ (Rn+ ).
If 1 ≤ q ≤ p or p = q = ∞, then the functional kf kp,q,γ is a norm. (see
[2], [5], [12]).
If p = q = ∞, then the space L∞,∞,γ (Rn+ ) is denoted by L∞,γ (Rn+ ).
In the case 1 < p, q < ∞, we give a functional k.k∗p,q,γ by
kf k∗p,q,γ =
 q

 R ∞ t p1 f ∗∗ (t)
γ
0


1
p
sup t fγ∗∗ (t),
dt
t
1/q
t>0
, 0 < p < ∞, 0 < q < ∞
0 < p ≤ ∞, q = ∞
(with the usual modification if 0 < p ≤ ∞, q = ∞) which is a norm on
Lp,q,γ (Rn+ ) for 1 < p < ∞, 1 ≤ q ≤ ∞ or p = q = ∞.
If 1 < p ≤ ∞, 1 ≤ q ≤ ∞, then
kf kp,q,γ ≤ kf k∗p,q,γ ≤
p
kf kp,q,γ ,
p−1
that is, the quasi-norms kf kp,q,γ and kf k∗p,q,γ are equivalent.
ON THE BOUUNDEDNESS OF FRACTIONAL B-MAXIMAL OPERATORS IN THE
LORENTZ SPACES Lp,q,γ (Rn
31
+)
Lemma 1. For any measurable set A = (A′ , An ) ⊂ Rn+ , An ⊂ (0, ∞), A′ =
A1 × ... × An−1 ⊂ Rn−1 , and y ∈ Rn+ , then the following equality holds
q
Z
Z
2
g z ′ , zn2 + zn+1
T y g(x)ynγ dy = Cγ
dµ (z, zn+1 ) ,
e
(x,0)+A
A
e = A′ × (−m, m) × [0, m), m = sup An and dµ (z, zn+1 ) =
where A
γ−1
zn+1 dz1 dz2 ...dzn dzn+1 .
The proof of Lemma 1 is straightforward, after applying the following
substitutions
z ′ = y ′ − x′ , zn = xn − yn cos α, zn+1 = yn sin α.
(3)
In the following lemma we give a relation between the generalized shift
operator T y and the γ-rearrangement of the function f .
Lemma 2. For any measurable set A ⊂ Rn+ and for any y ∈ Rn+ , the following
equality holds
Z
Z t
sup
fγ∗ (s)ds.
(4)
T y |f (x)|ynγ dy = Cγ
|A|γ =t
0
A
Proof. From Lemma 1 we have
Z
Z
y
γ
T |f (x)|yn dy = Cγ
(x,0)+Ã
A
q
2
2
zn + zn+1 dµ (z, zn+1 ) ,
f
(5)
e = A′ × (−m, m) × [0, m), m = sup An and dµ (z, zn+1 ) =
where A
p
γ−1
zn+1 dz1 dz2 ...dzn dzn+1 . For the function f˜(z, z1 ) = f ( z 2 + z12 ), the analogue
of the equality (1) is valid
Z
Z t
sup
(f˜)∗µ (s)ds,
(6)
|f˜(z, zn+1 )|dµ(z, zn+1 ) =
µ(Ã)=t
(x,0)+Ã
0
where (f˜)∗µ (s) = inf{t > 0 : µ({(z, zn+1 ) : |f˜(z, zn+1 )| > t}) ≤ s}.
e = |A|γ and (f˜)∗ (s) = f ∗ (s).
Note that µ((x, 0) + A))
µ
γ
Indeed, taking into account (3), we have
Z
n
˜
µ({(z, zn+1 ) ∈ R+ : |f (z, zn+1 )| > t}) =
xγn dx = f∗,γ (t).
{x∈R+ :|f (x)|>t}
Consequently,
(f˜)∗µ (s) = inf{t > 0 : f∗,γ (t) ≤ s} = fγ∗ (s).
32
Canay Aykol and Ayhan Serbetci
From (5) and (6) we have
Z
Z
y
γ
sup
T |f (x)|yn dy = Cγ sup
|A|γ =t
A
= Cγ
Z
0
Thus Lemma 2 is proved.
