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Revista de la Unión Matemática Argentina

On-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 A : [0,∞ ) → [0, ∞ ) is a Young function if it is continuous, convex, increasing and satisfies A (0) = 0 and A(t) → ∞ as t → ∞ .
Given a Young function A , we define the A -mean Luxemburg norm of a function f on a cube (or a ball) Q in ℝn by

 { 1 ∫ ( |f|) } ||f||A,Q = inf λ > 0 :---- A --- ≤ 1 . |Q | Q λ (1.1)

It is well known that if A (t) ≤ CB (t) for all t ≥ t 0 then ||f||A,Q ≤ C ||f ||B,Q , for all cubes Q and functions f . Thus, the behavior of A (t) for t ≤ t0 is not important. If A ≈ B , that is there are constants t0,c1,c2 > 0 such that c1A(t) ≤ B (t) ≤ c2A (t) for t ≥ t0 , the latter estimate implies that ||f||A,Q ≈ ||f||B,Q .
 Each Young function A has an associated complementary Young function A- satisfying

 - 1 --- 1 t ≤ A (t)A (t) ≤ 2t,

for all t > 0 . There is a generalization of Hölder's inequality

 1 ∫ ---- |fg| ≤ ||f||A,Q ||g||A,Q. |Q | Q (1.2)

A further generalization of Hölder's inequality (see [21]) that will be useful later is the following: If A,B and C are Young functions and

 -1 - 1 - 1 A (t)B (t) ≤ C (t)
then
∥f g∥C,Q ≤ 2∥f ∥A,Q∥g∥B,Q. (1.3)

When A (t) = t we understand that -- A (t) = 0 if 0 ≤ t ≤ 1 and -- A (t) = ∞ otherwise. Then -- A is not a Young function, but  -- L A = L∞ and the latter inequalities make sense if one of the functions is A or -- A .  

For each locally integrable function f and 0 ≤ α < n , the fractional maximal operator associated to the Young function A is defined by

 α ∕n M α,Af (x) = sup|Q | ||f||A,Q. Q∋x
For α = 0 we write MA instead of M0,A . When A(t) = t then M α,A = M α is the classical fractional maximal operator. For α = 0 and A (t) = t we obtain M0,A = M , the Hardy-Littlewood maximal operator. Consider the case α = 0 , for
 r r + β + k A (t) = t , A (t) = t(1 + log (t)) , A (t) = t(1 + log (t)) ,
the maximal operators associated to these Young functions are:
M (f ) = M (|f |r)1∕r, M r + β(f) and M + k(f) . r L (log L) L(log L)
If k ≥ 0 , k ∈ ℤ , then ML (log+L)k is pointwise equivalent to M k+1 , where  k times k ◜---◞◟----◝ M = M ∘ ... ∘ M , k ∈ ℕ. It is also easy to check that if k > 0 and r > 1 , then
M f (x) ≤ CML (log+L)kf(x) ≤ CMrf (x) .

The good weights for M are those in the A p classes of Muckenhoupt (see [19] and also [26] and [18] for the one-sided case).

The good weights for the M α maximal operator are the A (p,q) classes. It is proved in [20] ( see [1] for the one sided version) that ||(M αf)w ||q ≤ C ||fw ||p if and only if w ∈ A (p,q) , for 1 < p < q , 1∕p - 1 ∕q = α∕n , where

( 1 ∫ )1∕q ( 1 ∫ ′)1 ∕p′ ---- wq ---- w -p ≤ C, A(p,q) |Q| |Q | |Q | |Q|
for all cube Q .

Also observe that for the case p = q , w ∈ A(p,p ) is equivalent to say that  p w ∈ Ap .

