In this work we consider mixed weighted weak-type inequalities of the form. f(x) Mu(x)v(x) dx, v(x) : > t C

Transcription

1 MIXED WEAK TYPE ESTIMATES: EXAMPLES AND COUNTEREXAMPLES RELATED TO A PROBLEM OF E. SAWYER S.OMBROSI AND C. PÉREZ Abstract. In this paper we study mixed weighted weak-type inequalities for families of functions, which can be applied to study classic operators in harmonic analysis. Our main theorem extends the key result from [CMP].. Introduction and main results In this work we consider mixed weighted weak-type inequalities of the form (.) uv x T (fv)(x) : > t C f(x) Mu(x) dx, t where the operator T is either the Hardy-Littlewood maximal operator or any Calderón-Zygmund Operator. Versions of these type of inequalities were studied by Sawyer in [Sa] motivated by the work of Muckenhoupt and Wheeden [MW] (see also the works [AM] and [MOS]). E. Sawyer proved that inequality (.) holds in R when T = M is the Hardy-Littlewood maximal operator assuming that the weights u and v belong to the class A. This result can be seen as a very delicate extension of the classical weak type (, ) estimate. However, the reason why E. Sawyer considered (.) is due to the following interesting observation. Indeed, inequality (.) yields a new proof of the classical Muckenhoupt s theorem for M assuming that the A p weights can be factored (P. Jones s theorem). This means that if w A p then w = uv p for some u, v A. Now, define the operator f M(fv) which is bounded on L (uv) and it is of v weak type (, ) with respect to the measure uvdx by (.). Hence by the Marcinkiewicz interpolation theorem we recover Muckenhoupt s theorem. In the same paper, Sawyer conjectured that if T is instead the Hilbert transform the inequality also holds with the same hypotheses on the weights 000 Mathematics Subject Classification. 4B0, 4B5. Key words and phrases. Weights, maximal functions, singular integral operators, mixed weak-type inequalities. The second author is supported by the Severo Ochoa Excellence Programme at BCAM and the Spanish Government grant MTM P.

2 S.OMBROSI AND C. PÉREZ u and v. This conjecture was proved in [CMP]. In fact, it is proved in this paper that the inequality (.) holds for both the Hardy-Littlewood maximal operator and for any Calderón-Zygmund Operator in any dimension if either the weights u and v belong to A or u belongs to A and uv A. The method of proof is quite different from that in [Sa] (also from [MW]) and it is based on certain ideas from extrapolation that goes back to the work of Rubio de Francia (see [CMP] and also the expository paper [CMP3]). Applications of these results can be found in [LOPTT]. The authors conjectured in [CMP] that their results may hold under weaker hypotheses on the weights. To be more precise, they proposed that inequality (.) is true if u A and v A. Very recently, some quantitative estimates in term of the relevant constants of the weights have been obtained in [OPR] and some new conjectures have been formulated. Inequalities like (.), when T is the Hardy-Littlewood maximal operator, can also be seen as generalizations of the classical Fefferman-Stein inequality M(f) L, (u) c f L (Mu), where c depends is dimensional. However, in Section 3, we will see that (.) does not hold in general even for weights satisfying strong conditions like v RH A. In this work we generalize the extrapolation result in [CMP3] for a larger class of weights (see Theorem. below). This method of extrapolation is flexible enough whose scope goes beyond the classical linear operators. Indeed, it can be applied to square functions, vector valued operators as well as multilinear singular integral operators. See Section for some of these applications. In fact, the best way to state the extrapolation theorem is without considering operators and the result can be seen as a property of families of functions. Hereafter, F will denote a family of ordered pairs of non-negative, measurable functions (f, g). Also we are going to assume that this family F of functions, satisfies the following property: for some p 0, 0 < p 0 <, and every w A, (.) f(x) p 0 w(x) dx C g(x) p 0 w(x) dx, for all (f, g) F such that the left-hand side is finite, and where C depends only on the A constant of w. By the main theorem in [CMP], this assumption turns out to be true for any exponent p (0, ) and every w A, (.3) f(x) p w(x) dx C g(x) p w(x) dx,

