Maximal functions and the control of weighted inequalities for the

Indiana University Mathematics Journal, 54 (2005), 627–644.

Maximal functions and the control of weighted inequalities for the fractional integral operator Mar´ıa J. Carro

§

¶

´rez Carlos Pe

Fernando Soria

k

and Javier Soria

∗∗

Abstract We study weak-type (1, 1) weighted inequalities for the fractional integral operator Iα . We show that the fractional maximal operator Mα controls these inequalities when the weight is radially decreasing. However, we exhibit some counterexamples which show that Mα is not appropriate for this control on general weights. We do provide, nevertheless, some positive results related to this problem by considering other suitable maximal functions.

1

Introduction

The purpose of this paper is to study certain weighted estimates related to the fractional operator, defined by Z Iα f (x) = Rn

f (y) dy, |x − y|n−α

and to the fractional maximal operator Mα f (x) = sup x∈Q

1 |Q|1−α/n

Z |f (y)| dy, Q

where 0 < α < n. To be more precise, the problem we want to analyze is the following: § ¶ k ∗∗

Research partially supported by Grants MTM2004-02299 and 2001SGR00069. Research partially supported by Grant BFM2002-02204. Research partially supported by Grant MTM2004-00678. Research partially supported by Grants MTM2004-02299 and 2001SGR00069.

Keywords: Weighted inequalities, maximal operator, fractional integral. MSC2000: 42B25, 26A33.

Given a weight w (i.e., a positive, locally integrable function), find the best (“smallest”) weight W such that w

n o C Z x ∈ Rn : |Iα f (x)| > λ ≤ |f (x)| W (x)dx, λ Rn

(1)

for every λ > 0 and for suitable functions f . Similar results have been considered for other kind of operators like the HardyLittlewood maximal function ([FS]), the fractional maximal operator (see (4)), the Hilbert transform and Singular Integral operators ([CF], [P3]), the Kakeya maximal function ([M¨ uSo]), etc. We recall that, for p > 1, the weights (w, W ) for which the weak-type (p, p) version of (1) holds, were characterized by E. Sawyer ([S]). The characterization is as follows: the inequality n Z o C n w x ∈ R : |Iα f (x)| > λ ≤ p |f (x)|p W (x) dx λ Rn holds if and only if there is a constant C such that for every cube Q Z Q

p0

(Iα (χQ w)(x))

1−p0

W (x)

Z dx ≤ C

w(x)dx. Q

It is known that (1) is equivalent to checking the inequality on finite linear combinations of Dirac deltas (see [G] and [MeSo]). In particular, taking in (1) f = δx0 (the Dirac mass at x0 ), we obtain that Iα f (x) = |x − x0 |α−n , and hence, for regular weights W , n n o o n n α−n w x ∈ R : Iα δx0 (x) > λ = w x ∈ R : |x − x0 | >λ Z = w(x) dx {|x−x0 | λ ≤ λ Rn

(3)

where C is independent of λ, f and, possibly, of w too. This question was raised by the second author during the Spring School on Analysis: Function Spaces and Their Applications, held at Paseky nad Jizerou, Czech Republic, in 1999. An indication that this might be true can be found in the statement of Theorem 4.2, which says that there exists a constant C, depending only on dimension, so that for every λ > 0, all measurable functions f and all radial decreasing weights w, (3) holds. The main result of this paper (Theorem 2.1) shows that (3) is false, however, for general weights w and for any α ∈ (0, n). (A first result in this direction, for certain values of α and n, was given by Jan Mal´ y [M].) This should be compared with the following well known estimate, which follows as a consequence of classical covering lemmas ([FS]): n o C Z n w x ∈ R : Mα f (x) > λ ≤ |f (x)| Mα w(x)dx. λ Rn

(4)

These results naturally suggested the following problem: to determine which other maximal operators T may replace Mα so that inequality (1) holds with W = T w. Theorems 3.1 and 4.2 below are particular solutions to this question. The paper is organized as follows. We present in Section 2 the counterexamples to inequality (3). The remaining part of the paper is devoted to the study of the aforementioned question. In Section 3 we show that a good substitute for Mα is given by the composition of Mα with a generalized Hardy-Littlewood maximal function associated to logarithmic Orlicz norms. Finally, in Section 4 we prove Theorem 4.2 for radial weights as a consequence of a more general approach which also produces a different kind of “admissible” mappings w → T w = W for general weights. Acknowledgment. We are grateful to one of the referees for pointing out reference [R].

