Services on Demand
Journal
Article
Indicators
-
Cited by SciELO
Related links
-
Similars in SciELO
Share
Revista de la Unión Matemática Argentina
Print version ISSN 0041-6932On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008
Weighted inequalities for generalized fractional operators
María Silvina Riveros
Abstract. In this note we present weighted Coifman type estimates, and two-weight estimates of strong and weak type for general fractional operators. We give applications to fractional operators given by an homogeneous function, and by a Fourier multiplier. The complete proofs of these results appear in the work [5] done jointly with Ana L. Bernardis and María Lorente.
2000 Mathematics Subject Classification. 42B20, 42B25.
Key words and phrases. Fractional integrals, Hörmander's condition of Young type, Muckenhoupt weights, two-weight estimates.
The author is partially supported by CONICET, Agencia Nación, and SECYT-UNC
1. Introduction and preliminaries
I would like to dedicate this note in memory of Dr Carlos Segovia. First we will give some basic definitions and preliminaries needed to state the results. Let us recall some of the background on Orlicz spaces. (See [24] and [21] to complete this topic.)
A function is a Young function if it is continuous, convex, increasing and satisfies
and
as
.
Given a Young function , we define the
-mean Luxemburg norm of a function
on a cube (or a ball)
in
by
![]() |
It is well known that if for all
then
, for all cubes
and functions
. Thus, the behavior of
for
is not important. If
, that is there are constants
such that
for
, the latter estimate implies that
.
Each Young function has an associated complementary Young function
satisfying
![]() |
for all . There is a generalization of Hölder's inequality
![]() |
A further generalization of Hölder's inequality (see [21]) that will be useful later is the following: If and
are Young functions and
![-1 - 1 - 1 A (t)B (t) ≤ C (t)](/img/revistas/ruma/v49n2/2a0430x.png)
![]() |
When we understand that
if
and
otherwise. Then
is not a Young function, but
and the latter inequalities make sense if one of the functions is
or
.
For each locally integrable function and
, the fractional maximal operator associated to the Young function
is defined by
![α ∕n M α,Af (x) = sup|Q | ||f||A,Q. Q∋x](/img/revistas/ruma/v49n2/2a0443x.png)
![α = 0](/img/revistas/ruma/v49n2/2a0444x.png)
![MA](/img/revistas/ruma/v49n2/2a0445x.png)
![M0,A](/img/revistas/ruma/v49n2/2a0446x.png)
![A(t) = t](/img/revistas/ruma/v49n2/2a0447x.png)
![M α,A = M α](/img/revistas/ruma/v49n2/2a0448x.png)
![α = 0](/img/revistas/ruma/v49n2/2a0449x.png)
![A (t) = t](/img/revistas/ruma/v49n2/2a0450x.png)
![M0,A = M](/img/revistas/ruma/v49n2/2a0451x.png)
![α = 0](/img/revistas/ruma/v49n2/2a0452x.png)
![r r + β + k A (t) = t , A (t) = t(1 + log (t)) , A (t) = t(1 + log (t)) ,](/img/revistas/ruma/v49n2/2a0453x.png)
![M (f ) = M (|f |r)1∕r, M r + β(f) and M + k(f) . r L (log L) L(log L)](/img/revistas/ruma/v49n2/2a0454x.png)
![k ≥ 0](/img/revistas/ruma/v49n2/2a0455x.png)
![k ∈ ℤ](/img/revistas/ruma/v49n2/2a0456x.png)
![ML (log+L)k](/img/revistas/ruma/v49n2/2a0457x.png)
![M k+1](/img/revistas/ruma/v49n2/2a0458x.png)
![k times k ◜---◞◟----◝ M = M ∘ ... ∘ M , k ∈ ℕ.](/img/revistas/ruma/v49n2/2a0459x.png)
![k > 0](/img/revistas/ruma/v49n2/2a0460x.png)
![r > 1](/img/revistas/ruma/v49n2/2a0461x.png)
![M f (x) ≤ CML (log+L)kf(x) ≤ CMrf (x) .](/img/revistas/ruma/v49n2/2a0462x.png)
The good weights for are those in the
classes of Muckenhoupt (see [19] and also [26] and [18] for the one-sided case).