Let us define a generalization of the Hörmander condition, for a given kernel K . We used the notation: |x| ~ s for s < |x| ≤ 2 s and ∥f ∥A,|x|~s = ∥f χ ∥A,B(0,2s). {|x|~s}

Definition 1.1. Let A be a Young function and let 0 ≤ α < n . The kernel K α is said to satisfy the  α,A L -Hörmander type condition, we write K α ∈ H α,A , if there exist c ≥ 1 , C > 0 such that for any y ∈ ℝn and R > c |y |,

 ∞∑ (2m R )n- α∥K α(⋅ - y) - Kα (⋅)∥A,|x|~2mR ≤ C. m=1
We say that K α ∈ H α,∞ if K α satisfies the previous condition with ∥ ⋅ ∥L∞,|x|~2m R in place of ∥ ⋅ ∥A,|x|~2m R .

Definition 1.2. The kernel Kα is said to satisfy the  * H α,∞ condition, if there exist c ≥ 1 , C > 0 such that

 |y | |K α(x - y ) - K α(x)| ≤ C--------, |x | > c|y|. |x|n+1 -α

Observe that when α = 0 we obtain that H0,A = HA defined in [16].

If  r A (t) = t , for r ≥ 1 , then we write H α,A = H α,r . This H α,r condition appears implicitly in [12]. On the other hand, since t ≤ C A (t) for t ≥ 1 we have that H α,A ⊂ H α,1 . Also, it is easy to see that H *α,∞ ⊂ H α,∞ ⊂ H α,A .

Suppose that T is an operator given by convolution with a kernel K which satisfies some regularity condition and suppose that we know some behavior of T with respect to the Lebesgue's measure (weak or strong type inequalities for T ). Sometimes, in order to know how is the behavior of T when we change the measure, (i.e., when we consider the measure w (x)dx where w is a weight, (0 ≤ w ∈ L1loc(ℝn ) )) the following inequality is useful (we call it a Coifman type inequality)

∫ p ∫ p |Tf |w ≤ C (MT f )w . (1.4)

Here MT is a maximal operator related to the operator T which is normally easier to deal with. In general, M T is strongly related with the kernel K and its size is inverse to the smoothness of K : the rougher the kernel, the bigger the maximal.

For T a Calderón-Zygmund singular integral operator (i.e., K ∈ H *∞ , see Definition 1.2, for α = 0 ) inequality (1.4) holds with MT = M , the Hardy-Littlewood maximal function, 0 < p < ∞ , and w ∈ A ∞ (see [8]).

If T is a singular integral operator with less regular kernel, (see [13]) for example if the kernel K satisfies an Lr -Hörmander condition (Definition 1.1, for A (t) = tr and α = 0 ), then inequality (1.4) holds with MT = Mr ′ , with 1∕r + 1∕r′ = 1 , for all 0 < p < ∞ , and w ∈ A ∞ (see [25]).

For a Young function A , the  A L -Hörmander condition is introduced in [16], which generalized in the scale of the Orlicz spaces the  r L -Hörmander condition. In [16] the authors showed that, if the kernel K ∈ HA (Definition 1.1, for α = 0 ), then inequality (1.4) holds with MT = MA-- , where -- A is the complementary function of A , for all 0 < p < ∞ , and w ∈ A ∞ .

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 LA -Hörmander condition for  1 A (t) = e (t1+ε) - 1, (ε > 0 ). Therefore, this operator satisfies inequality (1.4) with the maximal operator  -- M A , where -- 1+ ε A (t) = t(log(e + t)) (actually they obtain a smaller operator since T is a one-sided operator, because suppK ⊂ (- ∞, 0) ).

The Coifman type inequality allows us to obtain, for general linear operators, two-weight inequalities of the type

