WEIGHTED NORM INEQUALITIES FOR THE

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WEIGHTED NORM INEQUALITIES FOR THE MAXIMAL OPERATOR ON VARIABLE LEBESGUE SPACES D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER Abstract. We prove weighted strong and weak-type norm inequalities for the Hardy-Littlewood maximal operator on the variable Lebesgue space Lp(·) . Our results generalize both the classical weighted norm inequalities on Lp and the more recent results on the boundedness of the maximal operator on variable Lebesgue spaces.

1. Introduction Given a measurable function p(·) : Rn → [1, ∞), hereafter referred to simply as an exponent, we define the variable Lebesgue space Lp(·) to be the set of measurable functions on Rn such that for some λ > 0, Z ρp(·) (f ) = ρ(f /λ) = |f (x)/λ|p(x) dx < ∞. Rn

This is a Banach space (see [7, 16, 20, 31]) when equipped with the norm kf kp(·) = inf{λ > 0 : ρ(f /λ) ≤ 1}. The variable Lebesgue spaces are a special case of the Musielak-Orlicz spaces [39] and generalize the classical Lebesgue spaces: when p(x) = p0 is constant, Lp(·) = Lp0 . Variable Lebesgue spaces were first studied by Orlicz [41] in 1931, but in the past two decades there has been a great deal of interest in them, particularly for their applications to partial differential equations with non-standard growth conditions and to modeling electrorheological fluids. See [18, 44] for the history and references. Much attention has been focused on finding conditions on the exponent function p(·) so that the Hardy-Littlewood maximal operator is bounded on Lp(·) . Recall that Date: April 19, 2012. 1991 Mathematics Subject Classification. 42B25,42B35. Key words and phrases. variable Lebesgue space, maximal operators, weights. The authors would like to thank Lars Diening and Peter H¨ast¨o for sharing with us many preprints of their work. 1

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D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

given a locally integrable function f on Rn , the maximal function M f is defined on Rn by Z Z 1 M f (x) = sup |f (y)| dy = sup − |f (y)| dy, Q3x |Q| Q Q3x Q where the supremum is taken over all cubes Q containing x whose sides are parallel to the coordinate axes. The following result first proved by the authors and their collaborators (see [3, 5, 8]) is nearly optimal. Hereafter, given a set E, let p− (E) = ess inf p(x), x∈E

p+ (E) = ess sup p(x). x∈E

n

If E = R , for brevity we will write simply p− and p+ . Theorem 1.1. Given p(·) : Rn → [1, ∞), suppose that 1 ≤ p− ≤ p+ < ∞ and that p(·) satisfies the local log-H¨older continuity condition C (1.1) LH0 : |p(x) − p(y)| ≤ , x, y ∈ Rn , |x − y| < 1/2, − log |x − y| and is log-H¨older continuous at infinity: there exists p∞ , 1 ≤ p∞ < ∞, such that C (1.2) LH∞ : |p(x) − p∞ | ≤ , x ∈ Rn . log(e + |x|) Then the Hardy-Littlewood maximal operator satisfies the weak-type inequality ktχ{x∈Rn :M f (x)>t} kp(·) ≤ Ckf kp(·) ,

t > 0.

If p− > 1, then it is bounded on Lp(·) : kM f kp(·) ≤ Ckf kp(·) . Theorem 1.1 was first proved by Diening [13] assuming that p(·) is constant outside of a large ball. It was proved independently by Nekvinda [40] (see also [3]) with (1.2) replaced by a somewhat more general condition. The log-H¨older continuity conditions are not necessary; see Lerner [32]. However, if they are relaxed it is possible to construct counter-examples: see [8, 42]. In this paper we generalize Theorem 1.1 to variable Lebesgue spaces with weights. For classical Lebesgue spaces the following result is due to Muckenhoupt [37] (see R R −1 − also [19, 21]). Hereafter, we use the notation Q f (x) dx = |Q| f (x) dx. Q Theorem 1.2. Given p, 1 ≤ p < ∞, and a locally integrable function w such that 0 < w(x) < ∞ almost everywhere, the following are equivalent: (a) w ∈ Ap : for every cube Q Z p−1 Z 1−p0 (1.3) − w(x) dx − w(x) dx ≤ K < ∞, Q

Q

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if p > 1, and Z − w(x) dx ≤ Kw(y), Q

for almost every y ∈ Q if p = 1. (b) The maximal operator is weak (p, p): for every t > 0, Z C n (1.4) w({x ∈ R : M f (x) > t}) ≤ p |f (x)|p w(x) dx; t Rn (c) If p > 1, the maximal operator is strong (p, p): Z Z p (1.5) M f (x) w(x) dx ≤ C |f (x)|p w(x) dx. Rn

Rn

Remark 1.3. In the definition of Ap , the smallest constant K is referred to as the Ap constant of w. There are two possible ways to generalize Theorem 1.2 to variable Lebesgue spaces. One way is to treat w dx as a measure, and define the weighted variable Lebesgue space Lp(·) (w) with respect to this measure. This approach is in one respect a natural generalization of Theorem 1.2 and was adopted by Diening and H¨asto [17]. They proved that if 1 < p− ≤ p+ < ∞ and p(·) is log-H¨older continuous locally and at infinity, then the maximal operator is bounded on Lp(·) (w) if and only if |Q|−p− (Q) w(Q)kw−1 χQ kp0 (·)/p(·) ≤ K < ∞. Here, p0 (·) is the conjugate exponent, defined pointwise by 1 1 + 0 = 1, p(x) p (x) where we let p0 (x) = ∞ if p(x) = 1. The second norm is defined as before even if p0 (·)/p(·) is less than 1. We take a different approach: we recast the inequalities so that the weight w acts as a multiplier. More precisely, in inequalities (1.4) and (1.5) we replace the weight w by wp and rewrite the integrals in terms of Lp norms to get ktχ{x∈Rn :M f (x)>t} wkp ≤ Ckf wkp , k(M f )wkp ≤ Ckf wkp . Similarly, with w replaced by wp , the Ap condition can be rewritten as (1.6)

kwχQ kp kw−1 χQ kp0 ≤ K|Q|.

While this not the standard way to write either the Ap condition or weighted norm inequalities for the maximal operator, it is the natural way to write the off-diagonal weighted inequalities for fractional integrals (see, for example, Muckenhoupt and

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Wheeden [38]) and has also been used in the study of two-weight norm inequalities (see [10]). Definition 1.4. Given an exponent function p(·) : Rn → [1, ∞) and a weight w— that is, a locally integrable function such that 0 < w(x) < ∞ a.e.—we say that w ∈ Ap(·) if there exists a constant K such that for every cube Q, kwχQ kp(·) kw−1 χQ kp0 (·) ≤ K|Q|. Our main result is the following weighted norm inequality. Theorem 1.5. Let p(·) : Rn → [1, ∞). If 1 < p− ≤ p+ < ∞ and p(·) satisfies the LH0 and LH∞ conditions (1.1) and (1.2), then given any w ∈ Ap(·) , k(M f )wkp(·) ≤ Ckf wkp(·) . If p− ≥ 1, then ktχ{x∈Rn :M f (x)>t} wkp(·) ≤ Ckf wkp(·) . Conversely, given a weight w and an exponent function p(·) such that the maximal operator satisfies the strong or weak-type inequality, then w ∈ Ap(·) . Remark 1.6. While this paper was in preparation, the strong-type inequality Theorem 1.5 when p− > 1 was proved using a very different approach by the first author, Diening, and H¨ast¨o [6]. That proof, while straightforward, depends heavily on the sophisticated machinery developed by Diening [14] (see also [16]) to give the necessary and sufficient condition for the boundedness of the maximal operator. Our approach here has two advantages: it is direct, and it makes clear the connection between the Ap(·) condition and the classical Muckenhoupt Ap condition. Remark 1.7. Separately, Diening and H¨ast¨o [17] proved when p− > 1 that the Ap(·) condition is necessary for the strong-type inequality. Remark 1.8. Particular results of this type for power weights (and generalizations of power weights) have been proved by Kokilashvili, Samko and Samko [25, 26, 27, 28, 29] and Khabazi [23]. Other authors have also considered weighted norm inequalities for the maximal operator on variable Lebesgue spaces. See, for instance, [1, 24, 35, 36]. In Theorem 1.5 the necessity of the Ap(·) condition is proved without using any continuity assumptions on the exponent function p(·). Given that the classical Ap condition is necessary and sufficient for the maximal operator to be bounded on weighted Lebesgue spaces, it was extremely tempting to conjecture that the Ap(·) condition, without additional continuity assumptions on p(·), is sufficient for the maximal operator to be bounded on variable Lebesgue spaces. However, even when