3

2

The counterexamples

We will prove in this section that (3) does not hold in general. The precise result is the following: Theorem 2.1 Given 0 < α < n, there exists a weight w such that for every finite constant C, one can find λ > 0 and a function f so that n o C Z n |f (x)|Mα (w)(x) dx. w x ∈ R : |Iα f (x)| > λ > λ Rn

(5)

Proof: We will construct initially the weight w depending on the value of C. A simple argument by iteration will allow us to remove later that constraint. The proof is based on the discretization results of [G] and [MeSo], and the main idea, for general α ∈ (0, n), is to choose the densities specially located on a suitable Cantor set. We also present a second example in the case α ∈ N. Let 0 < δ < 1 be the solution to the equation 2α (1 − δ)n−α = 1, and consider the following Cantor set (see [F]): Let E0 be the unit cube in Rn , and delete all but the 2n corner cubes {Q1k }, of side (1 − δ)/2 to obtain E1 . Continue in this way, at the N th stage replacing each cube of N to get E . Thus E EN −1 by the 2n corner cubes {QN N N contains k }, of side ((1 − δ)/2)

2nN cubes of volume ((1 − δ)/2)nN , and hence |EN | = (1 − δ)nN . (If E = ∩N EN , then it can be shown that E has Hausdorff dimension n − α.) For each k, l ∈ {1, · · · , 2nN }, we N say that QN k and Ql are j−relatives, j = 1, · · · , N , if j is the smallest index for which −j N −j N 0 there exists a (unique) cube QN such that QN m k , Ql ⊂ Qm , where Qm = E0 . Thus n n (j−1)n j−relatives, j = 2, · · · , N . Moreover, if QN k has (2 − 1) 1–relatives, and (2 − 1)2 N N N QN k and Ql are j−relatives, xk ∈ Qk and xl ∈ Ql , then it is easy to see that

|xk − xl | ≈

 1 − δ N −j 2

.

(6)

Now, in order to prove (5), we take w = χE , and by the result of [G] and [MeSo], it N

suffices to consider a finite sum of Dirac deltas, instead of a function f . For this, we

4

choose xk to be the center of the cube QN k , and f =

P2nN

k=1 δxk .

Given x ∈ EN we find a

lower bound for Iα (f )(x) as follows: if x ∈ QN k0 , nN

Iα (f )(x) =

2 X k=1

N X X 1 1 (2n − 1)2n(j−1) ≥ ≈ (N −j)(n−α)  |x − xk |n−α |x − xk |n−α 1−δ j=1

k6=k0

2

N  2 N (n−α) X  2 N (n−α) 2jα . ≈ ≈ N 1−δ 1−δ (1 − δ)−j(n−α) j=1

Therefore, if we take λN ≈ N



2 1−δ

N (n−α)

, we have

w({Iα (f ) > λN }) = |EN | = (1 − δ)nN . On the other hand, if we show that Mα χE (xk ) ≈ N

 2 −N α , 1−δ

(7)

then (5) follows by taking N >> C, since nN

2 1  2 N (α−n) nN  2 −N α |EN | 1 X 2 = Mα χE (xk ) ≈ . N λN N 1−δ 1−δ N k=1

To prove (7), fix xk and assume xk ∈ Qjm , j = 0, · · · , N . If j = 0,  2 −N α |Q0m ∩ EN | = |E | = . N 1−δ |Q0m |1−α/n j If j = 1, · · · N , then Qjm ∩ EN is equal to the union of all the 2n(N −j) cubes QN l ⊂ Qm ,

and hence |Qjm ∩ EN | |Qjm |1−α/n



1−δ 2

= 

nN

1−δ 2

2n(N −j)

nj(1−α/n)