3 MIXED WEIGHTED WEAK-TYPE INEQUALITIES 3 for all (f, g) F such that the left-hand side is finite, and where C depends only on the A constant of w. See the papers [CMP], [CGMP] and [CMP3] for more information and applications and the book [CMP4] for a general account. It is also interesting that both (.) and (.3) are equivalent to the following vector-valued version: for all 0 < p, q < and for all w A we have ( ) (f j ) q q ( C ) (g j ) q q (.4) L (w), p j L p (w) for any {(f j, g j )} j F, where these estimates hold whenever the left-hand sides are finite. Next theorem improves the corresponding Theorem from [CMP]. Theorem.. Let F be a family of functions satisfying (.) and let θ. Suppose that u A and that v is a weight such that for some δ > 0, v δ A. Then, there is a constant C such that (.5) f L C g L (f, g) F. v θ /θ, (uv) v (uv), θ /θ, Similarly, the following vector-valued extension holds: if 0 < p, q <, ) j (f j) q q ) L j (g j) q q L (.6) C, /θ, (uv) /θ, (uv) v θ for any {(f j, g j )} j F. Observe that the singular class of weights = x nr, r, is included in the hypothesis of the Theorem but not in the corresponding Theorem from [CMP]. The proof of (.6) is immediate since we can extrapolate using as initial hypothesis (.4) and then applying (.5). Corollary.. Let F, u and θ as in the Theorem. Suppose now that v i, i =,, m, are weights such that for some δ i > 0, v δ i i A, i =,, m. Then, if we denote v = m i= v i f L C g L (f, g) F. v θ /θ, (uv) v (uv), θ /θ, and similarly for 0 < p, q <, ) j (f j) q q L C /θ, (uv) v θ for any {(f j, g j )} j F. j v θ ) j (g j) q q v θ L, /θ, (uv)

4 4 S.OMBROSI AND C. PÉREZ The proof reduces to the Theorem by choosing δ > 0 small enough such that v δ = m i= vδ i A which follows by convexity since v δ i i A, i =,, m. To apply Theorem. above to some of the classical operators we need a mixed weak type estimate for the Hardy-Littlewood maximal operator. This is the content of next Theorem which was obtained in dimension one by Andersen and Muckenhoupt in [AM] and by Martín-Reyes, Ortega Salvador and Sarrión Gavián [MOS] in higher dimensions. In each case the proof follows as a consequence of a more general result with the additional hypothesis that u A. For the sake of completeness we will give an independent and direct proof with the advantage that no condition on the weight u is assumed. Theorem.3. Let u 0 and = x nr for some r >. Then there is a constant C such that for all t > 0, (.7) uv x : M(fv)(x) > t C f(x) Mu(x) dx. t Remark.4. We remark that the theorem could be false when r = even in the case u =, see [AM]. However, we already mentioned that the singular weight = x n is included in the extrapolation Theorem.. Acknowledgement. The authors are grateful to F. J. Martín-Reyes and P. Ortega-Salvador to point out reference [MOS].. Some applications In this section we show the flexibility of the method by giving two applications... The vector-valued case. Let T be any singular integral operator with standard kernel and let M is the Hardy-Littlewood maximal function. We are going to show that starting from the following inequality due to Coifman [Coi]: for 0 < p < and w A, (.) T f(x) p w(x) dx C Mf(x) p w(x) dx, which combined with the extrapolation Theorem. together with Theorem.3 yields the following corollary.

5 MIXED WEIGHTED WEAK-TYPE INEQUALITIES 5 Corollary.. Let u A and = x nr for some r >. Also let < q <. Then, there is a constant C such that for all t > 0, ( ) uv x j M(f jv)(x) q q : > t C ( ) f j (x) q q u(x) dx, t uv x : ( j T (f jv)(x) q ) q sup tuv x : t>0 > t C t j ( j f j (x) q ) q u(x) dx. Observe that in particular we have the following scalar version, uv x T (fv)(x) : > t C f(x) u(x) dx. t This scalar version was proved in [MOS]. The proof of the second inequality of the Corollary follows from the first one by applying inequality (.6) in Theorem. with initial hypothesis (.): ( ) j T (f j)(x) q q ( ) j M(f j)(x) q q > t C sup tuv x : > t. t>0 To prove the first inequality in Corollary. we first note that in [CGMP] was shown for < q < and for all 0 < p < and w A, ( ) ( (M(f j )) q q ( C M f j q) ) q. L p (w) Lp(w) j To conclude we apply Theorem. combined with Theorem.3... Multilinear Calderón-Zygmund operators: We now apply our main results to multilinear Calderón-Zygmund operator. We follow here the theory developed by Grafakos and Torres in [GT], that is, T is an m-linear operator such that T : L q L qm L q, where < q,..., q m <, 0 < q < and (.) q = + +. q q m The operator T is associated with a Calderón-Zygmund kernel K in the usual way: T (f,..., f m )(x) = K(x, y,..., y m ) f (y )... f m (y m ) dy... dy m, whenever f,..., f m are in C0 and x / m j= supp f j. We assume that K satisfies the appropriate decay and smoothness conditions (see [GT] for j