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w = 1 this is false; a counter-example is due independently to Diening [15] and Kopaliani [30]. There is a connection between the variable Ap(·) condition and the classical Ap condition: if w ∈ Ap(·) , then wp(·) ∈ A∞ —see Section 3 below—and this plays a critical role in our proofs. Earlier, Lerner [34] showed that there is a connection between the classical Ap condition and the boundedness of the maximal operator on Lp(·) (Ω) for bounded Ω. A deeper understanding of the inter-relationship between weights and variable Lebesgue spaces remains to be elucidated. Recently, there has been a great deal of interest in the sharp constant (in terms of the constant in the Ap condition) in weighted norm inequalities for various operators: see, for example, [11] and the references it contains. The sharp constant for the maximal operator in Theorem 1.2 is due to Buckley [2]; also see Lerner [33]. An open (and difficult) question is to determine the corresponding results in the variable Lebesgue spaces. For a discussion of a related problem, see [17]. Organization. The remainder of this paper is organized as follows. In Section 2 we gather together some basic results about variable Lebesgue spaces. In Section 3 we prove some basic properties of the Ap(·) condition. Finally, in Sections 4 and 5 we prove Theorem 1.5. Throughout this paper all notation is standard or will be defined as needed. In order to emphasize that we are dealing with variable exponents, we will always write p(·) instead of p to denote an exponent function. Unless otherwise specified, C and c will denote positive constants which will depend only on the dimension n, any underlying sets (such as a fixed cube Q), and the exponent p(·). If we write A ≈ B, then we mean that there exist positive constants c and C such that cA ≤ B ≤ CA. By cubes we will always mean cubes whose sides are parallel to the coordinate axes. Given a cube Q and r > 0, let rQ denote the cube with the same center as Q and such that `(rQ) = r`(Q). A weight w will always be assumed to be locally integrable and positive almost everywhere. 2. Variable Lebesgue spaces In this section we gather some basic results on variable Lebesgue spaces. Unless otherwise noted, proofs of these results can be found in [7, 16, 20, 31]. Lemma 2.1. Given p(·) : Rn → [1, ∞) such that p+ < ∞, then kf kp(·) ≤ C1 if and only if Z |f (x)|p(x) dx ≤ C2 . Rn

Moreover, if either constant equals 1 we can take the other equal to 1 as well.

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Lemma 2.2. Let p(·) : Rn → [1, ∞) be such that p+ < ∞. Then given any set Ω, Z p− (Ω) p+ (Ω) . |f (x)|p(x) dx ≤ kf χΩ kp(·) (a) if kf χΩ kp(·) ≤ 1, kf χΩ kp(·) ≤ ZΩ p− (Ω) p+ (Ω) (b) if kf χΩ kp(·) ≥ 1, kf χΩ kp(·) ≤ |f (x)|p(x) dx ≤ kf χΩ kp(·) . Ω

In particular, if kf kp(·) ≤ 1, Z

|f (x)|p(x) dx ≤ kf kp(·) .

Rn

Lemma 2.3. If p+ < ∞ and kf kp(·) = 1, then Z |f (x)|p(x) dx = 1. Rn

Lemma 2.4. If p+ < ∞, bounded functions of compact support are dense in Lp(·) . When the exponent p(·) equals +∞ on a set of positive measure, we modify the definition of the norm as follows. Let Ω∞ = {x ∈ Rn : p(x) = ∞}, and redefine the modular ρ by Z |f (x)|p(x) dx + kf kL∞ (Ω∞ ) .

ρ(f ) = Rn \Ω∞

With this definition k · kp(·) is still a norm and Lp(·) a Banach space. Moreover, we have the following generalization of H¨older’s inequality and an equivalent expression for the norm. Lemma 2.5. (H¨older’s Inequality) Given an exponent p(·), there exists a constant C such that Z |f (x)g(x)| dx ≤ Ckf kp(·) kgkp0 (·) . Rn 0

Lemma 2.6. Given an exponent p(·) and f ∈ Lp(·) , there exists g ∈ Lp (·) , kgkp0 (·) ≤ 1, such that Z kf kp(·) ≈ f (x)g(x) dx. Rn

The next two lemmas were proved in [3, Lemmas 2.7, 2.8] for Lebesgue measure. The exact same proofs work in general for non-negative measures µ. They are central to applying the LH∞ condition. Lemma 2.7. Given a set G and two exponent s(·) and r(·) such that C0 |s(y) − r(y)| ≤ , log(e + |y|)

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and given a non-negative measure µ, then for every t ≥ 1 there exists a constant C = C(t, C0 ) such that for all functions f with |f (y)| ≤ 1, Z Z Z 1 s(y) r(y) (2.1) |f (y)| dµ(y) ≤ C |f (y)| dµ(y) + dµ(y). tnr− (G) G G G (e + |y|) Lemma 2.8. Given a set G and two exponent s(·) and r(·) such that C0 0 ≤ r(y) − s(y) ≤ , log(e + |y|) and given a non-negative measure µ, then for every t ≥ 1 there exists a constant C = C(t, C0 ) such that for all functions f , inequality (2.1) holds. The final lemma is due to Diening [13] and is a crucial tool for applying the LH0 condition. His result is for balls instead of cubes, but the proof is essentially the same, changing only the dependence of the constant on the dimension n. Lemma 2.9. Given an exponent p(·), the following are equivalent: (a) p(·) satisfies the LH0 condition (1.1); (b) There exists a constant C such that for every cube Q, |Q|p− (Q)−p+ (Q) ≤ C. 3. The variable Ap condition In this section we prove some basic properties of the weights in Ap(·) . We begin with some equivalent definitions of A∞ ; for a proof of this well-known result, see [19, 21]. Lemma 3.1. Given a weight W , the following are equivalent: [ (a) W ∈ A∞ = Ap ; p≥1

(b) There exist constants 0 < α, β < 1 such that given any cube Q and measurable set E ⊂ Q, if |E| > α|Q|, then W (E) > βW (Q); (c) There exist constants δ > 0 and C1 > 1 such that given any cube Q and any measurable set E ⊂ Q,  δ W (E) |E| ≤ C1 ; W (Q) |Q| (d) There exist constants  > 0 and C2 > 1 such that given any cube Q and measurable set E ⊂ Q,   W (E) |E| ≤ C2 . |Q| W (Q)

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Weights in Ap(·) satisfy an A∞ -type condition in terms of the variable Lebesgue space norm. Lemma 3.2. Given an exponent p(·), if w ∈ Ap(·) , then there exists a constant C depending on p(·) and w such that given any cube Q and measurable set E ⊂ Q, kwχE kp(·) |E| ≤C . |Q| kwχQ kp(·) Proof. Fix Q and E ⊂ Q; then by H¨older’s inequality (Lemma 2.5) and the Ap(·) condition, Z |E| = w(x)χE w(x)−1 χQ dx Rn

≤ CkwχE kp(·) kw−1 χQ kp0 (·) ≤ CkwχE kp(·) kwχQ k−1 p(·) |Q|.  The Ap(·) condition and log-H¨older continuity together imply that a weighted version of Lemma 2.9 holds. Lemma 3.3. Given an exponent p(·) such that (1.1) and (1.2) hold, if w ∈ Ap(·) , then there exists a constant C0 depending on p(·) and w such that for all cubes Q, p (Q)−p+ (Q)

− kwχQ kp(·)

(3.1)

≤ C0 .