=

 2 −N α . 1−δ

Now, if xk ∈ QN k , take any cube Q centered at the point xk , and let j ∈ {1, · · · , N } be the largest index for which Q has nontrivial intersection with a cube QN l which is a j−relative N of QN k (if Q meets only Qk the estimate is trivial). Then by (6),

|Q| ≥ C

 1 − δ n(N −j) 2

5

,

N and Q ∩ EN contains only cubes QN l which are l−relatives of Qk , l ∈ {1, · · · , j}. Since

there are at most 2nj of such cubes, we get  nN nj 1−δ  2 −N α 2 2 |Q ∩ EN | , ≤ C ≈  n(N −j)(1−α/n) 1−δ |Q|1−α/n 1−δ 2

which proves (7). A second example for α ∈ N: When α is a positive integer, the proof simplifies considerably as can be seen in the following construction. Given n ≥ 2, let us take α ∈ {1, . . . , n − 1}. For large N ∈ N, define the rectangle in Rn n−α times

α times

z }| { z }| { R = (0, N ] × · · · × (0, N ] × (0, 1] × · · · × (0, 1], and consider the net {xk }k of points with integral coordinates inside R. Let µ =

P

k δxk .

Then, the total mass of µ and the volume of R coincide and equal ||µ|| = |R| = N n−α . It is easy to see that for x ∈ R one has N

X k≥1

X 1 1 ≥ c mn−α−1 ≈ log N, n−α |x − xk | |m|n−α m=1

where c denotes a small constant depending only on dimension, so that R ⊂ {x : Iα µ(x) > c log N }. Also, if we set w = χR , then Mα w(xk ) ≈ 1. Hence, w({x : Iα µ(x) > c log N }) = |R| = N n−α , whereas

P

k

Mα w(xk ) ≈ ||µ|| = N n−α . This shows, as in the first example, that  sup

λw({x : Iα µ(x) > λ}) R Mα w(x)dµ(x)

 = ∞,

the supremum taken over all λ > 0, all linear combinations of Dirac deltas µ and all weights w.

6

Let us now show how we can iterate the above examples in order to construct a weight w for which inequality (3) fails. We will only consider the second example, the construction for the first example being similar. j

Given j ∈ N, set N = 22 and let R be the rectangle constructed above for such N . Write Rj = R − zj , where zj ∈ R will be chosen later. Let {xjk }k be the net of points with P integral coordinates inside Rj and µj = k δxj . In this way Rj ⊂ {x : Iα µj (x) > c2j }. k

Put wj = χRj and dµ =

X

cj dµj ;

w=

j∈N

X

cj wj ,

with cj =

j∈N

1 . j|Rj |1/2

Choose finally the sequence {zj } sufficiently separated so that the rectangles {Rj } are disjoint and moreover for each j, we have Mα w(xjk ) ≈ Mα (cj wj )(xjk ) ≈ cj . We now have for every m ∈ N w({x : Iα µ(x) > c cm 2m }) ≥ w(Rm ) = cm |Rm |, whereas

R

Rn

Mα w(x)dµ(x) =

j k,j cj Mα w(xk )

P

≈

2 j cj ||µj ||

P

=

1 j j2

P

< ∞. If inequality

(3) were true we should have cm |Rm | ≤

C , cm 2m

for certain finite constant C independent of m. But this is clearly false, for the above would imply 2m ≤ Cm2 , ∀m ∈ N. This finishes the proof of Theorem 2.1.