6 6 S.OMBROSI AND C. PÉREZ complete details). Such an operator T is bounded on any other product of Lebesgue spaces with exponents < q,..., q m <, 0 < q < satisfying (.). Further, it also satisfies weak endpoint estimates when some of the q i s are equal to one. There are also weighted norm inequalities for multi-linear Calderón-Zygmund operators; these were first proved in [GT] using a goodλ inequality and fully characterized in [LOPTT] using the sharp maximal function M and a new maximal type function which plays a central role in the theory, M(f,..., f m )(x) = sup Q x Q cube m i= f i (z) dz, Q Q where the supremum is taken over cubes with sides parallel to the axes. Indeed, one of the main results from [LOPTT] is that for any 0 < p < and for any w A, T (f,..., f m ) C M(f,..., f m ) L p (w) L (w). p ( Beginning with these inequalities, ) we can apply Theorem. to the family F T (f,..., f m ), M(f,..., f m ). Hence, if u A and = x nr for some r >. (.3) T (f,..., f m ) L C M(f,..., f m ) L /m, (uv) /m, (uv) v m Corollary.. Let T be a multilinear Calderón-Zygmund operator as above. Let u A and = x nr for some r >. Then T (f,..., f m ) L m C f v m j u dx,. /m, (uv) To prove this corollary we will use the following version of the generalized Holder s inequality: for q,..., q m < with j= + + = q q m q, there is a constant C such that m h j L q, (w) C j= v m m h j L q j, (w). The proof of this inequality follows in a similar way that the proof of the classic generalized Holder s inequality in L p Theory. Now, if we combine this together with (.3) and of the trivial observation that m M(f,..., f m )(x) M(f i ), j= i=

7 we have MIXED WEIGHTED WEAK-TYPE INEQUALITIES 7 T (f,..., f m ) L C /m, (uv) v m m Mf j v j= L, (uv), Finally, an application of Theorem.3 concludes the proof of the corollary. 3. counterexamples An interesting point from Theorem.3 is that if = x nr, r >, the estimate (3.) uv x : M(fv)(x) > t C f(x) Mu(x) dx, t holds for any u 0. On the other hand we have already mentioned that the same inequality holds if u A and v A or uv A [CMP]. In particular, this is the case of u A and v RH. Assuming that v RH, a natural question is whether inequality (3.) holds with no assumption on u. This would improve the classical Fefferman-Stein inequality. However, we will show in next example that this is false in general. Example 3.. On the real line we let = k Z x k χ (x), where I I k k denote the interval x k /,. It is not difficult to see that v RH. If we choose u(x) = k log(k) χ (x), J k k N k>0 where J k = [ k +, k + 4k k], and f = χ[,]. Then, there is no finite constant C such that the inequality (3.) uvx : Mf (x) > C f M u holds. To prove this we will make use of the following observation: where and There is a geometric constant such that M w(x) M L log L w(x) x M L log L f(x) = sup f L log L,Q Q x f L log L,Q = inf{λ > 0 : Q Q Φ( f ) dx }. λ with Φ(t) = t log(e + t), see [PW] or [G]. Now, it is a computation to see that if x [, ], M u(x) M L log L u(x) C so the right hand side of (3.) is finite, and however the left hand side of (3.) is infinite. We will see that. For x > we have that Mf (x) and if x J x k I k for

