Directory UMM :Journals:Journal_of_mathematics:GMJ:
GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 2, 1998, 177-200
ON THE BOUNDEDNESS OF CLASSICAL OPERATORS
ON WEIGHTED LORENTZ SPACES
Y. RAKOTONDRATSIMBA
Abstract. Conditions on weights u(·), v(·) are given so that a classical operator T sends the weighted Lorentz space Lrs (vdx) into
Lpq (udx). Here T is either a fractional maximal operator Mα or
a fractional integral operator Iα or a Calder´
on–Zygmund operator. A
characterization of this boundedness is obtained for Mα and Iα when
the weights have some usual properties and max(r, s) ≤ min(p, q).
§ 0. Introduction
Let u(·), v(·), w1 (·), w2 (·) be weight functions on Rn , n ∈ N∗ , i.e., nonnegative locally integrable functions; and let T be a classical operator. The
purpose of this paper is to determine when T is bounded from the weighted
pq
Lorentz space Lrs
v (w1 ) into Lu (w2 ), i.e.,
w2 (·)(T f )(·) pq ≤ C w1 (·) f (·) rs for all functions f (·). (0.0)
Lv
Lu
Here C > 0 is a constant which depends only on n, p, q, r, s, and on the
weight functions. Recall that
kg(·)kqLpq
u
=q
Z∞
0
Z
u(y)dy
{y∈Rn ; |g(y)|>λ}
pq
λq−1 dλ,
for 1 ≤ p < ∞ and 1 ≤ q < ∞; and
= sup λ
kg(·)kLp∞
u
λ>0
Z
{y∈Rn ; |g(y)|>λ}
u(y)dy
p1
1991 Mathematics Subject Classification. 42B25, 42B20.
Key words and phrases. Maximal operators, fractional integral, Calder´
on–Zygmund
operators, weighted Lorentz spaces.
177
c 1998 Plenum Publishing Corporation
1072-947X/98/0300-0177$15.00/0
178
Y. RAKOTONDRATSIMBA
for 1 ≤ p < ∞. It is always assumed that 1 < r, s, p, q < ∞. For convenience, the embedding defined by (0.0) will be denoted by T : Lrs
v (w1 ) →
Lupq (w2 ).
The classical operator under consideration is a fractional maximal operator or a fractional integral operator or a Calder´on–Zygmund operator. The
fractional maximal operator Mα of order α, 0 ≤ α < n, is defined as
n α Z
o
(Mα f )(x) = sup |Q| n −1
|f (y)|dy; Q a cube with Q ∋ x .
Q
Here Q is a cube with sides parallel to the coordinate planes. Thus M =
M0 is the well-known Hardy–Littlewood maximal operator. The fractional
integral operator Iα , 0 < α < n, is given by
Z
|x − y|α−n f (y)dy.
(Iα f )(x) =
Rn
The Hilbert transform
(Hf )(x) = P.V.
Z
R1
f (y)
dy = lim
ε→0
x−y
Z
|x−y|>ε
f (y)
dy
x−y
is a particular case of the Calder´on–Zygmund operator.
pq
The boundedness M : Lrs
v (1) → Lu (1) was considered and studied by
many authors (see, for instance, [1], [2] and the references therein). However,
as mentioned by Kokilashvili and Krbec [1], easy necessary and sufficient
conditions on v(·), u(·) for which Mα : Lvrs (1) → Lupq (1), 0 ≤ α < n, are not
known. In this paper we find a sufficient condition for such a boundedness.
For weight functions having some special properties (generally shared by
usual weights), the condition found here is also a necessary one. One of the
reasons which lead to considering Mα : Lvrs (w1 ) → Lpq
u (w2 ) is the fact that
weights cannot be combined as in the Lebesgue case where, for instance,
1
= ku p (·)f (·)kLpp
.
kf (·)kLpp
u
1
A weight function w(·) is constant on annuli if for a constant c > 0
sup
R 0. In the proof of Lemma 1 below, it is observed
pq
pq
necessarily H : Lrs
that Mα : Lrs
v (1) → Lu (w) with
v (1) → Lu (1) implies
R
α−n
and (Hf )(x) = |y| 0.