∫ ∫ |Tf |pw ≤ C |f |pMT w , (1.5)

for 1 < p < ∞ and in the endpoint case p = 1 ,

 ∫ w ({x ∈ ℝn : |Tf (x)| > λ }) ≤ C |f(x)|MT w(x )dx, λ ℝn (1.6)

for every weight w , with no assumptions on w . The operators MT are again suitable maximal operators related with T 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 T is a Calderón-Zygmund operator (with kernel  * K ∈ H ∞ ), inequality (1.5) holds with  [p]+1 MT = M , where [p] is the integer part of p (see [22]). In the endpoint case p = 1 , inequality (1.6) for Calderón-Zygmund operators hold with MT = ML (logL)ɛ , for any ɛ > 0 , where ML (logL)ɛ is the maximal function associated to the Young function A (t) = t(1 + log+ t)ɛ . This result was proved by Carlos Pérez in [22]. For T a singular integral associated to a kernel K satisfying a general Hörmander's condition given by a Young function A , 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 T the classical Riesz potential I α and M T the fractional maximal function M α , defined for 0 < α < n and locally integrable function f by

 ∫ I f(x) = ---f(y)--- dy. α ℝn |x - y|n-α
 Observe that using the mean value theorem we can prove that, the kernel of the fractional integral, Iα ,  --1-- K α(x) = |x|n-α , belongs to H *α,∞ . For Iα , inequality (1.5) holds with MT = M αp(M [p]) (this result is also due to Carlos Pérez, see [23]). On the other hand, inequality (1.6) for I α holds with M = M (M ɛ) T α L(logL) in [6] (see also [2]).

There are fractional integrals with less regular kernel than the Riesz transform (see for example [7], [12], [27], [20], [9], [10]). Suppose that Ω is homogeneous of degree zero and Ω ∈ Ls(Sn -1) , where Sn- 1 denotes the sphere of ℝn and s > 1 . Define the fractional integral associated to Ω by

 ∫ Ω (y ∕|y|) T Ω,αf (x) = ---n--α-f (x - y)dy. ℝn |y|
In [10] inequalities with weights were established for this operator, which generalized the corresponding inequalities for Iα given by Muckenhoupt and Wheeden in [20]. In a more general context and with an aditional condition in Ω , that is, Ω satisfying the Ls(Sn -1) -Dini smoothness condition, Segovia and Torrea [27], studied the good weights for this operator and its commutators, using extrapolation theorems.

In this note we state and briefly sketch the proofs of the corresponding results for general fractional integrals T α , 0 < α < n , given by convolution with a kernel K α which satisfy a H α,A condition, for appropriate Young functions A (see Theorems 2.1, 2.3 and 2.5).

From now on, for 0 < α < n , Tα will be a fractional operator bounded from Lp(dx) to Lq (dx) , for all 1 < p < q < ∞ satisfying 1∕p - 1 ∕q = α∕n .

2. Statements of the Results

Theorem 2.1. Let Tαf = K α * f be a fractional operator given by a kernel K α . Suppose T α is of weak-type  n (1, n-α) .

(a)

If A be a Young function and Kα ∈ H α,A , then for any 0 < p < ∞ and w ∈ A ∞ ,

∫ ∫ p -- p ∞ n |T αf(x)| w (x)dx ≤ C n M α,Af (x) w (x )dx, f ∈ L c . ℝ ℝ (2.1)
(b)

Moreover, if the kernel K α is supported in (- ∞, 0) , then for any 0 < p < ∞ , w ∈ A+∞ , it follows that (2.1) holds with M +--f α,A in place of  -- M α,Af where  + α -- M α,Af(x) = supx <b(b - x ) ∥f∥A,(x,b) .

Remark 2.2. Observe that we can apply the theorem to I α and  + Iα (respectively) obtaining the result in [20] and [17], for -- A (t) = t .

Proof. To prove this Theorem we use the sharp operator of Tαf . Given  n x ∈ ℝ and a cube Q ∋ x , decompose f = f1 + f2 , where f1 = f χ2Q and f2 = f - f1 . For T αf1 we use Kolmogorov and that T α is of weak-type (1, -n-) n-α . For the global part we use that Tα is the convolution with the kernel K ∈ H α A and the generalized Hölders inequality. □

Theorem 2.3. Let A be a Young function and 1 < p < ∞ . Suppose that there exist Young functions E , D such that E ∈ Bp ′ and  -- E -1(t)F -1(t) ≤ A -1(t) with F (t) = D(tp) .  Let Tα be a linear operator such that its adjoint T* α satisfies

