Collect. Math. DOI 10.1007/s13348-015-0132-4

Sharp weighted bounds for one-sided maximal operators Francisco J. Martín-Reyes · Alberto de la Torre

Received: 13 May 2014 / Accepted: 1 January 2015 © Universitat de Barcelona 2015

Abstract In this note we prove some results about the best constants for the boundedness of the one-sided Hardy–Littlewood maximal operator in L p (μ), where μ is a locally finite Borel measure, that in the two-sided weights have been obtained by Buckley (Trans Am Math Soc 340(1):253–272, 1993) and more recently by Hytönen and Pérez (Anal PDE 6(4):777– 818, 2013) and Hytönen et al. (J Funct Anal 263(12):3883–3899, 2012). To prove Bucley’s theorem for one-sided maximal operators, we follow the ideas of Lerner (Proc Am Math Soc 136(8):2829–2833, 2008). To obtain a better estimate in terms of mixed constants we follow the steps in Hytönen and Pérez (Anal PDE 6(4):777–818, 2013) and Hytönen et al. (J Funct Anal 263(12):3883–3899, 2012) i.e., (a) getting a sharp estimate for the constant for the weak type type, in terms of the one-sided A p constant, (b) obtaining a sharp reverse Hölder inequality and (c) using Marcinkiewicz interpolation theorem. Our proofs of these facts are different from those in Hytönen and Pérez (Anal PDE 6(4):777–818, 2013) and Hytönen et al. (J Funct Anal 263(12):3883–3899, 2012) and apply to more general measures. Keywords

Weighted inequalities · One-sided maximal operator · Sharp bounds

Mathematics Subject Classification

Primary 42B25; Secondary 40A30 · 26D15

F. J. Martín-Reyes and A. de la Torre supported by grant MTM2011-28149-C02-02 of the Ministerio de Economía y Competitividad (Spain) and grants FQM-354 and FQM-01509 of the Junta de Andalucía. F. J. Martín-Reyes (B) · A. de la Torre Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Malaga, Spain e-mail: [email protected] A. de la Torre e-mail: [email protected]

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1 Introduction and main results Let μ be a Borel measure on the real line which is finite on bounded sets. For any measurable function f , let Mμ f be the (two-sided) Hardy–Littlewood maximal function defined by  1 | f | dμ Mμ f (x) = sup μ(I ) I where the supremum is taken over all (bounded) intervals I containing x and expressions of the form 0/0 are taken to be zero. The one-sided maximal operators Mμ+ and Mμ− are defined similarly except that the supremum is taken over intervals of the form [x, x + h) and (x − h, x] h > 0, respectively, that is,  1 + Mμ f (x) = sup | f | dμ, h>0 μ([x, x + h)) [x,x+h) and 1 h>0 μ((x − h, x])

Mμ− f (x) = sup

 (x−h,x]

| f | dμ.

If μ is the Lebesgue measure then we simply write M, M + and M − . If T is any of these operators then T is of weak type (1, 1) (see [2]) and of strong type ( p, p), 1 < p < ∞, that is, there exist C1 and C p such that for any measurable function f and all λ > 0  C1 μ({x : T f (x) > λ}) ≤ | f | dμ (1.1) λ and



 |T f | p dμ ≤ C p

| f | p dμ,

(1.2)

where C1 = 1 for T = Mμ+ and T = Mμ− , C1 = 2 for T = Mμ , C p = ( p/( p − 1)) p for T = Mμ+ and T = Mμ− and C p = 2 p−1 ( p/( p − 1)) p for T = Mμ . Let us consider now a nonnegative measurable function w. The weighted L p -space, 1 < p < ∞, is    1/ p p p p L (wdμ) = f : || f || L (wdμ) := | f | wdμ < +∞ . T is bounded on L p (wdμ) (or T satisfies the weighted strong type ( p, p) inequality for T ) if ||T f || L p (wdμ) ≤ C p || f || L p (wdμ) for all f ∈ L p (wdμ), (1.3) and, in this case, the norm of T is defined as ||T ||B(L p (wdμ)) :=

||T f || L p (wdμ) . 0= f ∈L p (wdμ) || f || L p (wdμ) sup

Andersen proved [1] (see [13] when μ is the Lebesgue measure) that (1.3) holds for T = Mμ if and only if w ∈ A p (μ), that is,   p−1    1 1  [w] A p (μ) := sup w dμ w 1− p dμ (1.4) μ((a, c)) (a,c) a
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Sharp weighted bounds for one-sided maximal operators

is finite, where p  is the exponent conjugate of p and the products of the form 0(+∞) are taken to be zero. Actually, Andersen obtained the result as a corollary of the corresponding results for T = Mμ+ and T = Mμ− (see [10,11,16] when μ is the Lebesgue measure). For T = Mμ+ , (1.3) holds if and only if w ∈ A+ p (μ), that is,   p−1    1 1  w dμ w 1− p dμ (1.5) [w]∗A+ (μ) := sup p μ((a, c)) [b,c) a
(1.6)

is finite. As before, we shall write simply [w]∗A+ and [w]∗A− when μ is the Lebesgue measure. p

p

If w ∈ A+ p (μ) then w(x) can be zero in a set of positive measure but the set {x : w(x) > 0} is an interval J and w is μ-integrable on each closed interval I ⊂ J . To avoid technical problems we shall assume in the rest of the paper that a weight w in R is an a.e. positive locally μ-integrable function. When μ is the Lebesgue measure, Buckley proved (see [3]) that if w ∈ A p then 1

||M||B(L p (wd x)) ≤ C( p)[w] Ap−1 p

(1.7)