Proof. Fix Q. Clearly it suffices to assume that kwχQ kp(·) ≤ 1. Let Q0 = Q(0, 1). The proof of (3.1) depends on the relative size of Q and Q0 and their distance from one another. We will consider the case |Q| ≤ |Q0 |; the case when |Q| > |Q0 | is proved in the same way, exchanging the roles of Q and Q0 . Suppose dist(Q, Q0 ) ≤ `(Q0 ). Then Q ⊂ 5Q0 , and so by H¨older’s inequality (Lemma 2.5) and the Ap(·) condition, Z |Q| = w(x)w(x)−1 dx ≤ CkwχQ kp(·) kw−1 χQ kp0 (·) Q

≤ C5n kwχQ kp(·) |5Q0 |−1 kw−1 χ5Q0 kp0 (·) ≤ CkwχQ kp(·) kwχ5Q0 k−1 p(·) . If we rearrange terms and raise both sides to the power p+ (Q)−p− (Q), by Lemma 2.9 we have that p (Q)−p+ (Q)

− kwχQ kp(·)

p (Q)−p+ (Q)

− ≤ C|Q|p− (Q)−p+ (Q) kwχ5Q0 kp(·)

p+ −p− ≤ C(1 + kwχ5Q0 k−1 . p(·) )

The right-hand side is a constant independent of Q and so we get (3.1).

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˜ such that Now suppose that dist(Q, Q0 ) ≥ `(Q0 ). Then there exists a cube Q ˜ and `(Q) ˜ ≈ dist(Q, Q0 ) ≈ dist(Q, 0) = dQ . If argue as we did above, with Q, Q0 ⊂ Q ˜ we get that 5Q0 replaced by Q, −1 −1 ˜ ˜ |Q| ≤ C|Q|kwχ ˜ kp(·) ≤ C|Q|kwχQ kp(·) kwχQ ˜ kp(·) . Q kp(·) kwχQ

We could continue the above argument and get (3.1) provided that ˜ p+ (Q)−p− (Q) ≤ C. |Q|

(3.2)

To see that this is the case: by the LH0 condition, p(·) is continuous, so p− (Q) = p(x1 ) and p+ (Q) = p(x2 ) where x1 , x2 ∈ Q. (More precisely, they may be in the closure of Q, but this makes no difference.) Since |x1 |, |x2 | ≈ dQ , by the LH∞ condition, p+ (Q) − p− (Q) ≤ |p(x2 ) − p∞ | + |p(x1 ) − p∞ | ≤

C . log(e + dQ )

˜ ≤ C(e + dQ )n , we get (3.2). If we combine this with the fact that |Q|



Lemma 3.4. Given an exponent p(·) such that (1.1) and (1.2) hold, if w ∈ Ap(·) , then W (·) = w(·)p(·) ∈ A∞ . Proof. Fix a cube Q and let E ⊂ Q be a measurable set. We will show that there exists a constant C (independent of E and Q) such that  1/p+ W (E) |E| ≤C ; |Q| W (Q) it will then follow at once from Lemma 3.1 that W ∈ A∞ . We consider three cases. If kwχQ kp(·) ≤ 1, then by Lemma 3.2, Lemma 2.2 (applied twice) and Lemma 3.3, kwχE kp(·) |E| ≤C p− (Q)/p+ (Q) 1−p (Q)/p+ (Q) |Q| kwχQ kp(·) kwχQ kp(·) −  1/p+ (Q)  1/p+ W (E) W (E) p− (Q)/p+ (Q)−1 kwχQ kp(·) ≤C . ≤C W (Q) W (Q) If kwχE kp(·) ≤ 1 ≤ kwχQ kp(·) , then again Lemmas 3.2 and 2.2 we get that |E| ≤C |Q|



W (E) W (Q)

1/p+ (Q)

 ≤C

W (E) W (Q)

1/p+ .

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Finally, suppose kwχQ kp(·) ≥ kwχE kp(·) ≥ 1. Let λ = kwχQ kp(·) . Since p(·) satisfies the LH∞ condition, by Lemma 2.7 with dµ = w(·)p(·) dx, for all t > 1, p(x) Z  Z Z w(x) w(x)p(x) −p∞ p(x) λ w(x) dx ≤ Ct dx + dx. tnp− λ Q Q Q (e + |x|) By Lemma 2.3, the first integral on the right-hand side equals 1. We claim that there exists t > 1 independent of Q such that the second integral is less than 1. To see this, let Qk = Q(0, 2k ). Then by Lemma 2.2, Z ∞ Z X w(x)p(x) w(x)p(x) −ntp− dx ≤ e W (Q ) + dx 0 tnp− tnp− (e + |x|) Rn (e + |x|) Q \Q k k−1 k=1 ≤e

−ntp−

W (Q0 ) + C

≤ e−ntp− W (Q0 ) + C

∞ X k=1 ∞ X

2−ktnp− W (Qk ) p− p+  2−ktnp− max kwχQk kp(·) , kwχQk kp(·) .

k=1

By Lemma 3.2, kwχQk kp(·) ≤ C

|Qk | kwχQ0 kp(·) ≤ C2nk . |Q0 |

Combining these two estimates we have that Z ∞ X w(x)p(x) −ntp− dx ≤ e 2nkp+ −ktnp− . (3.3) W (Q0 ) + C tnp− (e + |x|) n R k=1 For t > p+ /p− the sum converges, and by choosing t sufficiently large (depending only on w and p(·)) we can make the right-hand side less than 1. This gives us the desired bound. It follows from this that W (Q)1/p∞ ≤ (Ct + 1)1/p∞ kwχQ kp(·) . We now repeat the above argument, replacing Q with E and exchanging the roles of p∞ and p(·). By Lemma 2.3, Z Z Z w(x)p(x) −p(x) p(x) −p∞ p(x) dx. 1= λ w(x) dx ≤ Ct λ w(x) dx + tnp− E E E (e + |x|) Arguing as before we can make the second term on the right less than 1/2; if we rearrange terms, we get that kwχE kp(·) ≤ CW (E)1/p∞ .

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Therefore, by Lemma 3.2 kwχE kp(·) |E| ≤C ≤C |Q| kwχQ kp(·)



W (E) W (Q)

1/p∞

 ≤C

W (E) W (Q)

1/p+

This completes the proof.



As corollaries to the proof of Lemma 3.4 we get two lemmas that we will use repeatedly below. Lemma 3.5. Given an exponent p(·) such that (1.1) and (1.2) hold, if w ∈ Ap(·) , Q is a cube and E ⊂ Q is such that kwχE kp(·) ≥ 1, then  1/p∞ W (E) |E| ≤C . |Q| W (Q) Lemma 3.6. Given an exponent p(·) such that (1.1) and (1.2) hold, if w ∈ Ap(·) and Q is a cube such that kwχQ kp(·) ≥ 1, then kwχQ kp(·) ≈ W (Q)1/p∞ . The Ap(·) condition can be characterized in terms of averaging operators. Given any cube Q, define the operator AQ by Z AQ f (x) = − |f (y)| dyχQ (x). Q

Proposition 3.7. Given an exponent p(·) and a weight w, w ∈ Ap(·) if and only if sup k(AQ f )wkp(·) ≤ Ckf wkp(·) . Q

Proposition 3.7 is implicit in [16]. When p(·) is constant this is a lesser known property of the Muckenhoupt Ap weights; it was first given by Jawerth [22]. For completeness we sketch the short proof. Proof. First suppose that w ∈ Ap(·) . Then given any cube Q, by H¨older’s inequality (Lemma 2.5), Z k(AQ f )wkp(·) = − |f (x)| dxkwχQ kp(·) Q

≤ C|Q|−1 kf wkp(·) kw−1 χQ kp0 (·) kwχQ kp(·) ≤ Ckf wkp(·) . Since the constant C is independent of Q, we have that the AQ are uniformly bounded. Now assume that the AQ are uniformly bounded. Fix a cube Q; then by Lemma 2.6 (exchanging the roles of p(·) and p0 (·)) there exists a function g with kgwkp(·) ≤ 1 such that

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Z

−1

kwχQ kp(·) kw χQ kp0 (·) ≤ CkwχQ kp(·)

χQ g(x) dx Rn

= C|Q|k(AQ g)wkp(·) ≤ C|Q|kgwkp(·) ≤ C|Q|. It follows that w ∈ Ap(·) .