3

2

Logarithmic maximal functions

We are now going to consider alternative solutions to (1) in terms of weights W which are obtained by means of certain maximal operators defined in [P1]. In order to do that, we introduce some additional notation and give several definitions (see [BS]). We will write, as usual, the weak-Lp norm of a function, with respect to the measure

7

w(x) dx as: kf kLp,∞ (w) = sup t w({x ∈ Rn : |f (x)| > t})1/p . t>0

If B is a Young function, the mean Luxemburg norm of a measurable function f on a cube Q is defined by kf kB,Q

    Z 1 |f | = inf λ > 0 : B dx ≤ 1 . |Q| Q λ

(8)

When B(t) = t log(e + t)β , this norm is also denoted by k · kL(log L)β ,Q . When β = 0 we R 1 have the usual average |f |Q = |Q| Q |f |. We define the maximal function ML(log L)α f (x) = sup kf kL(log L)α ,Q . Q3x

For α = 0 we get the standard Hardy-Littlewood maximal function M f . More generally, for any B we define MB f (x) = sup kf kB,Q . Q3x

The main result in this section is contained in the following theorem. Here, M 2 denotes the composition of M with itself. Theorem 3.1 Let 0 < α < n and δ > 0. Then Z kIα f kL1,∞ (w) ≤ C

|f (x)| Mα (ML(log L)δ (w))(x)dx,

Rn

(9)

and in particular, Z kIα f kL1,∞ (w) ≤ C

|f (x)| Mα (M 2 (w))(x)dx.

Rn

That is, (1) holds with W = Mα (M 2 w). We now make the observation that (9) is not as sharp as Z kT f kL1,∞ (w) ≤ C

Rn

|f (x)| ML(log L)δ (w)(x)dx,

8

(10)

derived in [P3], where T is any classical Calder´on-Zygmund singular integral operator. Indeed, observe that Mα (ML(log L)δ (w)) ≥ Mα (ML (w)) = Mα (M (w)). Hence, when α = 0 we simply have M 2 which is bigger than ML(log L)δ (w). Probably, (9) holds for the weight Mα (M (w)) Theorem 3.1 is an immediate consequence of the next theorem, using also (4) and the fact that ML(log L)δ (w) ≤ CM 2 (w), for 0 ≤ δ ≤ 1 ([P3]). Theorem 3.2 Let 0 < α < n and δ > 0. Then kIα f kL1,∞ (w) ≤ C kMα f kL1,∞ (M

L(log L)δ

(w))

.

(11)

The method used to prove this theorem combines ideas from [P1] and [P2]. We begin by recalling the following decomposition lemma: Lemma 3.3 ([P1], Lemma 3.1) Let f and g be L∞ positive functions with compact support, and let µ be a nonnegative measure finite on compact sets. Let a > 2n , then there exist a family of cubes Qk,j and a family of pairwise disjoint subsets {Ek,j }, Ek,j ⊂ Qk,j , with |Qk,j |
0 such that for each cube Q C ess sup v(x) ≤ |Q| x∈Q

Z v dx. Q

It is very easy to check that RH∞ ⊂ A∞ (see [CN]).

9

Lemma 3.5 Let v be a weight satisfying the RH∞ condition. Then, there is a constant C such that for any weight w and all positive f , Z

Z Iα f (x) w(x) v(x) dx ≤ C

Mα (f )(x) M w(x) v(x) dx.

(14)

Rn

Rn

Proof: We start with inequality (13) with g replaced by w, and dµ replaced by v(x)dx: Z X |Qk,j |α/n Z f (y) dy w(y) v(y)dy |Qk,j | 3Q Q k,j k,j k,j Z X |Qk,j |α/n Z w(y)dy essupQk,j v ≤ C f (y) dy |Qk,j | Qk,j 3Qk,j k,j Z X |Qk,j |α/n Z 1 ≤ C f (y) dy w(y)dy v(Qk,j ). |Qk,j | |Qk,j | Qk,j 3Qk,j