∫ ∫ |Tα*f (x)|q w(x) dx ≤ C M α,Af (x)q w (x)dx, f ∈ L ∞c ℝn ℝn (2.2)

for all 0 < q < ∞ and w ∈ A ∞ . Then, for p > 1 and for any weight u ,

∫ ∫ |Tαf (x)|p u(x)dx ≤ C |f(x)|pM αp,Du(x) dx, f ∈ L∞ . ℝn ℝn c (2.3)

Remark 2.4. For the applications below, and since all our operators are of convolution type, proving (2.2) for  * T α or Tα 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  δ (MT w ) belongs to A1 , for all 0 < δ < 1 and any w . For the maximal operators MT that appears in this proof, has been proved in [4]. □

Theorem 2.5. Let T f = K * f α α be a fractional operator. Suppose that there exists δ > 0 such that for any p ∈ (1,1 + δ) , there exists a Young function Dp satisfying

∫ ∫ |Tαf|pu ≤ C |f|pM αp,Dpu, ℝn ℝn (2.4)

for all weight u . If K α ∈ H α,A , then for any weight u ,

 ∫ u({x ∈ ℝn : |T f(x)| > λ}) ≤ C- |f |(M u + M --u + M u), α λ ℝn α,A αp,Dp (2.5)

for all λ > 0 .

3. Applications

3.1. Fractional integrals associated to a homogeneous function. Denote by Sn -1 the unit sphere on ℝn . For x ⁄= 0 , we write x ′ = x∕|x| . Let us consider Ω ∈ L1(Sn -1) . This function can be extended to  n ℝ \ {0} as  ′ Ω (x) = Ω (x ) (abusing on the notation we call both functions Ω ). Thus Ω is a function homogeneous of degree 0 . Let 0 < α < n , and let A be a Young function such that  n-α B (t) = A (t n ) is also a Young function. Let Ω ∈ LA (Sn- 1) and satisfying the LA (Sn -1) -Dini smoothness condition, i.e.,

∫ 1 dt ϖA (t)-- < ∞, 0 t (3.1)

where

ϖA (t) = sup ∥Ω (⋅ + y) - Ω(⋅)∥A,Sn-1. |y|≤t
Consider the fractional operator
 ∫ -Ω(y-) TΩ,αf(x) = ℝn |y|n-αf (x - y)dy.
Using Hölder's inequality with B and -- B it is easy to see that Ω ∈ LA (Sn -1) implies  -n- Ω ∈ L n-α(Sn- 1) . Then, by the result in [7], TΩ,α is of weak type (1, nn-α) , with respect to the Lebesgue's measure and is bounded from Lp(dx ) to Lq (dx) , whenever 1∕p - 1∕q = α ∕n , 1 < p < q < ∞ . We can prove (as in [14]) that the kernel  -Ω(x) K α(x) = |x|n-α satisfies the H α,A condition. Therefore Theorems 2.1, 2.3 and 2.5 can be applied to the operator TΩ,α .

In the particular case that A (t) = tr with r ≥ -n-- n-α we get the following:

Theorem 3.1. Let Ω ∈ Lr (Sn-1) be as above and satisfying the Lr -Dini condition.

(a )

If 0 < p < ∞ and w ∈ A ∞ , then

∫ ∫ |TΩ,αf(x )|pw (x )dx ≤ C (M α,r′f)pw(x )dx, f ∈ L∞ . ℝn ℝn c (3.2)
(b)

If 1 < p < r and u a weight, then

∫ ∫ |T f(x)|pu(x)dx ≤ C |f|pM u (x)dx, f ∈ L∞ . ℝn Ω,α ℝn αp,Dp c (3.3)
(c)

If 1 < p < r and u is a weight, then

 ∫ u {x ∈ ℝn : |T f (x )| > λ } ≤ C |f(x)|(M ′ u(x) + M u (x ))dx. Ω,α λ ℝn r αp,Dp (3.4)

In both cases  (r∕p)′ + (r∕p)′(p- 1)+ε Dp (t) = t (1 + log t) and ε > 0 is small enough.