Let 1 < r <
n
α
and
1
1
r∗
=
satisfy the Wheeden–Muckenhoupt condition
1
≤ C2 (1.3)
11{|·|
ON THE BOUNDEDNESS OF CLASSICAL OPERATORS
ON WEIGHTED LORENTZ SPACES
Y. RAKOTONDRATSIMBA
Abstract. Conditions on weights u(·), v(·) are given so that a classical operator T sends the weighted Lorentz space Lrs (vdx) into
Lpq (udx). Here T is either a fractional maximal operator Mα or
a fractional integral operator Iα or a Calder´
on–Zygmund operator. A
characterization of this boundedness is obtained for Mα and Iα when
the weights have some usual properties and max(r, s) ≤ min(p, q).
§ 0. Introduction
Let u(·), v(·), w1 (·), w2 (·) be weight functions on Rn , n ∈ N∗ , i.e., nonnegative locally integrable functions; and let T be a classical operator. The
purpose of this paper is to determine when T is bounded from the weighted
pq
Lorentz space Lrs
v (w1 ) into Lu (w2 ), i.e.,
w2 (·)(T f )(·) pq ≤ C w1 (·) f (·) rs for all functions f (·). (0.0)
Lv
Lu
Here C > 0 is a constant which depends only on n, p, q, r, s, and on the
weight functions. Recall that
kg(·)kqLpq
u
=q
Z∞
0
Z
u(y)dy
{y∈Rn ; |g(y)|>λ}
pq
λq−1 dλ,
for 1 ≤ p < ∞ and 1 ≤ q < ∞; and
= sup λ
kg(·)kLp∞
u
λ>0
Z
{y∈Rn ; |g(y)|>λ}
u(y)dy
p1
1991 Mathematics Subject Classification. 42B25, 42B20.
Key words and phrases. Maximal operators, fractional integral, Calder´
on–Zygmund
operators, weighted Lorentz spaces.
177
c 1998 Plenum Publishing Corporation
1072-947X/98/0300-0177$15.00/0
178
Y. RAKOTONDRATSIMBA
for 1 ≤ p < ∞. It is always assumed that 1 < r, s, p, q < ∞. For convenience, the embedding defined by (0.0) will be denoted by T : Lrs
v (w1 ) →
Lupq (w2 ).
The classical operator under consideration is a fractional maximal operator or a fractional integral operator or a Calder´on–Zygmund operator. The
fractional maximal operator Mα of order α, 0 ≤ α < n, is defined as
n α Z
o
(Mα f )(x) = sup |Q| n −1
|f (y)|dy; Q a cube with Q ∋ x .
Q
Here Q is a cube with sides parallel to the coordinate planes. Thus M =
M0 is the well-known Hardy–Littlewood maximal operator. The fractional
integral operator Iα , 0 < α < n, is given by
Z
|x − y|α−n f (y)dy.
(Iα f )(x) =
Rn
The Hilbert transform
(Hf )(x) = P.V.
Z
R1
f (y)
dy = lim
ε→0
x−y
Z
|x−y|>ε
f (y)
dy
x−y
is a particular case of the Calder´on–Zygmund operator.
pq
The boundedness M : Lrs
v (1) → Lu (1) was considered and studied by
many authors (see, for instance, [1], [2] and the references therein). However,
as mentioned by Kokilashvili and Krbec [1], easy necessary and sufficient
conditions on v(·), u(·) for which Mα : Lvrs (1) → Lupq (1), 0 ≤ α < n, are not
known. In this paper we find a sufficient condition for such a boundedness.
For weight functions having some special properties (generally shared by
usual weights), the condition found here is also a necessary one. One of the
reasons which lead to considering Mα : Lvrs (w1 ) → Lpq
u (w2 ) is the fact that
weights cannot be combined as in the Lebesgue case where, for instance,
1
= ku p (·)f (·)kLpp
.
kf (·)kLpp
u
1
A weight function w(·) is constant on annuli if for a constant c > 0
sup
R 0. In the proof of Lemma 1 below, it is observed
pq
pq
necessarily H : Lrs
that Mα : Lrs
v (1) → Lu (w) with
v (1) → Lu (1) implies
R
α−n
and (Hf )(x) = |y| 0.
Let 1 < r <
n
α
and
1
1
r∗
=
satisfy the Wheeden–Muckenhoupt condition
1
≤ C2 (1.3)
11{|·|