Z

Iα f (x) w(x) v(x)dx ≤ C Rn

k,j

Since v ∈ A∞ and by the properties of the sets {Ek,j }, we have v(Qk,j ) ≤ C v(Ek,j ) for each k, j. Combining this with the fact that the family {Ek,j } is formed by pairwise disjoint subsets, with Ek,j ⊂ Qk,j , we continue with Z X |Qk,j |α/n Z 1 Iα f (x) w(x) v(x)dx ≤ C f (y) dy w(y)dy v(Ek,j ) |Qk,j | |Qk,j | Qk,j Rn 3Qk,j k,j XZ ≤ C Mα (f )(x) M w(x) v(x) dx

Z

k,j

Ejk

Z ≤ C

Mα (f )(x) M w(x) v(x) dx. Rn

2

The second lemma gives interesting examples of RH∞ weights. Lemma 3.6 ([CN]) Let g be any function such that M g is finite a.e. Then (M g)−α ∈ RH∞ , α > 0. Theorem 3.7 If 0 < p ≤ 1, then Z

Z

p

|Iα f (x)| w(x)dx ≤ C Rn

(Mα (f )(x))p M w(x) dx.

(15)

Rn

Proof: To prove (15) we will use the appropriate duality for the spaces Lp , p < 1: if f ≥0 kf kp = inf

nZ

fu

−1

o Z

−1

: u = 1 = f u−1 , p0

10

p p−1

for some u ≥ 0 such that u−1 p0 = 1, where p0 =

< 0. This follows from the

“reverse” H¨older’s inequality: Z f g ≥ kf kp kgkp0 , which is a consequence of the usual H¨older’s inequality.

We start the proof by choosing a nonnegative g, with g −1 Lp0 (M w) = 1, and such that Z kMα f kLp (M w) =

Mα (f )(x) Rn

M w(x) dx ≥ g(x)

Z Mα (f )(x) Rn

M w(x) dx, M (g δ )1/δ (x)

where we have used the Lebesgue differentiation theorem for any δ > 0. We apply both Lemma 3.5 and Lemma 3.6 to the weight M (g δ )−1/δ , to continue with Z kMα f kLp (M w) ≥

Iα (f )(x) Rn

w(x) dx ≥ kIα (f )kLp (w) M (g δ )−1/δ Lp0 (w) , 1/δ δ M (g )(x)

and everything is reduced to prove

−1

g p0

M (g δ )−1/δ p0 . ≥ L (M w) L (w) Since p0 < 0, this is equivalent to saying that Z

δ −p0 /δ

M (g )

Z

0

g −p (x) M w(x) dx.

(x) w(x) dx ≤

Rn

Rn

But, if we choose 0 < δ
1 and this follows from the classical

weighted norm inequality of C. Fefferman and E. Stein ([FS]): Z

p

Z

(M f ) (x) w(x) dx ≤ Rn

|f (x)|p M w(x) dx,

Rn

p > 1.

2

Proof of Theorem 3.2: By standard density arguments, we may assume that both f and the weight w are bounded, nonnegative functions and with compact support. Raising the quantity kIα f kL1,∞ (w) to the power 1/p, with p > 1 (which will be chosen at the end of the proof), then 1/p kIα f kL1,∞ (w)

Z

= (Iα f )1/p Lp,∞ (w) =

sup g∈L

p0 ,1

(w)

kgkLp0 ,1 (w)=1

11

Rn

(Iα f (x))1/p g(x)w(x)dx.