3
µ(Ã)=t
t
(x,0)+Ã
(f˜)∗µ (s)ds = Cγ
Z
|f˜(z, zn+1 )|dµ(z, zn+1 )
t
fγ∗ (s)ds.
0
Main results
We need the following lemma which is used in the proof of Theorem 1.
Lemma 3. Let 0 ≤ α < n + γ . Then there exists a positive constant C,
depending on α, n and γ, such that
Z
α
sup t1− n+γ (Mα,γ f )∗γ (t) ≤ C |f (x)|xγn dx
(7)
t>0
Rn
+
and
sup(Mα,γ f )∗γ (t) ≤ C sup t n+γ fγ∗ (t).
α
t>0
(8)
t>0
Proof. The estimate (7) follows from the Corollary 4 in [7].
For the estimate (8), by using (4), for every B(0, r) ⊂ Rn+ , we get
Z
α
n+γ −1
|B(0, r)|γ
T y |f (x)|ynγ dy
B(0,r)
α
n+γ
≤ Cγ |B(0, r)|γ
−1
where C = Cγ
|B(0,r)|γ
0
≤ C sup t n+γ fγ∗ (t),
α
Z
t n+γ fγ∗ (t)t− n+γ dt
α
α
t>0
n+γ
n+γ−α
.
Theorem 1. Let 0 ≤ α < n + γ. Then there is a positive constant C such
that
α
(9)
(Mα,γ f )∗γ (t) ≤ C sup τ n+γ fγ∗∗ (τ ), t > 0
t<τ <∞
n
Lloc
1 (R+ ),
+
holds for all f ∈
where C depends on α, n and γ. The inequality
(9) is sharp for all ϕ ∈ M (0, ∞; ↓) and there exists a function f on Rn+ such
that fα∗ = ϕ a.e. on (0, ∞) and
(Mα,γ f )∗γ (t) ≥ Cγ sup τ n+γ fγ∗∗ (τ ), t > 0.
α
t<τ <∞
(10)
ON THE BOUUNDEDNESS OF FRACTIONAL B-MAXIMAL OPERATORS IN THE
LORENTZ SPACES Lp,q,γ (Rn
33
+)
Proof. To prove (9), we may assume that
sup τ n+γ fγ∗∗ (τ ) < ∞
α
(11)
t<τ <∞
otherwise there is nothing to prove. Then
Z
E
|f (x)|xγn dx
≤
Zt
fγ∗ (s)ds
0
holds for for all E ⊂ Rn+ with |E|γ ≤ t. In particular, if we put E = {x ∈ Rn+ :
|f (x)| > fγ∗ (t)} then |E|γ ≤ t and so f ∈ L1,γ (E).
Then the function
gt (x) = max{|f (x)| − fγ∗ (t), 0}sgnf (x), x ∈ Rn+
belongs to L1,γ (Rn+ ). Also the function
ht (x) = min{|f (x)|, fγ∗ (t)}sgnf (x), x ∈ Rn+
satisfies
h∗tγ (τ ) = min{fγ∗ (τ ), fγ∗ (t)}, τ ∈ (0, ∞).
Hence,
sup τ n+γ (ht )∗γ (τ ) = max{ sup τ n+γ fγ∗ (t), sup τ n+γ fγ∗ (τ )}
α
α
0<τ <t
τ >0
α
t≤τ <∞
= sup τ n+γ fγ∗ (τ ) ≤ sup τ n+γ fγ∗∗ (t)
α
t≤τ <∞
α
(12)
t≤τ <∞
which, together with (11), implies that ht ∈ W L n+γ ,γ (Rn+ ).
α
Furthermore, since f = gt + ht and
gt∗ (τ ) = χ[0,t) (τ )(fγ∗ (τ ) − fγ∗ (t)), τ ∈ (0, ∞).
by using (2), (7), (8), (12) and (13), we get
t
t
∗
∗
∗
+ (Mα,γ ht )γ
(Mα,γ f )γ (t) ≤ (Mα,γ gt )γ
2
2
≤C
!