1

and the power [w] Ap−1 is best possible. Recently, Lerner [9] gave a very simple proof of this p estimate. More recently in [7,8] a better estimate for ||M||B(L p (wd x)) has been obtained in terms of mixed constants. It has been proved that 1/ p  ||M||B(L p (wd x)) ≤ C( p) [w] A p [σ ] A∞ , (1.8) 

where σ = w 1− p and

[σ ] A∞ := sup

I

Mμ (σ χ I ) dμ , I σ dμ

the supremum being taken over all the intervals I . Notice that original Buckley’s estimate p  −1 follows from (1.8) since [σ ] A∞ ≤ C[σ ] A p = C[w] A p . However, as it is explained in [7], 1  1/ p [w] A p [σ ] A∞ can be much smaller than [w] Ap−1 . p Our main results are the one sided version of Buckley’s estimate for Mμ+ using the ideas of Lerner adapted to the one-sided case and the one-sided version of the sharp estimate with mixed constants. The last estimate follows from a sharp p −  property which is obtained as a consequence of a sharp reverse Hölder inequality (the key result). One might expect that the constant [w]∗A+ (μ) in (1.5) would be the natural substitute for p

the A p constant of the weight. However, it is more convenient to use a different one. In order to define it, we introduce the following notations: given real numbers a ≤ b ≤ c, {a, b] and [b, c} will stand for (a, b] or [a, b] and [b, c) or [b, c], respectively, while {a, c} will denote the union {a, b] ∪ [b, c}. Definition 1.1 Let 1 < p < ∞. Let w be a weight. The one-sided constant [w] A+p (μ) is defined as   p−1    1 1  [w] A+p (μ) := sup w dμ w 1− p dμ , (1.9) μ({a, b]) {a,b] μ([b, c}) [b,c} T

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where the supremum is taken over the set T of triplets (a, b, c) such that 1 1 μ({a, c}) and μ([b, c}) ≥ μ({a, c}). 2 2 is defined reversing the orientation of the real line:

μ({a, c}) > 0, μ({a, b]) ≥ The one-sided constant [w] A−p (μ)  [w] A−p (μ) :=

sup

(a,b,c)∈T

1 μ([b, c})

 [b,c}

 w dμ

1 μ({a, b]))

 {a,b]

w

1− p 

 p−1 dμ . (1.10)

If μ is the Lebesgue measure then we simply write [w] A+p and [w] A−p . With these definitions, 

p−1

we have [w] A+p (μ) = [w 1− p ] A− (μ) . p

When μ is a continuous measure (μ({x}) = 0 for all x ∈ R) then    p−1  b  c 1 1  w dμ w 1− p dμ , [w] A+p (μ) = sup μ((a, b)) a μ((b, c)) b T where T is the set of intervals (a, b) and (b, c) such that μ((a, b)) = μ((b, c)). Notice that in the pioneer work of Sawyer [16] for the Lebesgue measure, the strong type ( p, p) of the one-sided maximal operator was characterized by the finiteness of this supremum. Observe that if w ∈ A+ < +∞ since [w] A+p (μ) ≤ 2 p [w]∗A+ (μ) . The p (μ) then [w] A+ p (μ) p

converse is not obvious but it is proved in the following theorem where we give a different characterization of A+ p (μ). Theorem 1.2 Let 1 < p < +∞. Let w be a weight in R. The following assertions are equivalent. ∗ (a) w ∈ A+ p (μ) ([w] A+ (μ) < +∞). p

(a ) [w] A+p (μ) < +∞. (b) There exists a constant C such that if μ([a, b)) > 0 then 

   p −1 σ dμ ≤ Cμ([a, b)) Mw+ dμ w −1 χ[a,b) (a) . [a,b)

(b )

There exists a constant C such that for all real numbers a < b

   p −1 Mμ+ (σ χ[a,b) )(a) ≤ C Mw+ dμ w −1 χ[a,b) (a) .