8 8 S.OMBROSI AND C. PÉREZ k > 0 >, then it is easy to see that (k +, k + ) {x J x k 4k k k : Mf(x) > } and therefore we obtain that uvx : Mf (x) > > k N k>0 > k N k>0 k log(k) k+ k k+ 4k 8k log(k) =. 4. Proof of Theorem. The following Lemmas will be useful: (x k) dx > Lemma 4.. If u A, w A, then there exists 0 < ɛ 0 < depending only on [u] A such that uw ɛ A for all 0 < ɛ < ɛ 0. Proof. Since u A, u RH s0 for some s 0 > depending on [u] A. Let ɛ 0 = /s 0 and 0 < ɛ < ɛ 0. This implies that u RH s with s = (/ɛ). Then since u, v A, for any cube Q and almost every x Q, ( ) /s ( ) /s u(y)w(y) ɛ dy u(y) s dy w(y) dy Q Q Q Q Q Q [u] ( ) /s RH s u(y) dy w(y) dy [u] RHs [u] A [w] ɛ A Q Q u(x)w(x) ɛ. Q Hence uw ɛ A with [uw ɛ ] A [u] RHs [u] A [w] ɛ A. Q We also need the following version of the Marcinkiewicz interpolation theorem in the scale of Lorentz spaces. In fact we need a version of this theorem with precise constants. The proof can be found in [CMP]. Proposition 4.. Given p 0, < p 0 <, let T be a sublinear operator such that T f L p 0, C 0 f L p 0, and T f L C f L. Then for all p 0 < p <, T f L p, /p ( C 0 (/p 0 /p) + C ) f L p,. Fix u A and v such that v δ A for some δ > 0. Then by the factorization theorem v δ = v v for some v A and v RH. Define the operator S λ by S λ f(x) = M(fuv/λδ ) uv /λδ for some large enough constant λ > that will be chosen soon.

9 MIXED WEIGHTED WEAK-TYPE INEQUALITIES 9 By Lemma 4., there exists 0 < ɛ 0 < (that depends only on [u] A ) such that u w ɛ A for all w A and 0 < ɛ < ɛ 0. Hence we choose λ > δɛ 0 such that uv /λδ A. Hence, S λ is bounded on L (uv) with constant C = [u] A. We will now show that for some larger λ, S λ is bounded on L m (uv). Observe that Sf(x) λ u(x) dx = M(fuv /λδ )(x) λ u(x) λ v (x) /δ dx. Since v = ṽ t for some ṽ A and t >. Hence, u λ v /δ = u λ ṽ t δ = ( u ṽ t ) δ(λ ) λ. By Lemma 4. there exists λ even bigger if necessary (λ > + t δɛ 0 ) such t δ(λ ) that u ṽ A and hence u λ v /δ A λ. By Muckenhoupt s theorem, M is bounded on L λ (u λ v /δ ) and therefore S is bounded on L λ (uv) with some constant C 0. Observe that λ depends upon the A constant of u. We fix one of these λ from now on. Thus by Proposition 4. above we have that S is bounded on L q, (uv), q > λ. Hence, Sf L q, (uv) /q ( C 0 (/λ /q) + C ) f L q, (uv). Thus, for all q λ we have that Sf L q, (uv) K 0 f L q, (uv) with K 0 = 4λ (C 0 + C ). We emphasize that the constant K 0 is valid for every q λ. Fix (f, g) F such that the left-hand side of (.5) is finite. We let r such that θ < r < θ(λ) that is going to be chosen soon. Now, by the duality of L r, and L r,, f v θ r = L /θ, (uv) (f v θ ) r = sup L f(x) r/θ, r (uv) h(x) u(x) θ/r dx, where the supremum is taken over all non-negative h L ( r θ ), (uv) with h L ( r θ ), (uv) =. Fix such a function h. We are going to build a larger function Rh using the Rubio de Francia s method such Rh uv θ/r A. Hence we will use the hypothesis (.3) with p = θ/r (recall that is equivalent to (.)) with the weight Rh uv θ/r A We let r such that ( r θ ) > λ and hence S ( r is bounded on θ ) L( r θ ), (uv) with constant bounded by K 0. Now apply the Rubio de Francia algorithm (see [GCRdF]) to define the operator R on h L ( r θ ), (uv), h 0, by Rh(x) = S j ( h(x) r θ ) j K j, 0 j=0