α
n+γ
−1 Z
α
t
∗
γ
n+γ
(ht )α (τ )
gt (y)yn dy + sup τ
n
2
τ >0
R+
(13)
34
Canay Aykol and Ayhan Serbetci
Z t
α
α
−1
∗
∗
∗∗
n+γ
n+γ
fγ (τ ) − fγ (t) dτ + sup τ
fγ (τ )
≤C t
t<τ <∞
0
≤ C sup τ n+γ fγ∗∗ (τ )
α
t<τ <∞
and (9) follows.
Furthermore, (10) holds for all t ∈ (0, ∞). Let ϕ ∈ M+ (0, ∞; ↓).
Putting f (x) = ϕ(ωn |x|n ), ωn is the volume of the unit ball in Rn+ , ωn =
|B(0, 1)|γ ; and x ∈ Rn+ , we have fγ∗ = ϕ(0, ∞). Moreover, given y ∈ Rn+ ,
denote by B(|y|) the ball with its center at the origin and having radius |y|.
Then, for x, y ∈ Rn+ such that |y| > |x|,
Z
α
−1
(Mα,γ f )(x) = sup |B(0, r)|γn+γ
T y |f (x)|ynγ dy
r>0
B(0,r)
Z
α
n+γ −1
T y |f (x)|ynγ dy
≥ |B(0, |y|)|γ
B(0,|y|)
α
n+γ
= Cγ |(B(0, |y|)|γ
α
Rt
0
Z
ωn |y|n
0
= Cγ H (ωn |y|n ) ,
where H (t) = t n+γ −1
Consequently,
−1
fγ∗ (τ )dτ
ϕ(τ )dτ, t ∈ (0, ∞).
(Mα,γ f )(x) ≥ Cγ
sup
H(τ )
τ >ωn |x|n
whence (10) follows on taking rearrangements.
From these estimates we find that Mα,γ f is bounded from L1,γ (Rn+ ) to
W L n+γ ,γ (Rn+ ) and from W L n+γ ,γ (Rn+ ) to L∞,γ (Rn+ ) .
n+γ−α
n+γ−α
In Theorem 1 if we take the limit as α → 0, then we get
lim (Mα,γ f )∗γ (t) = (Mγ f )∗γ (t)
α→0
and
lim sup τ n+γ fγ∗∗ (τ ) = fγ∗∗ (t),
α
α→0 t<τ <∞
and therefore we have the following:
Corollary 1. There exists a positive constant C, depending on α, n and γ,
such that
(Mγ f )∗γ (t) ≤ Cfγ∗∗ (t), t > 0,
(14)
ON THE BOUUNDEDNESS OF FRACTIONAL B-MAXIMAL OPERATORS IN THE
LORENTZ SPACES Lp,q,γ (Rn
35
+)
for every f ∈ Lloc
1 (0, ∞). Inequality (14) is sharp in the sense that for every
ϕ ∈ M + (0, ∞; ↓) there exists a function f on (0, ∞) such that fα∗ = ϕ a.e. on
(0, ∞) and
(Mγ f )∗γ (t) ≥ Cγ fγ∗∗ (t), t > 0.
(15)
Theorem 2. For 0 ≤ α < n + γ the following estimates are equivalent:
(i) For 1 < p ≤ q < ∞, 1 ≤ r ≤ s ≤ ∞, fractional B-maximal operator
Mα,γ : Lp,r,γ (Rn+ ) → Lq,s,γ (Rn+ ) is bounded.
(ii) For all ϕ ∈ M+ (0, ∞; ↓),
Z
∞
0
(iii)
sup τ
α
n+γ
−1
t<τ <∞
1
p
−
1
q
Z
τ
ϕ(σ)dσ
0
=
s
t
s
q −1
dt
1/s
≤C
Z
∞
p
ϕ (t)t
0
r
p −1
dt
1/r
.
α
n+γ .
Proof. The equivalence of (ii) and (iii) follows from [11].
It suffices to prove (i) ⇔ (ii).