Moreover, if A is the least possible constant in (b) or (b ) then  1

1 1  p−1 2− p [w] Ap−1 ≤ [w]∗A+ (μ) ≤ A ≤ [w] Ap−1 . + + (μ) (μ) p

p

p

Remark 1.3 For measures μ equivalent to Lebesgue measure this characterization was obtained in [12, Theorem 2]. Our next theorem shows that from statement (b) it follows easily that [w] A+p (μ) < +∞ is necessary and sufficient for Mμ+ to be of strong type ( p, p), 1 < p < +∞, and gives the sharp estimate for the norm of Mμ+ (the one-sided analogue of the result in [3]). Theorem 1.4 (Buckley’s theorem for one-sided maximal operators) Let 1 < p < +∞. Let w be a weight in R. The following assertions are equivalent.

123

Sharp weighted bounds for one-sided maximal operators ∗ (a) w ∈ A+ p (μ) ([w] A+ (μ) < +∞). p

(a ) [w] A+p (μ) < +∞. (b) There exists a constant C such that for all measurable f and all real numbers a

   p −1 . Mμ+ f (a) ≤ C Mw+ dμ (Mσ dμ ( f /σ )) p−1 w −1 (a) (c) Mμ+ is bounded on L p (wdμ) Moreover, if any of the above conditions hold then 1

1 1 1 p [w] Ap + (μ) ≤ [w]∗A+ (μ) ≤ Mμ+ B(L p (wdμ)) ≤ 2ep  [w] Ap−1 . + p p p (μ) 2 The pointwise inequality in statement (b) in Theorem 1.4 gives the strong type and, consequently, the weighted weak type inequality but with poor estimate on the constant for the weak type inequality. The next result gives the sharp estimate which will be needed in the proof of Theorem 1.9. This theorem can be found in [1] but we present here a different, and shorter proof and in this way the paper is self contained. Theorem 1.5 Let 1 < p < +∞. Let w be a weight in R. The following assertions are equivalent. (a) w ∈ A+ p (μ). (b) There exists a constant C such that for all measurable f and all real numbers a

1 p Mμ+ f (a) ≤ C Mw+ dμ (| f | p )(a) . (c) Mμ+ is bounded from L p (wdμ) into weak-L p (wdμ), that is, there exists a constant C such that for all measurable f + M f μ

 L p,∞ (wdμ)

:= sup λ λ>0

1

p

{x:Mμ+ f (x)>λ}

wdμ

≤ C|| f || L p (wdμ) .

Furthermore, if any of the above conditions hold and + M μ

B(L p (wdμ),L p,∞ (wdμ))

:=

+ M f μ

sup

0= f ∈L p (wdμ)

L p,∞ (wdμ)

|| f || L p (wdμ)

.

then 1

1 1 p ≤ Mμ+ B(L p (wdμ),L p,∞ (wdμ)) [w] Ap + (μ) ≤ [w]∗A+ (μ) p p 2 1

1 p ≤ 8 [w]∗A+ (μ) ≤ 8[w] Ap + (μ) . p

p

The result about the mixed constants is based on the reverse Hölder inequality. But it is known that there are measures μ for which the classes A+ p (μ) do not satisfy a reverse Hölder inequality (see [6,15]; see also [14] in the two-sided setting). This means that we have to impose some restriction in our measures Nevertheless, the measures μ we consider are quite general.

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Definition 1.6 Let μ be a Borel measure on the real line which is finite on bounded sets. It is said that μ is left-regular if there exists a positive constant Cμ such that μ([a, b)) ≤ Cμ μ((a, b)) for all bounded intervals (a, b) such that μ((a, b)) > 0. Notice that μ is left-regular if and only if there exists a constant Cμ such that for each atom {a} we have that μ((a, +∞)) = 0 or there exists another atom {b} which satisfies a < b, μ((a, b)) = 0 and μ({a}) ≤ Cμ μ({b}). Obviously, continuous measures are left-regular. The proof of the estimate with mixed constants involves the following one-sided constant. Definition 1.7 We define the A− ∞ (μ) constant of a weight w as Mμ+ (wχ I ) dμ [w] A−∞ (μ) := sup I , I w dμ where the supremum is taken over all bounded intervals I = [c, d) (or, equivalently, I = [c, d]) . It is said that w ∈ A− < +∞. ∞ (μ) if [w] A− ∞ (μ) and the reverse Hölder inequality Some relations among the union ∪ p>1 A− p (μ), [w] A− ∞ (μ) with exponent 0 < s < 1 are presented in Sect. 3 (for instance, we have ∪ p>1 A− p (μ) ⊂ − − (μ) and [w] ≤ e[w] ). In particular, we obtain in Theorem 3.4 (a key result) A− ∞ A∞ (μ) A p (μ) (μ) satisfy a reverse Hölder inequality with uniform constant. As a that the weights in A− ∞ consequence, the sharp p −  property for A+ (μ) classes is obtained. p Theorem 1.8 Assume that the measure μ is left-regular. Let 1 < p < +∞. If w ∈ A+ p (μ),  1 1− p σ =w , 0 < δ ≤ 2[σ ] − Cμ and q = ( p + δ)/(1 + δ) then 1 < q < p, w ∈ Aq+ (μ) A∞ (μ)

and

[w] Aq+ (μ) ≤ 2 p [w] A+p (μ) . The p −  property for measures equivalent to Lebegue measures was proved in [4,10, 11,16]. Once we have this theorem, Marcinkiewicz interpolation theorem gives immediately the following result which is the sharp Buckley’s theorem with mixed constants for one-sided maximal operators. Theorem 1.9 Assume that the measure μ is left-regular. Let 1 < p < +∞. If w ∈ A+ p (μ), then