10 0 S.OMBROSI AND C. PÉREZ Recall that the operator S ( r θ ) S ( r θ is defined by r θ M(fuv/( ) δ ) f(x) = ) uv /( r θ ) δ Recall that by the choice of r uv /( r θ ) δ A. It follows immediately from this definition that: (a) h(x) Rh(x); (b) Rh L ( r θ ), (uv) h L ( r θ ), (uv) ; (c) S ( r θ ) (Rh)(x) K 0 Rh(x). In particular, it follows from (c) and the definition of S that Rh uv /( r θ ) δ A and therefore Rh uv /( r θ ) = Rh uv /δ( r θ ) v /δ( r θ ) A. To apply the hypothesis (.3) we must first check that the left-hand side is finite, but this follows at once from Hölder s inequality and (b): f(x) r Rh(x) u(x) θ r dx (f v θ ) r Rh L r/θ, (uv) L (r/θ), (uv). f v θ r L /θ, (uv) h L ( r θ ), (uv) <. Thus since Rh uv /( r θ ) A by (.3) f(x) r h(x) u(x) θ r dx f(x) r Rh(x) u(x) θ r dx C g(x) r Rh(x) u(x) θ r dx C (g v θ ) r Rh L r/θ, (uv) L ( θ r ), (uv) C g v θ r L /θ, (uv). Since C is independent of h, inequality (.5) follows finishing the proof of the theorem. 5. Proof of Theorem Proof of (.7). The following lemma is important in the proof. Lemma 5.. Let f be a positive and locally integrable function. Then for r > there exists a positive real number a depending on f and λ such that the inequality ( ) f(y)dy a n = λ y a r holds.

11 MIXED WEIGHTED WEAK-TYPE INEQUALITIES Proof. Consider the function ( ) g(a) = f(y)dy a n, for a 0, y a r then by the hypothesis we have that g is a continuous and non decreasing function. Furthermore, g(0) = 0, and g(+ ) = +, and therefore, by the mean value theorem there exists a such that satisfies the conditions of lemma. Let u 0 and = x nr with r >. By homogeneity we can assume that λ =. Also, for simplicity we denote g = fv. Now, for each integer k we denote G k = { k < x k+}, I k = { k < x k+}, L k = { k+ < x }, C k = { x k }. It will be enough to prove the following estimates (5.) (5.) (5.3) k Z k Z k Z { uv x G k : M(gχ Ik )(x) > } x nr C r,n { uv x G k : M(gχ Lk )(x) > } x nr C r,n { uv x G k : M(gχ Ck )(x) > } x nr C r,n g Mu, g Mu, g Mu, Taking into account that in G k, = x nr knr, using the (, ) weak type inequality of M with respect to the pair of weights (u, Mu) and since the subsets I k overlap at most three times we obtain (5.). To prove inequality (5.) we will estimate M(gχ Lk )(x). Observe that if x belongs to G k and y L k = { k+ < y }, and if y x ρ, we have that y ρ, ρ n k Z y x ρ g(y)χ Lk (y) dy C n g(y) y k+ < y g(y) g(y) y n dy C n x < y y n dy. If we denote F (x) = n dy the left hand side of (5.) is bounded by x < y krn u { x : F (x) > C knr} tu {x : F (x) > t} dt t g(y) = F (x) u(x)dx = x < y y n dy u(x)dx = g(y) R y n u(x)dx dy C g(y) Mu(y)dy. n x < y 0

12 S.OMBROSI AND C. PÉREZ To prove (5.3) we estimate M(gχ Ck )(x) for x G k. Indeed, if y C k, y < x and since M(gχ Ck )(x) cn x C n k g(y)dy, we obtain M(gχ Ck )(x) C x n g C C k x n g, y x Thus, since the subsets G k are disjoints, the left hand side in (5.3) is bounded by { } uv x C : x n g > y x x nr. Now, if a denotes the( positive real number that appears in Lemma 5. ) (i.e., a satisfies that = y a r g a n, we express the last integral in the following way: (5.4) C uv x : x n + uv { k=0 y x g > x nr = uv x a C r : x n g > y x x nr x : k a r < x k+ a r and C x n If x a r, since y x x a r : C x n y x y x g > x nr we have that y a r, thus the set } { g > x nr x a r ( : x n(r ) > C g y a r Taking into account the last inclusion and since ( y a r g ) = a n, the first summand in the second term in (5.4) is bounded by uv x r > Ca = uv x > ca r. Using again Lemma 5., the last term can been estimated by uv dx C u(x) dx ( k a r ) nr x >C a r = C k= c k a r x <c k a r C u(x) dx k(r )n a n (c k a r ) n k= x c k a r g(y) dy u(x) dx k(r )n (c k a r ) n k= y a r And this term is bounded by C k(r )n k= y a r g(y)mu(y) dy C x c k a r g Mu + ) }.