The good weights for the maximal operator are the
classes. It is proved in [20] ( see [1] for the one sided version) that
if and only if
, for
,
, where
![( 1 ∫ )1∕q ( 1 ∫ ′)1 ∕p′ ---- wq ---- w -p ≤ C, A(p,q) |Q| |Q | |Q | |Q|](/img/revistas/ruma/v49n2/2a0471x.png)
![Q](/img/revistas/ruma/v49n2/2a0472x.png)
Also observe that for the case ,
is equivalent to say that
.
Let us define a generalization of the Hörmander condition, for a given kernel . We used the notation:
for
and
Definition 1.1. Let be a Young function and let
. The kernel
is said to satisfy the
-Hörmander type condition, we write
, if there exist
,
such that for any
and
![∞∑ (2m R )n- α∥K α(⋅ - y) - Kα (⋅)∥A,|x|~2mR ≤ C. m=1](/img/revistas/ruma/v49n2/2a0489x.png)
![K α ∈ H α,∞](/img/revistas/ruma/v49n2/2a0490x.png)
![K α](/img/revistas/ruma/v49n2/2a0491x.png)
![∥ ⋅ ∥L∞,|x|~2m R](/img/revistas/ruma/v49n2/2a0492x.png)
![∥ ⋅ ∥A,|x|~2m R](/img/revistas/ruma/v49n2/2a0493x.png)
Definition 1.2. The kernel is said to satisfy the
condition, if there exist
,
such that
![|y | |K α(x - y ) - K α(x)| ≤ C--------, |x | > c|y|. |x|n+1 -α](/img/revistas/ruma/v49n2/2a0498x.png)
Observe that when we obtain that
defined in [16].
If , for
, then we write
. This
condition appears implicitly in [12]. On the other hand, since
for
we have that
. Also, it is easy to see that
.
Suppose that is an operator given by convolution with a kernel
which satisfies some regularity condition and suppose that we know some behavior of
with respect to the Lebesgue's measure (weak or strong type inequalities for
). Sometimes, in order to know how is the behavior of
when we change the measure, (i.e., when we consider the measure
where
is a weight, (
)) the following inequality is useful (we call it a Coifman type inequality)
![]() |
Here is a maximal operator related to the operator
which is normally easier to deal with. In general,
is strongly related with the kernel
and its size is inverse to the smoothness of
: the rougher the kernel, the bigger the maximal.
For a Calderón-Zygmund singular integral operator (i.e.,
, see Definition 1.2, for
) inequality (1.4) holds with
, the Hardy-Littlewood maximal function,
, and
(see [8]).
If is a singular integral operator with less regular kernel, (see [13]) for example if the kernel
satisfies an
-Hörmander condition (Definition 1.1, for
and
), then inequality (1.4) holds with
, with
, for all
, and
(see [25]).
For a Young function , the
-Hörmander condition is introduced in [16], which generalized in the scale of the Orlicz spaces the
-Hörmander condition. In [16] the authors showed that, if the kernel
(Definition 1.1, for
), then inequality (1.4) holds with
, where
is the complementary function of
, for all
, and
.
The differential transform operator was studied in [11] and [3]. In [14] it is proved an inequality of the type (1.4), by showing that the kernel satisfies the -Hörmander condition for
(
). Therefore, this operator satisfies inequality (1.4) with the maximal operator
, where
(actually they obtain a smaller operator since
is a one-sided operator, because
).
The Coifman type inequality allows us to obtain, for general linear operators, two-weight inequalities of the type
![]() |
for and in the endpoint case
,
![]() |
for every weight , with no assumptions on
. The operators
are again suitable maximal operators related with
and not necessarily the same for inequalities (1.4), (1.5) and (1.6).
There is a great amount of works that deal with inequalities of the type (1.5) and (1.6). When is a Calderón-Zygmund operator (with kernel
), inequality (1.5) holds with
, where
is the integer part of
(see [22]). In the endpoint case
, inequality (1.6) for Calderón-Zygmund operators hold with
, for any
, where
is the maximal function associated to the Young function
. This result was proved by Carlos Pérez in [22]. For
a singular integral associated to a kernel K satisfying a general Hörmander's condition given by a Young function
, the corresponding results, that include as particular cases those of C. Pérez, has been proved in [14] and [15].