Finally, we note that when w ≡ 1, the log-H¨older continuity conditions LH0 and LH∞ imply the Ap(·) condition. This follows from the necessity in Theorem 1.5 but it is also possible to give a direct proof. This is done, for instance, in [16]; for the convenience of the reader we sketch the details. Proposition 3.8. Suppose the exponent p(·) satisfies (1.1) and (1.2) and p− > 1. Then 1 ∈ Ap(·) : there exists K such that kχQ kp(·) kχQ kp0 (·) ≤ K|Q|. Proof. Fix a cube Q. If |Q| ≤ 1, then by Lemma 2.2, kχQ kp(·) ≤ C|Q|1/p+ (Q) ,

0

kχQ kp0 (·) ≤ C|Q|1/p+ (Q) = C|Q|1−1/p− (Q) .

Therefore, kχQ kp(·) kχQ kp0 (·) ≤ C|Q||Q|1/p+ (Q)−1/p− (Q) ≤ K|Q|, where the last inequality follows from Lemma 2.9. Now suppose that |Q| > 1. Then by an argument that is essentially the same as that in the proof of Lemma 3.4, by Lemma 2.7 we have that kχQ kp(·) ≤ K|Q|1/p∞ ,

0

kχQ kp0 (·) ≤ K|Q|1/p∞ .

Multiplying the two inequalities we get the desired result.



4. The necessity of the Ap(·) condition In this section we show that the Ap(·) condition is necessary in Theorem 1.5. Proposition 4.1. Given an exponent p(·) and a weight w, if the maximal operator satisfies the strong-type inequality kM f wkp(·) ≤ Ckf wkp(·) or the weak-type inequality ktχ{x:M f (x)>t} wkp(·) ≤ Ckf wkp(·) , t > 0, then w ∈ Ap(·) . Proof. Since the strong-type inequality implies the weak-type inequality, it will suffice to show that the latter implies the Ap(·) condition. Our proof is modeled on the proof of necessity in Theorem 1.2.

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

13

Fix a cube Q; since the weak-type inequality and the Ap(·) condition are both homogeneous, we may assume without loss of generality that kw−1 χQ kp0 (·) = 1. Define the sets F0 = {x ∈ Q : p0 (x) < ∞}, F∞ = {x ∈ Q : p0 (x) = ∞}, and fix λ, 1/2 < λ < 1. Then by the definition of the norm,  −1  Z  p0 (x) w χQ w(x)−1 1 < ρp0 (·) = dx + λ−1 kw−1 χF∞ k∞ . λ λ F0 We consider two cases. First suppose that 1 λ−1 kw−1 χF∞ k∞ > . 2 −1 −1 Fix s, s > kw χF∞ k∞ = ess inf x∈F∞ w(x) (where we take 1/∞ = 0). Then there exists a set E ⊂ F∞ , |E| > 0, such that for all x ∈ E, w(x) ≤ s. Define f = χE . Since p(x) = 1 on F∞ , kf wkp(·) = kwχE kp(·) = w(E). Further, for all x ∈ Q, |E| M f (x) ≥ . |Q| Fix t < |E|/|Q|. Then by the weak-type inequality, Cw(E) = Ckf wkp(·) ≥ tkwχ{x:M f (x)>t} kp(·) ≥ tkwχQ kp(·) . Taking the supremum over all such t and re-arranging terms, we get that w(E) ≤ Cs. |Q|−1 kwχQ kp(·) ≤ C |E| If we now take the infimum over all such s we get 2C |Q|−1 kwχQ kp(·) ≤ Ckw−1 χF∞ k−1 ≤ 4C. ∞ ≤ λ Since kw−1 χQ kp0 (·) = 1, we get that the Ap(·) condition holds on Q. Now suppose p0 (x) w(x)−1 dx > 1/2. λ F0 If p0 (·) is unbounded, this integral could be infinite. To avoid this complication, define Z



FR = {x ∈ F0 : p0 (x) < R},

R > 1.

Then there exists R such that Z 1/2 < FR



w(x)−1 λ

p0 (x) dx < ∞.

14

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

(The lower bound is gotten by taking R sufficiently large. The upper bound comes 0 from the fact that by Lemma 2.1, w−p (·) χFR is integrable.) We claim that there exists E ⊂ FR such that p0 (x) Z  w(x)−1 dx ≈ 1/2, λ E and indeed this integral can be made as close to 1/2 as desired. To see this, fix  > 0; then by the continuity of the integral (cf. Rudin [43, Theorem 6.11]) we can decompose 0 FR into a finite number of disjoint sets such that the integral of (w(x)−1 /λ)p (x) on each set is less than . We can take E to be the union of some collection of these sets. Now define the function

0

w(x)−p (x) χE . λp0 (x)−1 Then p0 (x) Z  w(x)−1 ρp(·) (f w) = dx ≈ 1/2 < 1. λ E Hence, kf wkp(·) ≤ 1. On the other hand, for all x ∈ Q, p0 (x) Z  Z λ w(x)−1 λ dx ≈ . M f (x) ≥ − f (x) dx = |Q| E λ 2|Q| Q f (x) =

Fix t
0 there exists a set of pairwise disjoint dyadic cubes {Qj } (called the Calder´on-Zygmund (CZ) cubes of f at height λ) such that [ Ωλ = {x ∈ Rn : M f (x) > 4n λ} ⊂ 3Qj . j

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

15

Further, these cubes have the property that for all j, Z Z n λ < − |f (x)| dx ≤ 3 − |f (x)| dx. Qj

3Qj

Lemma 5.2. Given any a > 2n , let {Qkj }, k ∈ Z, be the CZ cubes of a function f ek ⊂ Qk such that the sets at height ak . Then for each j and k there exists a set Q j j ek are pairwise disjoint (for all j and k), and there exists α, 0 < α < 1, such that Q j e |Qkj | ≥ α|Qkj |. The final lemma generalizes the classical inequalities for the Hardy-Littlewood maximal operator. For a proof, see [21, p. 146]. Definition 5.3. Given a weight σ and a locally integrable function f , define the maximal operator Mσ by Z 1 Mσ f (x) = sup |f (y)|σ(y) dy, Q3x σ(Q) Q where the supremum is taken over all cubes containing x. Lemma 5.4. Given a weight σ ∈ A∞ , for 1 < p < ∞ and f ∈ Lp (σ), Z Z p |f (x)|p (x) σ(x) dx. Mσ f (x) σ(x) dx ≤ C Rn

Rn

The constant depends on p, n and σ. Our proof of sufficiency is loosely adapted from the proof of Theorem 1.2 due to Christ and Fefferman [4]. Fix f ; without loss of generality we may assume that f is non-negative and kf wkp(·) = 1. We wantR to form the Calder´on-Zygmund cubes of f ; in order to do so, we need to show that −Q f (y) dy → 0 as |Q| → ∞. By Lemma 3.4, W (x) = w(x)p(x) is in A∞ . Therefore, given cubes Q1 ⊂ Q2 , |Q2 | = r|Q1 |, by Lemma 3.1 we have that for some δ > 0, W (Q2 ) ≥ crδ W (Q1 ). Therefore, it follows that W (Q) → ∞ as |Q| → ∞. Hence, by H¨older’s inequality (Lemma 2.5) the Ap(·) condition and Lemma 2.2, for all cubes sufficiently large Z −1/p+ − f (y) dy ≤ Ckf wkp(·) |Q|−1 kw−1 χQ kp0 (·) ≤ CkwχQ k−1 . p(·) ≤ CW (Q) Q

This gives us the desired limit.