0

The last equality follows since Lp ,1 (w) and Lp,∞ (w) are associate spaces (this equality can also be proved as a consequence of Kolmogorov’s identity). Fixing one of these g’s we use (15) to continue with 1/p kIα f kL1,∞ (w)

Z

(Mα f (x))1/p M (gw)(x)dx

≤ C Rn

Z

M (gw)(x) f M w(x)dx, fw(x) M

(Mα f (x))1/p

= C Rn

f is an appropriate (maximal type) operator to be chosen soon. We continue with where M fw(x)dx): H¨older’s inequality for Lorentz spaces (the underlying measure is now M



M (gw) 1/p kIα f kL1,∞ (w) ≤ C (Mα f )1/p Lp,∞ (M fw) M fw Lp0 ,1 (M fw)

(gw) 1/p = C kMα f k 1,∞ f Mf

p0 ,1 f . Mw L (M w) L

(M w)

To conclude we just need to show that

M (gw)

M fw

fw) Lp0 ,1 (M

≤ c kgkLp0 ,1 (w) ,

or equivalently 0

0

fw), S : Lp ,1 (w) → Lp ,1 (M

(16)

where Sf =

M (f w) . f(w) M

f pointwise bigger than M ; that is, such that M w ≤ M fw for each To do this we choose M w. With this choice we trivially have fw). S : L∞ (w) → L∞ (M Therefore by the Marcinkiewicz’s interpolation theorem for Lorentz spaces due to R. Hunt ([BS]) it will be enough to show that for some  > 0 0

0

fw), S : L(p+) (w) → L(p+) (M

12

(17)

which amounts to prove Z Rn



Z (p+)0  1−(p+)0 f dy ≤ C M (wf )(y) M w(y)



(p+)0 w(y)dy, f (y)

(18)

Rn

for any f ≥ 0 bounded and with compact support. But this result follows from [P2]: indeed it is shown there that for r > 1 and η > 0 Z



Rn

Z r 0  1−r0 dy ≤ C M (f )(y) ML(log L)r−1+η (w)(y)



r 0 0 f (y) w(y)1−r dy.

(19)

Rn

We finally choose the appropriate parameters and weight. Let r = p + , η = , and pick the weight fw = M M (w). L(log L)p−1+2 This shows that for any p > 1 and  > 0, kIα f kL1,∞ (w) ≤ c kMα f kL1,∞ (M

L(log L)p−1+2

(w)) .

We conclude the proof of (11) by choosing p = 1+δ −2 and  such that 0 < 2 < δ.

4

2

Local and global parts. Radial weights

The purpose of this section is to prove Theorem 4.2 below. In fact, we are a going to show a more general result from which the theorem easily follows. Following arguments from [SW] we shall construct a collection of operators T such that (1) holds for any weight w, with W = T w. Let k ∈ Z and set Jk = {y ∈ Rn : 2k ≤ |y| < 2k+1 }, and Jk∗ = {y ∈ Rn : 2k−1 ≤ |y| < 2k+2 }. Let f be a positive function and, for each k ∈ Z, let us write f = fk,0 + fk,1 ,

fk,0 = f χJ ∗ . k

13

Then, Iα f (x) =

X

 Iα f (x) χJ (x) k

k∈Z

≤

X

  X Iα fk,1 (x) χJ (x) = Iα0 f (x) + Iα1 f (x). Iα fk,0 (x) χJ (x) + k

k

k∈Z

k∈Z

Iα0 is called the local part and Iα1 the global part of the operator Iα . Clearly, n n n o o o n 0 1 w x ∈ R : Iα f (x) > λ ≤ w x : Iα f (x) > λ/2 + w x : Iα f (x) > λ/2 o X n o X n = w x ∈ Jk : Iα fk,0 (x) > λ/2 + w x ∈ Jk : Iα fk,1 (x) > λ/2 k∈Z

k∈Z

=L(w) + G(w). We shall estimate L(w) and G(w) separately. Global Part: To estimate the global part, we observe that if x ∈ Jk and y ∈ / Jk∗ , then |x − y| ≈ |x| + |y|, and hence, for every x ∈ Jk , Z Iα fk,1 (x) ≤ C Rn

f (y) dy ≤ CMα f (x) + C (|x| + |y|)n−α

Z {|y|>|x|}

f (y) dy. |y|n−α

Therefore, G(w) ≤ w

n o x ∈ Rn : Mα f (x) > λ/(2C) n Z o f (y) n ¯ +w x∈R : dy > λ/(2C) := G(w). n−α {|y|>|x|} |y|

Using (4) in the first term and Chebyshev’s inequality in the second, we get that C G(w) ≤ λ

Z f (y) Mα w(y) dy. Rn

This shows that estimate (3) does hold for general weights w, if we replace Iα by its global part Iα1 .