In 1974, Muckenhoupt and Wheeden [20] proved inequality (1.4) for the classical Riesz potential
and
the fractional maximal function
, defined for
and locally integrable function
by
![∫ I f(x) = ---f(y)--- dy. α ℝn |x - y|n-α](/img/revistas/ruma/v49n2/2a04181x.png)
![Iα](/img/revistas/ruma/v49n2/2a04182x.png)
![--1-- K α(x) = |x|n-α](/img/revistas/ruma/v49n2/2a04183x.png)
![H *α,∞](/img/revistas/ruma/v49n2/2a04184x.png)
![Iα](/img/revistas/ruma/v49n2/2a04185x.png)
![MT = M αp(M [p])](/img/revistas/ruma/v49n2/2a04186x.png)
![I α](/img/revistas/ruma/v49n2/2a04187x.png)
![M = M (M ɛ) T α L(logL)](/img/revistas/ruma/v49n2/2a04188x.png)
There are fractional integrals with less regular kernel than the Riesz transform (see for example [7], [12], [27], [20], [9], [10]). Suppose that is homogeneous of degree zero and
, where
denotes the sphere of
and
. Define the fractional integral associated to
by
![∫ Ω (y ∕|y|) T Ω,αf (x) = ---n--α-f (x - y)dy. ℝn |y|](/img/revistas/ruma/v49n2/2a04195x.png)
![Iα](/img/revistas/ruma/v49n2/2a04196x.png)
![Ω](/img/revistas/ruma/v49n2/2a04197x.png)
![Ω](/img/revistas/ruma/v49n2/2a04198x.png)
![Ls(Sn -1)](/img/revistas/ruma/v49n2/2a04199x.png)
In this note we state and briefly sketch the proofs of the corresponding results for general fractional integrals ,
, given by convolution with a kernel
which satisfy a
condition, for appropriate Young functions
(see Theorems 2.1, 2.3 and 2.5).
From now on, for ,
will be a fractional operator bounded from
to
, for all
satisfying
.
Theorem 2.1. Let be a fractional operator given by a kernel
. Suppose
is of weak-type
.
-
If
be a Young function and
, then for any
and
,
(2.1)
-
Moreover, if the kernel
is supported in
, then for any
,
, it follows that (2.1) holds with
in place of
where
.
Remark 2.2. Observe that we can apply the theorem to and
(respectively) obtaining the result in [20] and [17], for
.
Proof. To prove this Theorem we use the sharp operator of . Given
and a cube
, decompose
, where
and
. For
we use Kolmogorov and that
is of weak-type
. For the global part we use that
is the convolution with the kernel
and the generalized Hölders inequality. □
Theorem 2.3. Let be a Young function and
. Suppose that there exist Young functions
,
such that
and
with
. Let
be a linear operator such that its adjoint
satisfies
![]() |
for all and
. Then, for
and for any weight
,
![]() |
Remark 2.4. For the applications below, and since all our operators are of convolution type, proving (2.2) for or
turns out to be equivalent.
Proof. To prove this Theorem we use duality and apply Theorem 2.1. To do this we need the fact that the weight belongs to
, for all
and any
. For the maximal operators
that appears in this proof, has been proved in [4]. □
Theorem 2.5. Let be a fractional operator. Suppose that there exists
such that for any
, there exists a Young function
satisfying
![]() |
for all weight . If
, then for any weight
,
![]() |
for all .
3.1. Fractional integrals associated to a homogeneous function. Denote by the unit sphere on
. For
, we write
. Let us consider
. This function can be extended to
as
(abusing on the notation we call both functions
). Thus
is a function homogeneous of degree
. Let
, and let
be a Young function such that
is also a Young function. Let
and satisfying the
-Dini smoothness condition, i.e.,
![]() |
where
![ϖA (t) = sup ∥Ω (⋅ + y) - Ω(⋅)∥A,Sn-1. |y|≤t](/img/revistas/ruma/v49n2/2a04291x.png)
![∫ -Ω(y-) TΩ,αf(x) = ℝn |y|n-αf (x - y)dy.](/img/revistas/ruma/v49n2/2a04292x.png)
![B](/img/revistas/ruma/v49n2/2a04293x.png)
![-- B](/img/revistas/ruma/v49n2/2a04294x.png)
![Ω ∈ LA (Sn -1)](/img/revistas/ruma/v49n2/2a04295x.png)
![-n- Ω ∈ L n-α(Sn- 1)](/img/revistas/ruma/v49n2/2a04296x.png)
![TΩ,α](/img/revistas/ruma/v49n2/2a04297x.png)
![(1, nn-α)](/img/revistas/ruma/v49n2/2a04298x.png)
![Lp(dx )](/img/revistas/ruma/v49n2/2a04299x.png)
![Lq (dx)](/img/revistas/ruma/v49n2/2a04300x.png)
![1∕p - 1∕q = α ∕n](/img/revistas/ruma/v49n2/2a04301x.png)
![1 < p < q < ∞](/img/revistas/ruma/v49n2/2a04302x.png)
![-Ω(x) K α(x) = |x|n-α](/img/revistas/ruma/v49n2/2a04303x.png)
![H α,A](/img/revistas/ruma/v49n2/2a04304x.png)
![TΩ,α](/img/revistas/ruma/v49n2/2a04305x.png)
In the particular case that with
we get the following:
Theorem 3.1. Let be as above and satisfying the
-Dini condition.