1/ p + 7  M p + − C ≤ 2 p [w] [σ ] . μ μ B(L (wdμ)) A p (μ) A∞ (μ) If μ is left-regular then Buckley’s estimate for one-sided maximal operators contained in Theorem 1.4, up to constants, is a consequence of the last theorem since [σ ] A−∞ (μ) ≤ p  −1

e[σ ] A− (μ) = e[w] A+ (μ) (see Theorem 3.3). p

p

As we said, to prove Theorems 1.8 and 1.9 we follow the approach in [7,8] but we simplify it.

All our results are presented for Mμ+ (or Mμ− ). The corresponding results for Mμ− (or Mμ+ ) and Mμ hold also but we omit them.

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2 Proof of Theorems 1.2, 1.4 and 1.5 We shall need two lemmas in the proof of Theorem 1.2. Lemma 2.1 Let a, b ∈ R, a < b. Then there exists α ∈ [a, b) such that μ([a, α]) ≥ 1 1 2 μ([a, b)) ≥ μ([a, α)) and, consequently, μ([α, b)) ≥ 2 μ([a, b)). Proof We may assume μ([a, b)) > 0. Let   1 E = y ∈ [a, b) : μ([a, b)) ≤ μ([a, y]) . 2 It is clear that E  = ∅ since μ([a, b)) > 0. Let α = inf E. Obviously, 21 μ([a, b)) ≤ μ([a, α]) and μ([a, α)) ≤ 21 μ([a, b)) (which gives α < b). Finally, μ([a, b)) = μ([a, α))+

μ([α, b)) ≤ 21 μ([a, b)) + μ([α, b)) and the lemma is completely proved. Lemma 2.2 Let a, b ∈ R such that μ([a, b)) > 0. Then there exists a finite or infinite strictly N +1 N [x decreasing sequence {xk }k=0 (N ∈ N, 0 ≤ N ≤ +∞) such that [a, b) = ∪k=1 k+1 , x k ) 1 up to a set of measure zero, μ([a, xk+1 ]) ≥ 2 μ([a, xk )) and μ([xk+1 , xk )) ≥ 21 μ([a, xk )) for all k. Proof Let x0 = b. Assume that x0 , . . . , xk , k ≥ 0, have been chosen. If μ([a, xk )) > 0 then we apply Lemma 2.1 to the interval [a, xk ). In this way we choose xk+1 = α. Therefore, we have μ([a, xk+1 ]) ≥ 21 μ([a, xk )) and μ([xk+1 , xk )) ≥ 21 μ([a, xk )). If μ([a, xk )) = 0 then we do not choose more elements and the process finishes with N + 1 = k (notice that k = N + 1 is necessarily greater than or equal to 1). We only have to show that N [x [a, b) = ∪k=1 k+1 , x k ) up to a set of measure zero. It is obvious when N is finite. Otherwise, since the intervals [xk+1 , xk ) are disjoint we have ∞ k=1 μ([x k+1 , x k )) ≤ μ([a, b)) < +∞. Therefore, limk→∞ μ([xk+1 , xk )) = 0. It follows from μ([xk+1 , xk )) ≥ 21 μ([a, xk )) that limk→∞ μ([a, xk )) = 0. Since the sequence is strictly decreasing we have that there exists a  := limk→∞ xk and the previous equality means that μ([a, a  ]) = 0. Since (a  , b) = ∪∞

k=1 [x k+1 , x k ), we are done. Proof of Theorem 1.2 (a)⇒(a ): It follows from [w] A+p (μ) ≤ 2 p [w]∗A+ (μ) < +∞. p

N +1 (a )⇒(b): Let {xk }k=0 be the sequence given by Lemma 2.2 for the interval [a, b).  By (a ),

 p −1   p −1

μ([a, xk+1 ]) σ ≤ [w] A+p (μ) μ([xk+1 , xk )) [xk+1 ,xk ) [a,xk+1 ] w dμ  p −1



   p −1 ≤ [w] A+p (μ) μ([xk+1 , xk )) Mw+ dμ w −1 χ[a,b) (a) .