14

Local Part: To estimate this part we use that Iα is self adjoint: Z Z 2X 2X L(w) ≤ Iα fk,0 (x)w(x) dx = f (y)Iα (wχJ )(y) dy k ∗ λ λ J J k k k∈Z k∈Z Z 2 = f (y)T (w)(y) dy, λ Rn where T (w)(y) =

XZ k∈Z Jk

w(x) dx χJ ∗ (y). k |x − y|n−α

(20)

Now, observe that if x ∈ Jk and y ∈ Jk∗ , then |x − y| ≤ 2k+3 , and hence the inner integral can be estimated by Z |x−y|≤2k+3

Z k+3 X w(x) 1 dx ≤ w(x) dx := Sk . |x − y|n−α 2j(n−α) |x−y|≤2j j=−∞

Let (h(j))j∈Z be any positive sequence such that, for every k ∈ Z, H(k) =

k+3 X

1 < ∞. h(j)

j=−∞

(21)

Then, Sk ≤ H(k) sup j∈Z

Z

h(j) 2j(n−α)

w(x) dx, x∈Jk ,|x−y|≤2j

and if we denote by Mα,h f (x) = sup j∈Z

Z

h(j)

f (y) dy,

2j(n−α)

|x−y|≤2j

we obtain T (w)(y) ≤

X

H(k)Mα,h (wχJ )(y) χJ ∗ (y). k

k∈Z

k

Thus, if y ∈ Jk∗ , Mα,h (wχJ )(y) ≤ sup k

j∈Z

Z

h(j) 2j(n−α)

|x−y|≤2j ,

|y| ≤|x| λ ≤ λ Rn holds, with ¯ W (y) = Mα w(y) + H(y)C α,h (w)(y).

(23)

EXAMPLES. We present here several interesting consequences of the above theorem. (i) If h(j) = 2−αj , then Mα,h ≤ CM , where M is the Hardy-Littlewood maximal operator, and we get that (1) holds for W (y) = Mα w(y) + |y|α M (w)(y). Now, if w is a radial decreasing weight, then 1 M (w)(y) ≈ n |y|

Z w(x)dx, |x|≤|y|

and, therefore, |y|α M w(y) ≤ Mα w(y). A corollary of this gives the following result announced in the introduction. Theorem 4.2 There is a constant C such that if w is a radial decreasing weight, then

n o C Z n w x ∈ R : |Iα f (x)| > λ ≤ |f (x)|Mα (w)(x) dx. λ Rn

16

(24)

Remark 4.3 It should be mentioned that reference [R] contains a particular form of Theorem 4.1, closely related to the result above, Theorem 4.2. We are thankful to the referee for pointing out to us this reference. We want to stress, nonetheless, that Theorem 4.2 as well as some of the arguments of this Section 4, are implicitly described and based on ideas from the paper [SW], where the case of “radially decreasing weights” for singular integrals is considered. A similar argument to that in the proof of Theorem 4.1, shows that in the case h(j) = 2−αj we are considering, one can also see that (1) holds for W (y) = Mα w(y) + M (w|x|α )(y), as well as for W (y) = sup r>0

max(r, |y|)α rn

Z w(x) dx. |y−x|≤r

(ii) If 0 < β < α and we take h(t) = 2(β−α)t , then Mα,h w(y) ≤

1 |y|α−β Cβ w(y), α−β

where Cβ w(y) = sup r>0

1 rn−β

Z |x−y|≤r,

|y| ≤|x| 0, h(t) = (1 + t)(1 + log(1 + t))1+δ , then we get that (1) holds with      1+δ + |y| Z 1 + log+ |y| 1 + log 1 + log r r w(x) dx. W (y) = sup rn−α r>0 |x−y|≤r

(iv) Finally, we can also prove that, for every 0 < β < α, (1) holds for W (y) =

1 Cβ w(y) + Mα ((1 + log+ | · |)w(·))(y). α−β

To see this, we observe that   k+3 X X X    T w(y) ≤ + k≤0

k>0

j=−∞

 1 2j(n−α)

Z |x−y|≈2j

w(x)χJ (x)dx χJ ∗ (y) = I + II.