In both cases and
is small enough.
Proof. We only have to apply the theorems with the following Young functions: ,
, and
, where
is some small enough number that is related with
. Observe that in part (c) we obtain
on the right hand side, but it is easy to see that
and
. □
For as above, we obtain the following weighted inequality as in [10] (see also [27]).
Corollary 3.2. Suppose that we are under the same hypothesis as in Theorem 3.1. Let ,
and
. Then
![( ∫ )1 ∕q (∫ )1∕p q q p p n |TΩ,αf (x)|w (x)dx ≤ C n |f| w (x )dx . ℝ ℝ](/img/revistas/ruma/v49n2/2a04336x.png)
Proof. First of all observe that implies
. Then by part (a) of Theorem 3.1
![∫ ∫ |T Ω,αf (x)|qwq (x)dx ≤ C (M α,r′f)qwq(x )dx. ℝn ℝn](/img/revistas/ruma/v49n2/2a04339x.png)
![( r′ )1∕r′ M α,r′f (x) = M αr′|f| (x )](/img/revistas/ruma/v49n2/2a04340x.png)
![r′ ′ ′ w ∈ A (p∕r ,q∕r )](/img/revistas/ruma/v49n2/2a04341x.png)
3.2. Fractional integrals associated to a multiplier. Let . Given a function
defined in
we consider the multiplier operator
defined a priori for functions
in the Schwartz class by
. Given
and
we say that
if there exists a constant
such that
and
![|β|+ α β sRu>p0 R ∥D m ∥Ls,|ξ|~R < + ∞, for all |β| ≤ l.](/img/revistas/ruma/v49n2/2a04354x.png)
![n∕s < l ≤ n](/img/revistas/ruma/v49n2/2a04355x.png)
![m ∈ M (s,l,α)](/img/revistas/ruma/v49n2/2a04356x.png)
![T α](/img/revistas/ruma/v49n2/2a04357x.png)
![p L (dx )](/img/revistas/ruma/v49n2/2a04358x.png)
![q L (dx)](/img/revistas/ruma/v49n2/2a04359x.png)
![1 < p < n∕α](/img/revistas/ruma/v49n2/2a04360x.png)
![1∕q = 1∕p - α ∕n](/img/revistas/ruma/v49n2/2a04361x.png)
![K α](/img/revistas/ruma/v49n2/2a04362x.png)
![Tα](/img/revistas/ruma/v49n2/2a04363x.png)
![K α ∈ H α,r](/img/revistas/ruma/v49n2/2a04364x.png)
![1 < r < (n∕l)′](/img/revistas/ruma/v49n2/2a04365x.png)
![ɛ > 0](/img/revistas/ruma/v49n2/2a04366x.png)
![0 < p < ∞](/img/revistas/ruma/v49n2/2a04367x.png)
![w ∈ A ∞](/img/revistas/ruma/v49n2/2a04368x.png)
![]() |
Now we can apply Theorems 2.3 and 2.5 to this operator.
Theorem 3.3. If and
a weight, then
![]() |
and
![]() |
where and
is small enough.
Observe that as is at our choice, we can write
, for all
. Therefore, we can write
in (3.6) and (3.7).