16

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER 0

Define σ(x) = w(x)−p (x) . If we write f = f1 + f2 , where f1 = f χ{f σ−1 >1} and f2 = f χ{f σ−1 ≤1} , then M f ≤ M f1 + M f2 and for i = 1, 2, by Lemma 2.2, Z (5.1) |fi (x)|p(x) w(x)p(x) dx ≤ kfi wkp(·) ≤ kf wkp(·) = 1. Rn

Hence, by Lemma 2.1, to prove the desired inequality it will suffice to show that there exists a constant C depending on p(·) and w such that Z M fi (x)p(x) w(x)p(x) dx ≤ C, i = 1, 2. (5.2) Rn

Estimate (5.2) for f1 . Let a = 4n and for each k ∈ Z let Ωk = {x ∈ Rn : M f1 (x) > ak+1 }. R Since f ∈ L1loc and −Q f (y) dy → 0 as |Q| → ∞, 0 < M f1 (x) < ∞ a.e., and Rn = S k Ωk \ Ωk+1 (up to a set of measure 0). By Lemma 5.1, let {Qkj } be the CZ cubes of f1 at height ak . Then for all k, [ Ωk ⊂ 3Qkj . j

For each k define the sets Ejk inductively: E1k = Ωk \ Ωk+1 ∩ 3Qk1 , E2k = (Ωk \ Ωk+1 ∩ 3Qk2 ) \ E1k , E3k = (Ωk \ Ωk+1 ∩ 3Qk3 ) \ (E1k ∪ E2k ), etc. The sets Ejk are pairwise disjoint S for all j and k and Ωk \ Ωk+1 = j Ejk . We now estimate as follows: Z M f1 (x)p(x) w(x)p(x) dx Rn XZ = M f1 (x)p(x) w(x)p(x) dx Ωk \Ωk+1

k

≤a

2p+

XZ Ωk \Ωk+1

k

≤C

XZ Ejk

k,j

=C

XZ k,j

akp(x) w(x)p(x) dx

Ejk

1 |3Qkj | Z

!p(x)

Z f1 (y) dy

w(x)p(x) dx

3Qkj

!p(x) f1 (y)σ(y)−1 σ(y) dy

3Qkj

Since f1 σ −1 ≥ 1 or f1 σ −1 = 0, by (5.1),

|3Qkj |−p(x) w(x)p(x) dx.

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

Z

17

k

(f1 (y)σ(y)−1 )p(y)/p− (3Qj ) σ(y) dy

3Qkj

Z

−1 p(y)



(f1 (y)σ(y) )

Z σ(y) dy =

3Qkj

f1 (y)p(y) w(y)p(y) dy ≤ 1.

3Qkj

Therefore, we can first raise f1 σ −1 to the power p(y)/p− (3Qkj ) and then decrease the exponent of the whole integral to p− (3Qkj ) to get (5.3) XZ k,j

Ejk



!p(x)

Z

f1 (y)σ(y)−1 σ(y) dy

3Qkj

X Z k,j

(f1 (y)σ(y)−1 )

|3Qkj |−p(x) w(x)p(x) dx

p(y)/p− (3Qkj )

!p− (3Qkj ) Z σ(y) dy

3Qkj

Ejk

|Qkj |−p(x) w(x)p(x) dx.

If we multiply and divide by σ(3Qkj ) and then apply H¨older’s inequality, we get



X k,j

!p− (3Qkj ) Z 1 k (f1 (y)σ(y)−1 )p(y)/p− (3Qj ) σ(y) dy σ(3Qkj ) 3Qkj Z k σ(3Qkj )p− (3Qj ) |Qkj |−p(x) w(x)p(x) dx × Ejk



X k,j

1 σ(3Qkj )

!p −

Z

−1 p(y)/p−

(f1 (y)σ(y) )

σ(y) dy

3Qkj

(5.4) Z × Ejk

k

σ(3Qkj )p− (3Qj ) |Qkj |−p(x) w(x)p(x) dx.

Assume for the moment that Z k (5.5) σ(3Qkj )p− (3Qj ) |Qkj |−p(x) w(x)p(x) dx ≤ Cσ(3Qkj ); Ejk

By Lemma 3.4 applied to w−1 ∈ Ap0 (·) , σ ∈ A∞ , so by Lemmas 3.1 and 5.2, (5.6)

˜ k ). σ(3Qkj ) ≤ Cσ(Qkj ) ≤ Cσ(Q j

18

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

Therefore, (5.4) is bounded by !p − Z X 1 ˜k) C σ(Q (f1 (y)σ(y)−1 )p(y)/p− σ(y) dy j k k σ(3Q ) 3Qj j k,j XZ ≤C Mσ ((f1 σ −1 )p(·)/p− )(x)p− σ(x) dx k,j

Z

˜k Q j

Mσ ((f1 σ −1 )p(·)/p− )(x)p− σ(x) dx;

≤C Rn

by Lemma 5.4 and (5.1), Z

f1 (x)p(x) σ(x)−p(x) σ(x) dx

≤C Rn

Z

f1 (x)p(x) w(x)p(x) dx

=C Rn

≤ C. We will now show that (5.5) holds for all j and k. If p(·) = p is constant, then this reduces to the Ap condition (1.6). We will show that given our hypotheses it is implied by the Ap(·) condition. For brevity, let Q = 3Qkj . Since w ∈ Ap(·) , we have that

(C|Q|)−1 kw−1 χQ kp0 (·) wχQ ≤ 1, p(·) so Z (5.7) Q

p(x)

kw−1 χQ kp0 (·) |Q|−p(x) w(x)p(x) dx ≤ C.

The lefthand side of (5.5) is dominated by  p− (Q) Z σ(Q) p (Q)−p(x) p(x) kw−1 χQ kp−0 (·) kw−1 χQ kp0 (·) |Q|−p(x) w(x)p(x) dx, kw−1 χQ kp0 (·) Q so by (5.7) it will suffice to show that p− (Q)  σ(Q) (5.8) ≤ Cσ(Q) kw−1 χQ kp0 (·) and (5.9)

p (Q)−p(x)

kw−1 χQ kp−0 (·)

≤ C.

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

19

We first show (5.9); we may assume that kw−1 χQ kp0 (·) ≤ 1 since otherwise the inequality is trivial with C = 1. Let (p0 )+ be the supremum of p0 (·) and (p0 )− be the infimum. By the definition of p0 (·), p0 (x) (p0 )+ (Q) (5.10) p(x) − p− (Q) = 0 − p (x) − 1 (p0 )+ (Q) − 1 (p0 )+ (Q) − p0 (x) (p0 )+ (Q) − (p0 )− (Q) = 0 ≤ . [p (x) − 1][(p0 )+ (Q) − 1] [(p0 )− − 1]2 Therefore, by Lemma 3.3 (applied to w−1 ∈ Ap0 (·) ) we have that (5.9) holds. We now prove (5.8). If kw−1 χQ kp0 (·) > 1, then by Lemma 2.2,  p− (Q)  p− (Q) σ(Q) 1−1/(p0 )+ (Q) ≤ σ(Q) = σ(Q). kw−1 χQ kp0 (·) If kw−1 χQ kp0 (·) ≤ 1, then by Lemma 2.2 (applied twice) and Lemma 3.3, p− (Q)   p (Q) σ(Q) (p0 )− (Q)−1 − −1 ≤ kw χQ kp0 (·) kw−1 χQ kp0 (·) p (Q)  (p0 )− (Q)−1+(p0 )+ (Q)−(p0 )− (Q) − −1 ≤ Ckw χQ kp0 (·) p (Q)  (p0 ) (Q)−1 − ≤ C kw−1 χQ kp0 (·)+  p− (Q) (p0 )+ (Q)−1 (p0 )+ (Q) ≤ C σ(Q)  p− (Q) p− (Q)0 −1 0 = C σ(Q) p− (Q) = Cσ(Q). This completes the proof of (5.5) and so the proof of (5.2) for f1 . Estimate (5.2) for f2 . This argument is considerably more technical. We begin with a geometric observation. Since by Lemma 3.4, σ and W = w(·)p(·) are in A∞ , by the properties of A∞ weights given Lemma 3.1, we can find a cube n