Assume Mα,γ : Lp,r,γ (Rn+ ) → Lq,s,γ (Rn+ ) is bounded. Then
kMα,γ f kLq,s,γ (Rn+ ) ≤ Ckf kLp,r,γ (Rn+ )
holds. From Theorem 1 and by the definition of quasi-norm in Lorentz spaces,
for every ϕ = fγ∗ (t) ∈ M+ (0, ∞; ↓), fγ∗ = ϕ a.e. on (0, ∞),
Z
∞
0
=
Z
Cγ sup τ n+γ −1
α
t<τ <∞
∞
0
≤
Z
≤C
0
∞
Z
Z
τ
0
Cγ sup τ
α
n+γ
t<τ <∞
fγ∗ (σ)dσ
s
fγ∗∗ (τ )
s
((Mα,γ f )∗γ (t))s t q −1 dt
0
∞
fγ∗ (t)p t p −1 dt
r
1/r
.
1/s
t
s
s
q −1
t q −1 dt
s
dt
1/s
1/s
36
Canay Aykol and Ayhan Serbetci
Conversely, for every ϕ = fγ∗ (t) ∈ M+ (0, ∞; ↓), fγ∗ = ϕ a.e. on (0, ∞),
Z
∞
0
≤
=
Z
s s
( Mα,γ f )∗γ (t) t q −1 dt
∞
0
Z
∞
0
≤C
Z
∞
0
1/s
1/s
s
s
α
C sup τ n+γ fγ∗∗ (τ ) t q −1 dt
t<τ <∞
C sup τ
α
n+γ
t<τ <∞
r
fγ∗ (t)p t p −1 dt
−1
Z
1/r
τ
0
fγ∗ (σ)dσ
s
t
s
q −1
dt
1/s
and
kMα,γ f kLq,s,γ (Rn+ ) ≤ Ckf kLp,r,γ (Rn+ )
holds.
From Theorem 2 and Corollary 1, we can now give the boundedness conditions of maximal operator
Z
(Mγ f )(x) = sup |B(0, r)|−1
T y |f (x)| ynγ dy, x ∈ Rn+ ,
γ
r>0
B(0,r)
in Lp,q,γ (Rn+ ) spaces.
Theorem 3. For f ∈ L1,γ (Rn+ ) and C being a positive constant independent
of f ,
kMγ f k1,∞,γ ≤ Ckf k1,γ
holds.
Proof. For f ∈ L1,γ (Rn+ ) we have
kMγ f k1,∞,γ = sup t(Mγ f )∗γ (t)
t>0
≤ C sup tfγ∗∗ (t)
t>0
= C sup
t>0
Z
t
0
≤ Ckf kL1,γ .
fγ∗ (s)ds
ON THE BOUUNDEDNESS OF FRACTIONAL B-MAXIMAL OPERATORS IN THE
LORENTZ SPACES Lp,q,γ (Rn
37
+)
Theorem 4. If 1 < p < ∞ and 1 ≤ q ≤ ∞ or p = q = ∞, then there is a
positive constant C independent of f and for all f ∈ Lp,q,γ such that
kMγ f kp,q,γ ≤ Ckf kp,q,γ .
Proof. In the case 1 < p < ∞ and 1 ≤ q < ∞, we have
kMγ f kp,q,γ
q dt 1/q
=
t
t
0
Z ∞ q dt 1/q
1
t p fγ∗∗ (t)
≤ C1
t
0
Z
∞
1
p
(Mγ f )∗γ (t)
= C1 kf k∗p,q,γ
p
≤ C1
kf kp,q,γ ,
p−1
= Ckf kp,q,γ .
In the case 1 < p < ∞ and q = ∞,
1
kMγ f kp,q,γ = sup t p (Mγ f )∗γ (t)
t>0
1
≤ C2 sup t p fγ∗∗ (t) = C2 kf k∗p,q,γ
t>0
≤ C2
p
kf kp,q,γ = Ckf kp,q,γ .
p−1
If p = q = ∞, then the following inequalities hold
sup(Mγ f )∗γ (t) ≤ sup Cfγ∗∗ (t)
t>0
t>0
= Ckf k∗p,q,γ ≤ Ckf kp,q,γ .
This completes the proof.
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Ankara University,
Department of Mathematics
Ankara, Turkey
Email: Canay.Aykol@science.ankara.edu.tr, serbetci@science.ankara.edu.tr