Since ∪k [xk+1 , xk ) = [a, b) up to sets of measure zero, summing up on k we obtain (b). It 1

also follows from this proof that A ≤ [w] Ap−1 . + (μ) p

(b)⇔(b ): It is obvious and the constants are the same. (b)⇒(a): We are going to see that [w]∗A+ (μ) is finite. Let a, b and c be real numbers such p

that a < b < c and μ([b, c)) > 0 (if μ([b, c)) = 0 there is nothing to prove). Then, (b) implies that for all x ∈ (a, b],

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F. J. Martín-Reyes, A. de la Torre

λ0 :=

[b,c) σ

dμ



[x,c) σ

≤

dμ

   p −1 ≤ C Mw+ dμ w −1 χ[x,c) (x)

μ((a, c)) μ([x, c))

   p −1 ≤ C Mw+ dμ w −1 χ(a,c) (x)

Applying (1.1) with T = Mw+ dμ we obtain

(a,b]





 w dμ ≤

 p−1 w dμ λ0 + −1 {x:Mw } dμ (w χ(a,c) )(x)> C

≤

C λ0

 p−1 μ((a, c)),

∗ p−1 . Therefore, which shows that w ∈ A+ p (μ) and the constant C in (b) satisfies [w] A+ (μ) ≤ C p

[w] A+p (μ) ≤ 2 p [w]∗A+ (μ) ≤ C p−1 .



p

Proof of Theorem 1.4 (a)⇔(a ) is established in Theorem 1.2. (a )⇒(b): Let [a, b) be an interval such that μ([a, b)) > 0. By (b) in Theorem 1.2 we have

   p −1 [a,b) | f | dμ [a,b) | f | dμ Mw+ dμ w −1 χ[a,b) (a) ≤C μ([a, b)) [a,b) σ dμ

 p −1  ≤ C Mw+ dμ (Mσ dμ ( f /σ )) p−1 w −1 χ[a,b) )(a) . (b)⇒(c): It follows immediately from inequality (1.2) applied to T = Mw+ dμ and T = Mσ dμ . Actually, we have 

 +    p + p−1 −1 M f (x) p wdμ ≤ C M (M (x) ( f /σ )) w wdμ σ dμ μ w dμ  ≤C  ≤C

p  p −1 p p − 1

 p  (Mσ dμ ( f /σ )) p σ dμ 

 p 2

p−1

p p−1

p  | f | p wdμ.

It follows from the proofs of the implications and Theorem 1.2 that   p −1 1 1 1 + p M p p ≤ 2 p  [w] Ap−1 ≤ 2ep  [w] Ap−1 . + + μ B(L (wdμ))  (μ) p p (μ) p −1 (c)⇒(a): The proof is as usual by testing the inequality in (c) with functions σ χ[b,c) (see 1

[1]). This proof gives ([w]∗A+ (μ) ) p ≤ ||Mμ+ ||B(L p (wdμ)) .



p

Remark 2.3 If μ is the Lebesgue measure then the power in 1

||M||B(L p (wdμ)) ≤ C( p)[w] Ap−1 p is the best posible one [3]. To prove that, Buckley used the power weights w(x) = |x|α . If we consider a measure μ = g(x)d x, where g is a positive measurable function then one can 1

prove that the power in ||Mμ+ ||B(L p (wdμ)) ≤ C( p)[w] Ap−1 is the best possible. The proof is + p (μ) x α the same as the one in Buckley’s paper but using wα (x) = 0 g instead of power weights.

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Sharp weighted bounds for one-sided maximal operators

The result in Theorem 1.4 can be proved for fractional maximal operators. The result is the following. Theorem 2.4 Let 0 ≤ α < 1, 1 < p < 1/α, fractional operator defined as

1 q

=

1 p

− α, r = 1 +

1 1−α h>0 (μ([x, x + h)))

+ Mα,μ f (x) = sup

q p  . Let

+ the one-sided Mα,μ

 [x,x+h)

| f | dμ,

Let w be a weight in R. The following assertions are equivalent. (a) wq ∈ Ar+ (μ) ([wq ]∗A+ (μ) < +∞). r (b) There exists a constant C such that for all measurable f and all real numbers a

+ Mα,μ f (a) ≤

pα C|| f || L p (w p dμ)

Mw+q dμ



  p (1−α) q  p−1 1−α Mw− p dμ ( f w ) w −q (a) . p

+ is bounded from L p (w p dμ) into L q (w q dμ). (c) Mα,μ

If any of the above conditions hold then 1

+ 1 q q1 q p p ≤ Mα,μ [w ] A+ (μ) ≤ [wq ]∗A+ (μ) B(L (w dμ),L q (wq dμ)) r r 2 p (1−α)

≤ 2e1−α ( p  ) p/q [wq ] A+q(μ) , r

where

+ || ||Mα,μ B(L p (w p dμ),L q (wq dμ))

is defined as usual.