P =

2 [

Pi

i=1

that is the union of 2n dyadic cubes adjacent to the origin and such that for each i, |Pi | ≥ C, W (Pi ) ≥ C, and σ(Pi ) ≥ C, where C > 1 is chosen so that if Q is any cube

20

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

adjacent to one of the Pi and the same size, then W (Q), σ(Q) ≥ 1. Below we will make repeated use of the fact that W, σ ∈ A∞ without further reference. We now decompose the integral of M f2 as we did above for M f1 to get, with the same notation as before, Z M f2 (x)p(x) w(x)p(x) dx Rn

≤C

XZ k,j

1 |3Qkj |

Ejk

!p(x)

Z f2 (y) dy

w(x)p(x) dx

3Qkj



 X

=C

X

+

(k,j)∈F

X

+

(k,j)∈G

 (k,j)∈H

= C(I1 + I2 + I3 ), where F = {(k, j) : Qkj ⊂ P }, G = {(k, j) : Qkj 6⊂ P, dist(0, 3Qkj ) = 0}, H = {(k, j) : Qkj 6⊂ P, dist(0, 3Qkj ) > 0}. We estimate each sum in turn, which is also in order of increasing difficulty. ˜ k be the pairwise disjoint sets associated with the Estimate of I1 . As before, let Q j k cubes Qj (by Lemma 5.2). If (k, j) ∈ F, then 3Qkj ⊂ 3P . Therefore, since f2 σ −1 ≤ 1, by inequalities (5.5) and (5.6), !p(x) Z X Z − σ(y) dy w(x)p(x) dx I1 ≤ (k,j)∈F



X Z (k,j)∈F



Ejk

X

Ejk

(1 +

3Qkj k

k

σ(3Qkj )p(x)−p− (3Qj ) σ(3Qkj )p− (3Qj ) |3Qkj |−p(x) w(x)p(x) dx k k σ(3Qkj ))p+ (3Qj )−p− (3Qj )

(k,j)∈F

≤ C(1 + σ(3P ))p+ −p−

X

σ(3Qkj )

(k,j)∈F

≤ C(1 + σ(3P ))p+ −p−

X

˜k) σ(Q j

(k,j)∈F p+ −p−

≤ C(1 + σ(3P ))

σ(3P ).

Z Ejk

k

σ(3Qkj )p− (3Qj ) |3Qkj |−p(x) w(x)p(x) dx

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

21

The last term is a constant depending only on w and p(·) as required. Estimate of I2 . Since the cubes Qkj are dyadic, either dist(Qkj , 0) = 0 or dist(Qkj , 0) ≥ `(Qkj ). Hence, if (k, j) ∈ G, then we must have that for some i, Pi ⊂ 3Qkj . In particular, we have that W (3Qkj ), σ(3Qkj ) ≥ 1. Hence, by Lemmas 3.5 and 3.6 (applied to w−1 ∈ Ap0 (·) ) 0

0

|3Qkj |−1 ≤ C|Pi |−1 σ(Pi )1/p∞ σ(3Qkj )−1/p∞ ≤ Ckw−1 χ3Qkj k−1 p0 (·) .

(5.11)

The final constant C depends on σ and Pi , and so on w and p(·). Hence, by H¨older’s inequality (Lemma 2.5), Z Z −1 −1 f2 (y) dy − f2 (y) dy ≤ Ckw χ3Qkj kp0 (·) 3Qkj

3Qkj

−1 0 ≤ Ckw−1 χ3Qkj k−1 p0 (·) kf2 wkp(·) kw χ3Qkj kp (·) ≤ C.

Therefore, by Lemma 2.7 we can estimate as follows: !p(x) Z X Z w(x)p(x) dx I2 ≤ C C −1 − f2 (y) dy (k,j)∈G

3Qkj

Ejk

X Z

≤ Ct

(k,j)∈G

Z −

Ejk

!p∞ f2 (y) dy

p(x)

w(x)

dx +

3Qkj

X Z (k,j)∈G

Ejk

w(x)p(x) dx. (e + |x|)ntp−

Arguing exactly as in the proof of Lemma 3.4, inequality (3.3), since the sets Ejk are disjoint we can choose t > 1 (depending only on p(·) and w) so that Z X Z w(x)p(x) w(x)p(x) (5.12) dx ≤ dx ≤ 1. ntp− ntp− Ejk (e + |x|) Rn (e + |x|) (k,j)∈G

Therefore, to complete the estimate of I2 we only have to show that the first sum is bounded by a constant. But we have that !p∞ Z X Z − f2 (y) dy w(x)p(x) dx (k,j)∈G

=

Ejk

X (k,j)∈G

3Qkj

1 σ(3Qkj )

Z 3Qkj

!p ∞ f2 (y)σ(y)−1 σ(y) dy

σ(3Qkj ) |3Qkj |

!p∞ W (Ejk ).

22

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

Again by Lemma 3.6 and by the Ap(·) condition, σ(3Qkj )p∞ −1

=

0 σ(3Qkj )p∞ /p∞

≤ Ckw

−1

χ3Qkj kpp∞ 0 (·)

|3Qkj | kwχ3Qkj kp(·)

≤C

!p∞

|3Qkj |p∞ ≤C . W (3Qkj )

Therefore, if we apply this estimate, by Lemmas 5.4 and 2.7 and inequality (5.6), !p∞ !p ∞ Z k X σ(3Q ) 1 j f2 (y)σ(y)−1 σ(y) dy W (Ejk ) σ(3Qkj ) 3Qkj |3Qkj | (k,j)∈G !p ∞ Z X 1 f2 (y)σ(y)−1 σ(y) dy σ(3Qkj )W (3Qkj )−1 W (Ejk ) ≤C k σ(3Qj ) 3Qkj (k,j)∈G

(5.13) 1 σ(3Qkj )

X

≤C

(k,j)∈G

X Z

≤C

(k,j)∈G

Z ≤C

˜k Q j

!p ∞

Z

f2 (y)σ(y)−1 σ(y) dy

3Qkj

˜ kj ) σ(Q

Mσ (f2 σ −1 )(x)p∞ σ(x) dx

Mσ (f2 σ −1 )(x)p∞ σ(x) dx

Rn

Z ≤C

(f2 (x)σ −1 (x))p∞ σ(x) dx

Rn

Z

Z

σ(x) σ(x) dx + dx tnp− Rn Rn (e + |x|) Z Z σ(x) p(x) ≤ Ct f2 (x)w(x) dx + dx. tnp− Rn (e + |x|) Rn ≤ Ct

−1

p(x)

(f2 (x)σ (x))

By inequality (5.1), the first term is bounded by a constant; using an argument identical to that used to prove (5.12), replacing wp(·) by σ, we can find t so that the second term is bounded by 1. This completes the estimate of I2 . Estimate of I3 . If (k, j) ∈ H, because the cubes Qkj are dyadic, since 3Qkj is not adjacent to the origin, dist(3Qkj , 0) ≥ `(Qkj ). Therefore, |x| is essentially constant on 3Qkj : more precisely, there exists a constant R > 1 independent of (k, j) such that (5.14)

sup |x| ≤ R inf |x|. x∈3Qkj

x∈3Qkj

To estimate I3 we actually need to divide H into two subsets: H1 = {(k, j) ∈ H : σ(3Qkj ) ≤ 1},

H2 = {(k, j) ∈ H : σ(3Qkj ) > 1}.