13 k=0 MIXED WEIGHTED WEAK-TYPE INEQUALITIES 3 To finish, we must estimate the series in (5.4). It is clear that sum is bounded by uv x k a r < x k+ a r C u dx ( k a r ) nr k a r x < k a r k=0 and arguing as before we conclude the proof of (5.3). Remark 5.. We observe that the proof only uses the following conditions for a sublinear operator T : a) T is of weak type (, ) with respect to the pair of weight (u, Mu) and b) T is a convolution type operator such that the associate kernel satisfies the usual standard condition: K(x) c x. n In particular if u A, this observation can be applied to the usual Calderón- Zygmund Singular Integral Operators and moreover to the Strongly Singular Integral Operators (see [Ch] and [F]). [AM] [Ch] [Coi] [CMP] [CMP] [CMP3] [CMP4] References K. Andersen, B. Muckenhoupt Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 7 (98), no., 9-6. S. Chanillo, Weighted norm inequalities for strongly singular convolution operators. Trans. Am. Math. Soc. V 8, (984). R.R. Coifman, Distribution function inequalities for singular integrals, Proc. Acad. Sci. U.S.A. 69 (97), D. Cruz-Uribe, J.M. Martell and C. Pérez, Extrapolation results for A weights and applications, J. Funct. Anal. 3 (004) D. Cruz-Uribe, J.M. Martell and C. Pérez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., , D. Cruz-Uribe, J.M. Martell y C. Pérez, Extensions of Rubio de Francia s extrapolation theorem, Collect. Math., 57 (006) D. Cruz-Uribe, J.M. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Birkhauser Basel, (0). [CGMP] G.P. Curbera, J. García-Cuerva, J.M. Martell and C. Pérez, Extrapolation with Weights, Rearrangement Invariant Function Spaces, Modular inequalities and applications to Singular Integrals, Adv. Math., 03 (006) [F] C. Fefferman. Inequalities for strongly singular convolution operator, Acta Math. 4 (970) [GCRdF] J. García-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland Math. Studies 6, North Holland, Amsterdam, 985. [G] [GT] [GT] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, New Jersey, 004. L. Grafakos and R. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 65 (00), L. Grafakos and R. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J. 5 (00), no. 5, 6 76.

14 4 S.OMBROSI AND C. PÉREZ [LOPTT] A.K. Lerner, S. Ombrosi, C. Pérez, R.H. Torres and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 0, -64 (009). [MOS] F. J. Martín-Reyes, P. Ortega Salvador, M. D. Sarrión Gavilán, Boundedness of operators of Hardy type in Λ p,q spaces and weighted mixed inequalities for singular integral operators, Proc. Roy. Soc. Edinburgh Sect. A 7 (997), no., [MW] B. Muckenhoupt and R. Wheeden, Some weighted weak-type inequalities for the Hardy-Littlewood maximal function and the Hilbert transform, Indiana Univ. Math. J. 6 (977), [OPR] S. Ombrosi, C. Pérez and J. Recchi, Quantitative weighted mixed weak-type inequalities for classical operators, Indiana Univ. Math. J. 65 (06), [PW] C. Pérez and R. Wheeden, Uncertaitny principle estimates for vector fields, Journal of Functional Analysis 8 (00), [Sa] E.T. Sawyer, A weighted weak type inequality for the maximal function, Proc. Amer. Math. Soc. 93 (985), Department of Mathematics, Universidad Nacional del Sur, Bahía Blanca, Argentina. address: sombrosi@uns.edu.ar Carlos Pérez, Department of Mathematics, University of the Basque Country, Ikerbasque, and BCAM, Bilbao, Spain. address: carlos.perezmo@ehu.es