We omit the proof since it is similar to the proof of Theorem 1.4 Proof of Theorem 1.5 (a)⇒(b): We proceed as in the proof of (a )⇒(b) in Theorem 1.2 but N +1 using the sequence {xk }k=0 given by Lemma 2.2 for the measure μw = w dμ instead of μ. In what follows, if N is finite then we define x N +2 := a. By Hölder’s inequality, 1/ p  1/ p  N   | f | dμ ≤ | f | p w dμ σ [a,b)

[xk+1 ,xk )

k=0

≤

N

 k=0

[w]∗A+ (μ) p

1/ p

[xk+1 ,xk )



[xk+1 ,xk ) | f |



p w dμ

[xk+2 ,xk+1 ] w dμ

1/ p μ([xk+2 , xk )).

Since μw ([xk+2 , xk+1 ]) = μw ([a, xk+1 ]) − μw ([a, xk+2 )) 1 1 1 1 ≥ μw ([a, xk ))− μw ([a, xk+1 )) = μw ([xk+1 , xk )) ≥ μw ([a, xk )), 2 2 2 4 we have 1/ p

 p N  [xk+1 ,xk ) | f | w dμ ∗ 1/ p | f | dμ ≤ (4[w] ) μ([xk+2 , xk )) [a,b) [a,xk ) w dμ k=0 1/ p

1/ p

μ([a, b)) Mw+ dμ (| f | p )(a) ≤ 2 4[w]∗A+ (μ) p 1/ p



1/ p ≤ 8 [w]∗A+ (μ) μ([a, b)) Mw+ dμ (| f | p )(a) , p

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F. J. Martín-Reyes, A. de la Torre

which proves (b). (b)⇒(c): It follows immediately from (b) and (1.1) for T = Mw+ dμ . (c)⇒(a): It suffices to test the inequality with functions f = σ χ[b,c) . The relations between the constants follow from the proofs and the relations 2− p [w] A+p (μ) ≤ [w]∗A+ (μ) ≤ [w] A+p (μ) obtained in Theorem 1.2.

p

3 One-sided A∞ (µ) constants and one-sided sharp reverse Hölder inequality 

1− p ∈ A− (μ). The Given a weight w, it is clear that w ∈ A+ p (μ) if and only if σ = w p results of this section will be applied to σ . That is the reason why we work in this section with A− p (μ) classes. We introduce some definitions and previous results.

Definition 3.1 Let μ be a locally finite Borel measure. A weight w belongs to A− ∞,exp (μ) if the one-sided exponential constant of w       1 1 w dμ exp log(1/w) dμ [w] A−∞,exp (μ) := sup μ([b, c} [b,c} μ({a, b]) {a,b] T is finite, where, as in Definition 1.1, T is the set of triplets (a, b, c) such that μ({a, c}) > 0, μ({a, b]) ≥ 21 μ({a, c}) and μ([b, c}) ≥ 21 μ({a, c}). Lemma 3.2 Let μ be a locally finite Borel measure. A weight w belongs to A− ∞,exp (μ) if and only if there exists C > 0 such that for all s ∈ (0, 1) and all triplets (a, b, c) ∈ T    1   s 1 1 w dμ ≤ C w s dμ . (3.1) μ([b, c}) [b,c} μ({a, b]) {a,b] Moreover, the least possible constant A in the last inequality is [w] A−∞,exp (μ) . Proof The proof is an immediate consequence of the following remark: for fixed I = {a, b],

1/s s 1 the function ϕ(s) = μ(I is increasing in (0, 1) and lims→0+ ϕ(s) = ) I w dμ 1 exp μ(I ) I log(w) dμ. − ≤ [w] A+p (μ) . Theorem 3.3 (a) If w ∈ A− p (μ) then w ∈ A∞,exp (μ) and [w] A− ∞,exp (μ) − − (b) If w ∈ A∞,exp (μ) then w ∈ A∞ (μ) and [w] A−∞ (μ) ≤ e[w] A−∞,exp (μ) .

Proof (a) By Hölder’s inequality, we have [w] Aq− (μ) ≤ [w] A−p (μ) for all q > p. Letting q tend to +∞ we obtain [w] A−∞,exp (μ) ≤ [w] A+p (μ) . (b) Let I = [c, d). Let us fix a ∈ I then  1 Mμ+ (wχ I )(a) = sup w dμ b:a 0, we choose a sequence as in Lemma 2.2. Then, using Lemma 3.2,  1   s 1 w dμ ≤ [w] A−∞,exp (μ) μ([xk+1 , xk )) w s dμ μ([a, xk+1 ]) [a,xk+1 ] [xk+1 ,xk )  + 1 ≤ [w] A−∞,exp (μ) μ([xk+1 , xk )) Mμ (χ I w s )(a) s .

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Adding up we get  [a,b)

 1 w dμ ≤ [w] A−∞,exp (μ) μ([a, b)) Mμ+ (χ I w s )(a) s .

1  Therefore, Mμ+ (wχ I )(a) ≤ [w] A−∞,exp (μ) Mμ+ (χ I w s )(a) s for all a ∈ I . By using (1.2) with T = Mμ+ we obtain   + −1/s Mμ (wχ I ) dμ ≤ [w] A−∞,exp (μ) (1 − s) w dμ. I

I



Letting s tend to 0 we are done.