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

23

We will estimate the sum in I3 by first summing over H1 and then over H2 . For the first estimate, we want to apply Lemma 2.8 to replace p(·) by p+ (3Qkj ). Since p(·) is continuous (as a consequence of the LH0 condition) there exists x+ ∈ 3Qkj such that p(x+ ) = p+ (3Qkj ). (More precisely, x+ is in the closure of 3Qkj , but this has no effect since we may replace all the cubes by their closures.) Hence, by the LH∞ condition and (5.14), for all x ∈ 3Qkj , |p+ (3Qkj ) − p(x)| ≤ |p(x+ ) − p∞ | + |p(x) − p∞ | C∞ C∞ C ≤ + ≤ . log(e + |x+ |) log(e + |x|) log(e + |x|) Therefore, we can estimate as follows: by Lemma 2.8 and (5.12), !p(x) Z X Z − f2 (y) dy w(x)p(x) dx (k,j)∈H1

≤ Ct

Ejk

3Qkj

X Z (k,j)∈H1

≤ Ct

Ejk

X Z (k,j)∈H1

!p+ (3Qkj )

Z −

f2 (y) dy

w(x)

p(x)

dx +

3Qkj

(k,j)∈H1

Ejk

w(x)p(x) dx (e + |x|)tnp−

!p+ (3Qkj )

Z −

Ejk

X Z

w(x)p(x) dx + 1.

f2 (y) dy

3Qkj

To estimate the sum, by Lemma 2.9 we have that !p+ (3Qkj ) Z X Z − f2 (y) dy w(x)p(x) dx (k,j)∈H1

Ejk

≤C

3Qkj

X Z (k,j)∈H1

Ejk

1 σ(3Qkj )

Z

!p+ (3Qkj ) f2 (y)σ(y)−1 σ(y) dy

3Qkj

k

× σ(3Qkj )p+ (3Qj ) |3Qkj |−p(x) w(x)p(x) dx; since f2 σ −1 ≤ 1, by Lemma 2.7 we have that ≤C

X Z (k,j)∈H1

Ejk

1 σ(3Qkj ) k

Z

!p∞ f2 (y)σ(y)−1 σ(y) dy

3Qkj

× σ(3Qkj )p+ (3Qj ) |3Qkj |−p(x) w(x)p(x) dx

24

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

X Z

+

(k,j)∈H1

Ejk

k

σ(3Qkj )p+ (3Qj ) |3Qkj |−p(x)

w(x)p(x) dx (e + |x|)tnp−

= J1 + J2 . To estimate J2 we use (5.14) (since Ejk ⊂ 3Qkj ), the fact that σ(3Qkj ) ≤ 1, (5.5) and (5.6) to get Z X k −tnp− σ(3Qkj )p− (3Qj ) |3Qkj |−p(x) w(x)p(x) dx J2 ≤ sup (e + |x|) Ejk

k (k,j)∈H1 x∈Ej

X

≤C

sup (e + |x|)−tnp− σ(3Qkj )

k (k,j)∈H1 x∈Ej

X Z

≤C

(k,j)∈H1

˜k Q j

σ(x) dx (e + |x|)tnp−

≤ C; the last inequality follows as it did above at the end of the estimate for I2 . To estimate J1 we again use the fact that σ(3Qkj ) ≤ 1 and (5.5) to get !p∞ Z X 1 f2 (y)σ(y)−1 σ(y) dy J1 ≤ σ(3Qkj ) 3Qkj (k,j)∈H1 Z k σ(3Qkj )p− (3Qj ) |3Qkj |−p(x) w(x)p(x) dx × Ejk

(5.15)

X

≤C

(k,j)∈H1

1 σ(3Qkj )

Z

!p∞ f2 (y)σ(y)−1 σ(y) dy

3Qkj

σ(3Qkj ).

The final sum is bounded by a constant. We can show this arguing exactly as we did ˜ k ) and then argue as we in the estimate for I2 : use (5.6) to replace σ(3Qkj ) with σ(Q j did from (5.13). This completes the bound for the sum over H1 . Finally, we estimate the sum over H2 . By H¨older’s inequality (Lemma 2.5), Z f2 (y) dy ≤ ckf2 wkp(·) kw−1 χ3Qkj kp0 (·) ≤ ckw−1 χ3Qkj kp0 (·) . 3Qkj

Therefore, by Lemma 2.7, !p(x) Z X Z − f2 (y) dy w(x)p(x) dx (k,j)∈H2

Ejk

3Qkj

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

X Z

≤C

(k,j)∈H2

ckw−1 χ3Qkj k−1 p0 (·)

Ejk

kw−1 χ3Qkj kp0 (·)

× ≤C

(k,j)∈H2

kw−1 χ3Qkj kp0 (·)

×

+

(k,j)∈H2

Ejk

!p∞

Z f2 (y) dy 3Qkj

!p(x)

|3Qkj |

X Z

f2 (y) dy 3Qkj

w(x)p(x) dx

kw−1 χ3Qkj k−1 p0 (·)

Ejk

!p(x)

Z

!p(x)

|3Qkj | X Z

25

w(x)p(x) dx

kw−1 χ3Qkj kp0 (·)

!p(x)

|3Qkj |

w(x)p(x) dx (e + |x|)ntp−

= K1 + K2 . ˜ kj ) >  > 0, where  does To estimate K2 , note that since σ(3Qkj ) ≥ 1, by (5.6), σ(Q not depend on (k, j). Therefore, by (5.7) and (5.14) we have that !p(x) Z X kw−1 χ3Qkj kp0 (·) K2 ≤ sup (e + |x|)−ntp− w(x)p(x) dx k k |3Q | k 3Qj j (k,j)∈H2 x∈Qj X ˜ kj ) ≤C sup (e + |x|)−ntp− σ(Q k (k,j)∈H2 x∈Qj

Z ≤C Rn

σ(x) dx (e + |x|)ntp−

≤ C; the final bound is gotten as in the estimate of J2 above. To estimate K1 , we use Lemma 3.6 to get 0

k p∞ ∞ ≤ Cσ(3Qkj )−p∞ /p∞ +p∞ = Cσ(3Qkj ). kw−1 χ3Qkj k−p p0 (·) σ(3Qj )

Therefore, by (5.7) and (5.6) we have that !p∞ Z X Z 1 K1 = f2 (y) dy σ(3Qkj ) 3Qkj Ejk (k,j)∈H2

k p∞ p(x)−p∞ σ(3Qj ) w(x)p(x) |3Qkj |p(x)

× kw−1 χ3Qkj kp0 (·)

dx

26

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

≤C

X (k,j)∈H2

1 σ(3Qkj ) Z ×

≤C

(k,j)∈H2

1 σ(3Qkj )

σ(3Qkj )

f2 (y) dy 3Qkj

p(x)

3Qkj

X

!p ∞

Z

kw−1 χ3Qkj kp0 (·) |3Qkj |−p(x) w(x)p(x) dx !p ∞

Z

˜ k ). σ(Q j

f2 (y) dy 3Qkj

We can estimate the final term as we did in the estimate for J1 , inequality (5.15). This completes the bound for the sum over H2 , and so gives us the desired estimate for I3 . This completes the estimate for f2 and so the proof of the sufficiency of the Ap(·) condition for the strong-type inequality. The Weak-Type Inequality. The proof of the weak-type inequality is very similar to the proof of the strong-type inequality. In the proof of the latter we use that p− > 1 only to apply the strong-type norm inequality for Mσ . We can readily modify the proof to avoid this. Define f1 and f2 as before. Then for all t, {x ∈ Rn : M f (x) > t} ⊂ {x ∈ Rn : M f1 (x) > t/2} ∪ {x ∈ Rn : M f2 (x) > t/2}. Therefore, it will suffice to prove that f1 and f2 each satisfy the weak-type inequality. We first consider f1 . Fix t > 0 and form the CZ cubes {Qj } of f at height t/4n . Then [ {x ∈ Rn : M f1 (x) > t} ⊂ 3Qj , j

and by Lemma 5.1 we have that Z XZ p(x) p(x) t χ{x∈Q0 :M f1 (x)>t} (x)w(x) dx ≤ C Rn

j

3Qj

!p(x) Z − f1 (y) dy w(x)p(x) dx. Qj

We can then modify the integral of f1 as in (5.3) (here using p− = 1) and then use inequality (5.5) (again with p− = 1), the fact that σ ∈ A∞ , and the fact that the cubes Qj are disjoint to get that !p(x) Z XZ − f1 (y) dy w(x)p(x) dx j