Let us take the following sequence {h(j)}j∈Z    2−j(α−β) , j < 0 h(j) =   1, j ≥ 0.

18

k

k

Then, we have I≤

X 2k(α−β) α−β

k≤0

sup j

1 2j(n−β)

!

Z |x−y|≈2j

w(x)χJ (x)dx χJ ∗ (y) ≤ k

k

1 Cβ w(y). α−β

On the other hand, II can be estimated as follows II ≤

∞ X 0 X

2j(α−β) Cβ w(y)χJ ∗ (y) +

k=1 j=−∞

≈

k

∞ X

(k + 3)Cα w(y)χJ ∗ (y)

k=1

k

1 Cβ w(y) + Cα ((1 + log+ | · |)w(·))(y), α−β

where we have used that if x ∈ Jk , with k ≥ 1, then k + 3 ≈ 1 + log |x|.

References [BS]

C. Bennett and R.C. Sharpley, Interpolation of Operators, Pure and Appl. Math. 129, Academic Press, Boston, MA, 1988.

[CF]

A. C´ordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57 (1976), 97–101.

[CN]

D. Cruz-Uribe and C.J. Neugebauer, The structure of the reverse H¨ older classes, Trans. Amer. Math. Soc. 347 (1995), 2941–2960.

[F]

K.J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics 85, Cambridge University Press, Cambridge, 1986.

[FS]

C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115.

[GR]

J. Garc´ıa-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics, North Holland Math. Studies 116, North Holland, Amsterdam, 1985.

[G]

M. de Guzm´an, Real variable methods in Fourier analysis, North-Holland Math. Studies 46, North Holland, Amsterdam, 1981.

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[M]

J. Mal´ y, Personal communication to the second author.

[MeSo] M.T. Men´arguez and F. Soria, Weak type (1, 1) inequalities of maximal convolution operators, Rend. Circ. Mat. Palermo (2) 41 (1992), 342–352. [M¨ uSo] D. M¨ uller and F. Soria, A double-weight L2 -inequality for the Kakeya maximal function, J. Fourier Anal. Appl. (Special Issue) (1995), 467–478. [P1]

C. P´erez, Sharp Lp –weighted Sobolev inequalities, Ann. Inst. Fourier 45 (1995), 809–824.

[P2]

—, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp -spaces with different weights, Proc. London Math. Soc. 71 (1995), 135–57.

[P3]

—, Weighted norm inequalities for singular integral operators, J. London Math. Soc. 49 (1994), 296–308.

[R]

Y. Rakotondratsimba, Weighted inequalities of weak type for the fractional integral operator, Z. Anal. Anwendungen 17 (1998), 115–134.

[S]

E.T. Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), 339–345.

[SW]

F. Soria and G. Weiss, A remark on singular integrals and power weights, Indiana Univ. Math. J. 43 (1994), 187–204.

Mar´ıa J. Carro Departament de Matem`atica Aplicada i An`alisi Universitat de Barcelona 08071 Barcelona (Spain) E-mail: [email protected] Carlos P´erez Departamento de An´alisis Matem´atico Universidad de Sevilla 41080 Sevilla (Spain) E-mail: [email protected] 20

Fernando Soria Departamento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid (Spain) E-mail: [email protected] Javier Soria Departament de Matem`atica Aplicada i An`alisi Universitat de Barcelona 08071 Barcelona (Spain) E-mail: [email protected]

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