Theorem 3.4 (Sharp reverse Hölder inequality) Assume that the measure μ is left-regular with constant Cμ (see Definition 1.6) and w ∈ A− ∞ (μ). (a) If 0 < ε ≤

1 2[w] A− (μ) Cμ −1 ∞



 I

(b) If 0 < ε ≤

then for all intervals I = [a, b)

1+ε ε  dμ ≤ 2[w] A−∞ (μ) Mμ+ (wχ I )(a) Mμ+ (wχ I )

1 2[w] A− (μ) Cμ ∞



I

 w dμ I

then for all intervals I = [a, b)  ε w 1+ε dμ ≤ 2 Mμ+ (wχ I )(a)

 w dμ. I

 1+ε Consequently, Mμ+ (w 1+ε χ I )(x) ≤ 2 Mμ+ (wχ I )(x) for x ∈ I . (c) If 0 < ε ≤

1 2[w] A− (μ) Cμ ∞

then for all real numbers a < b < c

(μ((a, b]))ε

 [b,c)

  w 1+ε dμ ≤ 2

(a,c)

1+ε w dμ , 1

−1 ∗ ε that is, w −1 ∈ A+ p (wdμ), p = (1 + ε)/ε, and [w ] A+ (wdμ) ≤ 2 . p

We shall need the following two lemmas. Lemma 3.5 Let u and v be nonnegative measurable functions on an interval I . Let 0 < L < N , u N = min{u, N } and v N = min{v, N }. For each λ > 0, consider the sets Uλ = {x ∈ I : u(x) > λ} and Vλ = {x ∈ I : v(x) > λ}. If there exists A > 0 such that E λ u dμ ≤ Aλμ(Vλ ) for all λ ∈ (L , N ) then for any ε > 0     ε u N u εN − L ε dμ ≤ A v ε+1 dμ. 1+ε I N I Proof of Lemma 3.5 Multiply both sides of the inequality Uλ u dμ ≤ Aλμ(Vλ ) by λε−1 and integrate in λ ∈ (L , N ).

Lemma 3.6 Assume that the measure μ is left-regular with constant Cμ . Let w be a weight, I = [a, b) an interval, λ0 = Mμ+ (wχ I )(a), Wλ = {x ∈ I : w(x) > λ} and Uλ = {x ∈ I : Mμ+ (wχ I )(x) > λ}.

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F. J. Martín-Reyes, A. de la Torre

+ (a) If w ∈ A− Cμ λμ(Uλ ) for all λ > λ0 . ∞ (μ) then Uλ Mμ (wχ I ) dμ ≤ [w] A− ∞ (μ) (b) Wλ w dμ ≤ Cμ λμ(Uλ ) for all λ > λ0 . Proof of Lemma 3.6 Let Oλ = {x : Mμ+ (wχ I )(x) > λ}. Then Uλ = Oλ ∩ I . Let {I j } j be the connected components of Oλ . Since Mμ+ (wχ I ) is upper continuous then each I j is an interval of the form (a j , b j } which means that I j = (a j , b j ) or I j = (a j , b j ] (see [1]). For λ > λ0 we have that a ≤ a j for all j. Therefore (a j , b j } ⊂ I and, consequently, Uλ = Oλ = ∪ j I j . Using that a j  ∈ Oλ and that the measure μ is left-regular, we have   w dμ ≤ w dμ ≤ λμ([a j , b j }) ≤ Cμ λμ((a j , b j }). Ij

[a j ,b j }

Summing up on k, we obtain

 Uλ

w dμ ≤ Cμ λμ(Uλ ).

(3.2)

This inequality proves (b) because Wλ ⊂ Uλ (observe that we have not used that w ∈ A− ∞ (μ)). By the maximality of the intervals I j we have that Mμ+ (wχ I )(x) = Mμ+ (wχ I j )(x) for all x ∈ I j . This remark, w ∈ A− ∞ (μ) and (3.2) give    Mμ+ (wχ I ) dμ = Mμ+ (wχ I j ) dμ ≤ [w] A−∞ (μ) w dμ Uλ

j

Ij

= [w] A−∞ (μ)

 Uλ

j

Ij

w dμ ≤ [w] A−∞ (μ) Cμ λμ(Uλ )



and the lemma is completely proved.