3Qj

≤C

Qj

XZ j

3Qj

1 σ(Qj )

Z Qj

! (f1 (y)σ(y)−1 )p(y) σ(y) dy σ(3Qj )|3Qj |−p(x) w(x)p(x) dx

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp



1 σ(Qj )

X j

Z ≤C

Z

27

! (f1 (y)σ(y)−1 )p(y) σ(y) dy σ(3Qj )

Qj

f1 (y)p(y) w(y)p(y) dy

Rn

≤ C. The estimates for f2 , though longer, can be adapted in exactly the same way to complete the proof of the weak-type inequality. References [1] M. Asif, V. Kokilashvili, and A. Meskhi. Boundedness criteria for maximal functions and potentials on the half-space in weighted Lebesgue spaces with variable exponents. Integral Transforms Spec. Funct., 20(11-12):805–819, 2009. [2] S. M. Buckley. Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc., 340(1):253–272, 1993. [3] C. Capone, D. Cruz-Uribe, and A. Fiorenza. The fractional maximal operator and fractional integrals on variable Lp spaces. Rev. Mat. Iberoam., 23(3):743–770, 2007. [4] M. Christ and R. Fefferman. A note on weighted norm inequalities for the Hardy-Littlewood maximal operator. Proc. Amer. Math. Soc., 87(3):447–448, 1983. [5] D. Cruz-Uribe, L. Diening, and A. Fiorenza. A new proof of the boundedness of maximal operators on variable Lebesgue spaces. Boll. Unione Mat. Ital. (9), 2(1):151–173, 2009. [6] D. Cruz-Uribe, L. Diening, and P. H¨ast¨o. The maximal operator on weighted variable Lebesgue spaces. Frac. Calc. Appl. Anal., 14(3):361–374, 2011. [7] D. Cruz-Uribe and A. Fiorenza. Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Birkh¨ auser, Basel, forthcoming. [8] D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. The maximal function on variable Lp spaces. Ann. Acad. Sci. Fenn. Math., 28(1):223–238, 2003. See also errata [9]. [9] D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. Corrections to: “The maximal function on variable Lp spaces” [Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223–238; ]. Ann. Acad. Sci. Fenn. Math., 29(1):247–249, 2004. [10] D. Cruz-Uribe, J. M. Martell, and C. P´erez. Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture. Adv. Math., 216(2):647– 676, 2007. [11] D. Cruz-Uribe, J. M. Martell, and C. P´erez. Sharp weighted estimates for classical operators. Adv. Math., 229:408–441, 2011. [12] D. Cruz-Uribe, J. M. Martell, and C. P´erez. Weights, extrapolation and the theory of Rubio de Francia, volume 215 of Operator Theory: Advances and Applications. Birkh¨auser/Springer Basel AG, Basel, 2011. [13] L. Diening. Maximal function on generalized Lebesgue spaces Lp(·) . Math. Inequal. Appl., 7(2):245–253, 2004. [14] L. Diening. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129(8):657–700, 2005. [15] L. Diening. Lebesgue and Sobolev spaces with variable exponent. Habilitation, Universit¨ at Freiburg, 2007.

28

D. CRUZ-URIBE, SFO, A. FIORENZA, AND C.J. NEUGEBAUER

[16] L. Diening, P. Harjulehto, P. H¨ ast¨o, and M. R˚ uˇziˇcka. Lebesgue and Sobolev spaces with variable exponents, volume 2017 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011. [17] L. Diening and P. H¨ ast¨ o. Muckenhoupt weights in variable exponent spaces. Preprint, 2010. [18] L. Diening, P. H¨ ast¨ o, and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings (Drabek and Rakosnik (eds.); Milovy, Czech Republic, pages 38–58. Academy of Sciences of the Czech Republic, Prague, 2005. [19] J. Duoandikoetxea. Fourier analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. [20] X. Fan and D. Zhao. On the spaces Lp(x) (Ω) and W m,p(x) (Ω). J. Math. Anal. Appl., 263(2):424– 446, 2001. [21] J. Garc´ıa-Cuerva and J.L. Rubio de Francia. Weighted norm inequalities and related topics, volume 116 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1985. [22] B. Jawerth. Weighted inequalities for maximal operators: linearization, localization and factorization. Amer. J. Math., 108(2):361–414, 1986. [23] M. Khabazi. Maximal operators in weighted Lp(x) spaces. Proc. A. Razmadze Math. Inst., 135:143–144, 2004. [24] V. Kokilashvili and A. Meskhi. Two-weighted inequalities for Hardy-Littlewood functions and singular integrals in Lp(·) spaces. J. Math. Sci., 173(6):656–673, 2011. [25] V. Kokilashvili, N. Samko, and S. Samko. The maximal operator in variable spaces Lp(·) (Ω, ρ) with oscillating weights. Georgian Math. J., 13(1):109–125, 2006. [26] V. Kokilashvili, N. Samko, and S. Samko. The maximal operator in weighted variable spaces Lp(·) . J. Funct. Spaces Appl., 5(3):299–317, 2007. [27] V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted Lp(x) spaces. Proc. A. Razmadze Math. Inst., 129:145–149, 2002. [28] V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted Lp(x) spaces. Rev. Mat. Iberoamericana, 20(2):493–515, 2004. [29] V. Kokilashvili and S. Samko. Operators of harmonic analysis in weighted spaces with nonstandard growth. J. Math. Anal. Appl., 352(1):15–34, 2009. [30] T. Kopaliani. On the Muckenchaupt condition in variable Lebesgue spaces. Proc. A. Razmadze Math. Inst., 148:29–33, 2008. [31] O. Kov´ aˇcik and J. R´ akosn´ık. On spaces Lp(x) and W k,p(x) . Czechoslovak Math. J., 41(116)(4):592–618, 1991. [32] A. K. Lerner. Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces. Math. Z., 251(3):509–521, 2005. [33] A. K. Lerner. An elementary approach to several results on the Hardy-Littlewood maximal operator. Proc. Amer. Math. Soc., 136(8):2829–2833, 2008. [34] A. K. Lerner. On some questions related to the maximal operator on variable Lp spaces. Trans. Amer. Math. Soc., 362(8):4229–4242, 2010. [35] F. Mamedov and Y. Zeren. On a two-weighted estimation of the maximal operator in the Lebesgue space with variable exponent. Ann. Mat. Pura Appl., 190(2):263–275, 2011. [36] F. Mamedov and Y. Zeren. Two-weight inequalities for the maximal operator in a Lebesgue space with variable exponent. J. Math. Sci., 173(6):701–716, 2011. [37] B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972.

WEIGHTED NORM INEQUALITIES ON VARIABLE Lp

29

[38] B. Muckenhoupt and R.L. Wheeden. Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc., 192:261–274, 1974. [39] J. Musielak. Orlicz spaces and modular spaces, volume 1034 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1983. [40] A. Nekvinda. Hardy-Littlewood maximal operator on Lp(x) (Rn ). Math. Inequal. Appl., 7(2):255–265, 2004. ¨ [41] W. Orlicz. Uber konjugierte Exponentenfolgen. Stud. Math., 3:200–211, 1931. [42] L. Pick and M. R˚ uˇziˇcka. An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math., 19(4):369–371, 2001. [43] W. Rudin. Real and complex analysis. McGraw-Hill Book Co., New York, third edition, 1987. [44] S. Samko. On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct., 16(5-6):461–482, 2005. Dept. of Mathematics, Trinity College, Hartford, CT 06106-3100, USA E-mail address: [email protected] ` di Napoli, Via Monteoliveto, 3, I-80134 Dipartimento di Architettura, Universita Napoli, Italy, and Istituto per le Applicazioni del Calcolo ”Mauro Picone”, sezione di Napoli, Consiglio Nazionale delle Ricerche, via Pietro Castellino, 111, I-80131 Napoli, Italy Dept. of Mathematics, Purdue University, West Lafayette, IN 47907-1395 E-mail address: [email protected]