Proof. We only have to apply the theorems with the following Young functions: A (t) = tr , E (t) = tp′(1 + log+ t)-1-ɛ , and F (t) = trr-pp(1 + log+ t)(r∕p)′(p-1)+ε , where ɛ > 0 is some small enough number that is related with ε > 0 . Observe that in part (c) we obtain M u + M α,r′u + M αp,Dpu on the right hand side, but it is easy to see that M α,r′u ≤ Mr ′u + M αp,Dpu and M u ≤ Mr′u . □

 For TΩ,α 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  ′ n- r < p < α , 1 1 α- q = p - n and  r′ ′ ′ w ∈ A (p∕r ,q∕r ) . Then

( ∫ )1 ∕q (∫ )1∕p q q p p n |TΩ,αf (x)|w (x)dx ≤ C n |f| w (x )dx . ℝ ℝ

Proof. First of all observe that wr ′ ∈ A (p∕r′,q∕r′) implies wq ∈ A ∞ . 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
Since  ( r′ )1∕r′ M α,r′f (x) = M αr′|f| (x ) and  r′ ′ ′ w ∈ A (p∕r ,q∕r ) (see [20]) we have that

∫ ∫ ( )q∕r′ ( ∫ ( )p∕r′ )q∕p (M α,r′f )qwq = M αr′|f |r′ (wr ′)q∕r′ ≤ C |f|r′ (wr′)p∕r′ . ℝn ℝn ℝn

3.2. Fractional integrals associated to a multiplier. Let 0 < α < n . Given a function m defined in  n ℝ we consider the multiplier operator T α defined a priori for functions f in the Schwartz class by ^Tαf(ξ) = m (ξ)f^(ξ) . Given 1 < s ≤ 2 and 0 ≤ l ∈ ℕ we say that m ∈ M (s,l,α) if there exists a constant B such that |m (x)| ≤ B|x|-α and

 |β|+ α β sRu>p0 R ∥D m ∥Ls,|ξ|~R < + ∞, for all |β| ≤ l.
In [12], Kurtz proved that if n∕s < l ≤ n and m ∈ M (s,l,α) then T α is bounded from  p L (dx ) to  q L (dx) , for 1 < p < n∕α and 1∕q = 1∕p - α ∕n . If K α is the kernel of Tα , he proved that K α ∈ H α,r for all 1 < r < (n∕l)′ and, as a consequence, he obtained the following Coifman type inequality: for all ɛ > 0 , 0 < p < ∞ and w ∈ A ∞ ,
∫ ∫ |Tαf(x )|pw (x)dx ≤ C M α,n∕l+ɛf (x)pw (x)dx. ℝn ℝn (3.5)

Now we can apply Theorems 2.3 and 2.5 to this operator.

Theorem 3.3. If  ′ 1 < p < r < (n∕l) and u a weight, then

∫ ∫ p p ∞ n |Tαf (x)|u (x )dx ≤ C n |f| M αp,Dpu (x )dx, f ∈ Lc . ℝ ℝ (3.6)

and

 ∫ u{x ∈ ℝn : |Tαf(x)| > λ} ≤ C- |f(x)|(Mr ′ u (x ) + M αp,Dpu(x)) dx, λ ℝn (3.7)

where  (r∕p)′ + (r∕p)′(p-1)+ ε Dp(t) = t (1 + log t) and ε > 0 is small enough.

Observe that as ε is at our choice, we can write  ′ ′ Dp (t) = t(r∕p) (1 + log+ t)(r∕p)(p-1)+ε  ′ ≲ t(˜r∕p) , for all 1 < p < ˜r < (n∕l)′ . Therefore, we can write Dp (t) = t(r∕p)′ in (3.6) and (3.7).

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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