Proof of Theorem 3.4 (a) Let λ0 = Mμ+ (wχ I )(a), u = v = Mμ+ (wχ I ). Taking into account statement (a) in Lemma 3.6, Lemma 3.5 gives that    ε 1+ε ε 1− [w] A−∞ (μ) Cμ u N dμ ≤ λ0 u N dμ. ε+1 I I for all N > λ0 and all ε > 0. Letting N tend to ∞ we have     + 1+ε ε Mμ (wχ I ) dμ ≤ λε0 Mμ+ (wχ I ) dμ 1− [w] A−∞ (μ) Cμ 1+ε I I  ε ≤ λ0 [w] A−∞ (μ) w dμ. I

If 0 < ε ≤ 1/(2[w] A−∞ (μ) Cμ − 1) then the quantity in the bracket is greater than or equal to 1/2 and we are done. (b) Let λ0 = Mμ+ (wχ I )(a), u = w, v = Mμ+ (wχ I ). It follows from statement (b) in Lemmas 3.6 and 3.5 that for all N > λ0 and all ε > 0     ε u N u εN − λε0 dμ ≤ Cμ v 1+ε dμ. 1+ε I N I Since 0 < ε ≤ 1/(2[w] A−∞ (μ) Cμ ) ≤ 1/(2[w] A−∞ (μ) Cμ − 1), the last inequality and the inequality in statement (a) give     ε 2[w] A−∞ (μ) λε0 w dμ. u N u εN − λε0 dμ ≤ Cμ 1+ε I I

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Letting N tend to ∞ and keeping in mind that 0 < ε ≤  I

1 2[w] A− (μ) Cμ ,

we obtain

∞

  ε λε0 w 1+ε I   

ε ≤ 1 + 2Cμ [w] A−∞ (μ) ε λ0 w ≤ 2λε0 w.

 w 1+ε dμ ≤ 1 + 2Cμ [w] A−∞ (μ)

I

I

(c) It follows from statement (b) and Theorem 1.2 applied to the measure w dμ, the weight w −1 and p = (1 + ε)/ε.

Actually, at least for continuous measures, we have proved several characterizations of the union of the classes A+ p (μ). We collect them in the following result. Theorem 3.7 Let μ be a continuous Borel measure on the real line which is finite on bounded sets. The following statements are equivalent. (a) (b) (c) (d) (e)

w ∈ ∪ p>1 A− p (μ). w ∈ A− ∞,exp (μ) There exist C > 0 and s ∈ (0, 1) such that (3.1) holds for all triplets (a, b, c) ∈ T . w ∈ A− ∞ (μ). There exist ε > 0 and C > 0 such that for all intervals I = [a, b)    + 1+ε  ε Mμ (wχ I ) dμ ≤ C Mμ+ (wχ I )(a) w dμ I

I

(f) There exist ε > 0 and C > 0 such that for all intervals I = [a, b)    ε w 1+ε dμ ≤ C Mμ+ (wχ I )(a) w dμ. I

I

(g) There exist ε > 0 and C > 0 such that for all real numbers a < b < c    1+ε w 1+ε dμ ≤ 2 w dμ , (μ((a, b]))ε [b,c)

(a,c)

(h) w −1 ∈ ∪ p>1 A+ p (wdμ). Notice that the implications (a)⇒(b)⇒(c) ⇒(d)⇒(e)⇒(f)⇒(g)⇒(h) have already been proved. In particular, we have (a)⇒(h). Since wdμ is continuous, the analogous result, changing the orientation of the real line, shows that (h)⇒(a).

4 Proof of Theorems 1.8 and 1.9 − − Proof of Theorem 1.8 Since w ∈ A+ p (μ) ⇔ σ ∈ A p  (μ), we have that σ ∈ A∞ (μ). Statement (b) in Theorem 3.4 gives that for all intervals I = [a, b)    1+δ . Mμ+ σ 1+δ χ I (a) ≤ 2 Mμ+ (σ χ I )(a)

Since w ∈ A+ p (μ), Theorem 1.2 allows us to obtain

1  −1   p −1 + Mμ+ (σ χ[a,b) )(a) ≤ [w] Ap−1 M χ . w (a) [a,b) + w dμ (μ) p

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F. J. Martín-Reyes, A. de la Torre

Putting together both inequalities and using that ( p − 1) = (1 + δ)(q − 1),



1  −1  q  −1  + w (a) Mμ+ w 1−q χ I (a) ≤ 2[w] Aq−1 M χ . [a,b] + w dμ (μ) p

Applying again Theorem 1.2 we have that w ∈ Aq+ (μ) and  q−1 1 q−1 q = 2q [w] A+p (μ) ≤ 2 p [w] A+p (μ) . [w] Aq+ (μ) ≤ 2 [w] A+ (μ) p

Proof of Theorem 1.9 Let δ = Aq+ (μ)

2 p [w]

1

2[σ ] A− (μ) Cμ ∞

and q = ( p + δ)/(1 + δ). By Theorem 1.8, w ∈

and [w] Aq+ (μ) ≤ . Using Theorem 1.5 and Marcinkiewicz interpolation A+ p (μ) theorem (see [5, Theorem 2.4]), 1/ p

 q/ p + p 1/q M p 8[w] A+ (μ) μ B(L (wdμ)) ≤ 2 q p−q  1/ p

1/ p 4 1+δ p 2 [w] A+p (μ) ≤2 p ≤ 27 Cμ p  [σ ] A−∞ (μ) [w] A+p (μ) . δ



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