[1] K. F. Andersen and E. T. Sawyer, Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), 547-558. [ Links ]
[2] A.L. Bernardis, M. Lorente, Sharp two weight inequalities for commutators of Riemann-Liouville and Weyl fractional integral operator, Preprint. [ Links ]
[3] A.L. Bernardis, M. Lorente, F.J. Martín-Reyes, M.T. Martínez, A. de la Torre and J.L. Torrea, Differential transforms in weighted spaces, J. Fourier Anal. Appl. 12 (2006), no. 1, 83-103. [ Links ]
[4] A.L. Bernardis, M. Lorente, G. Pradolini, M.S. Riveros,Composition of fractional Orlicz maximal operators and -weights on spaces of homogeneous type, Preprint. [ Links ]
[5] A.L. Bernardis, M. Lorente, M.S. Riveros.On Weighted inequalities for generalized fractional integrals operators Preprint. [ Links ]
[6] M.J. Carro, C. Pérez, F. Soria and J. Soria, Maximal functions and the control of weighted inequalities for the fractional integral operator, Indiana Univ. Math. J. 54 (3) (2005), 627-644. [ Links ]
[7] S. Chanillo, D.K. Watson and R.L. Wheeden, Some integral and maximal operators related to starlike sets, Studia Math. 107(3) (1993), 223-255. [ Links ]
[8] R. Coifman, Distribution function inequalities for singular integrals, Proc. Acad. Sci. U.S.A. 69 (1972), 2838-2839. [ Links ]
[9] Y. Ding, Weak type bounds for a class of rough operators with power weights, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2939-2942. [ Links ]
[10] Y. Ding and S. Lu, Weighted norm inequalities for fractional integral operators with rough kernel, Can. J. Math. 50 (1998), no. 1, 29-39. [ Links ]
[11] R.L. Jones and J. Rosenblatt, Differential and ergodic transform, Math. Ann. 323 (2002), 525-546. [ Links ]
[12] D.S. Kurtz, Sharp function estimates for fractional integrals and related operators, Trans. Amer. Math. Soc. 255 (1979), 343-362. [ Links ]
[13] D.S. Kurtz and R.L. Wheeden, Results on weighted norm inequalities for multipliers, J. Austral. Math. Soc. A 49 (1990), 129-137. [ Links ]
[14] M. Lorente, J.M. Martell, M.S. Riveros and A. de la Torre Generalized Hörmander's condition, commutators and weights, J. Math. Anal. Appl. (2008), doi:10.1016/j.jmaa.2008.01.003. [ Links ]
[15] M. Lorente, J.M. Martell, C. Pérez and M.S. Riveros Generalized Hörmander's conditions and weighted endpoint estimates, Preprint 2007. [ Links ]
[16] M. Lorente, M.S. Riveros and A. de la Torre, Weighted estimates for singular integral operators satisfying Hörmander's conditions of Young type, J. Fourier Anal. Apl. 11 (2005), no. 5, 497-509. [ Links ]
[17] F.J. Martín-Reyes and A. de la Torre, One Sided BMO Spaces, J. London Math. Soc. 2 (49) (1994), no. 3, 529-542. [ Links ]
[18] F.J. Martín-Reyes, P. Ortega and A. de la Torre, Weighted inequalities for one-sided maximal functions, Trans. Amer. Math. Soc. 319 (1990), no. 2, 517-534. [ Links ]
[19] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165, (1972) 207-226. [ Links ]
[20] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192, (1974) 261-274. [ Links ]
[21] R. O'Neil, Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc. 115, (1963) 300-328. [ Links ]
[22] C. Pérez, Weighted norm inequalities for singular integral operators,J. London Math. Soc.49 (1994) 296-308. [ Links ]
[23] C. Pérez, Sharp -weighted Sobolev inequalities, Ann. Inst. Fourier (Grenoble) 45 (3), (1995) 809-824. [ Links ]
[24] M. Rao and Z.D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, Inc., New York, 1991. [ Links ]
[25] J.L. Rubio de Francia, F.J. Ruiz and J. L. Torrea, Calderón-Zygmund theory for vector-valued functions, Adv. in Math. 62 (1986), 7-48. [ Links ]
[26] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Trans. Amer. Math. Soc. 297 (1986), 53-61. [ Links ]
[27] C. Segovia and J.L. Torrea, Higher order commutators for vector-valued Calderón-Zygnund operators, Trans. Amer. Math. Soc. 336 (1993), 537-556. [ Links ]
M. S. Riveros
FaMAF,
Universidad Nacional de Córdoba,
CIEM (CONICET),
5000 Córdoba, Argentina
sriveros@mate.uncor.edu
Recibido: 10 de abril de 2008
Aceptado: 5 de junio de 2008