Endpoint Estimates for Commutators of Singular Integral Operators (original) (raw)

David Cruz-Uribe; Alberto Fiorenza
Weighted endpoint estimates for commutators of fractional integrals

Czechoslovak Mathematical Journal, Vol. 57 (2007), No. 1, 153-160

Persistent URL: http://dml.cz/dmlcz/128162

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WEIGHTED ENDPOINT ESTIMATES FOR COMMUTATORS OF FRACTIONAL INTEGRALS

D. Cruz-Uribe, SFO, Hartford, A. Fiorenza, Napoli

(Received November 9, 2004)

Abstract

Given α,0<α<n\alpha, 0<\alpha<n, and b∈BMOb \in \mathrm{BMO}, we give sufficient conditions on weights for the commutator of the fractional integral operator, [b,Iα]\left[b, I_{\alpha}\right], to satisfy weighted endpoint inequalities on Rn\mathbb{R}^{n} and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on Rn\mathbb{R}^{n}.

Keywords: fractional integrals, commutators, BMO, weights, Orlicz spaces, maximal functions

MSC 2000: 42B20, 42B25

1. InTRODUCTION

Given α,0<α<n\alpha, 0<\alpha<n, define the fractional integral operator IαI_{\alpha} by

Iαf(x)=∫Rnf(y)∣x−y∣n−αdyI_{\alpha} f(x)=\int_{\mathbb{R}^{n}} \frac{f(y)}{|x-y|^{n-\alpha}} \mathrm{d} y

for b∈BMOb \in \mathrm{BMO}, define the commutator [b,Iα]\left[b, I_{\alpha}\right] by

[b,Iα]f(x)=b(x)Iαf(x)−Iα(bf)(x)=∫Rn(b(x)−b(y))f(y)∣x−y∣n−αdy\left[b, I_{\alpha}\right] f(x)=b(x) I_{\alpha} f(x)-I_{\alpha}(b f)(x)=\int_{\mathbb{R}^{n}}(b(x)-b(y)) \frac{f(y)}{|x-y|^{n-\alpha}} \mathrm{d} y

Commutators were first introduced by Chanillo [1] who proved LpL^{p} estimates, 1<1< p<∞p<\infty. In [3] we proved the following endpoint estimate:

∣{x∈Rn:∣[b,Iα]f(x)∣>t}∣⩽CΨ(∫RnB(∥b∥BMO∣f(x)∣t)dx)\left|\left\{x \in \mathbb{R}^{n}:\left|\left[b, I_{\alpha}\right] f(x)\right|>t\right\}\right| \leqslant C \Psi\left(\int_{\mathbb{R}^{n}} B\left(\|b\|_{\mathrm{BMO}} \frac{|f(x)|}{t}\right) \mathrm{d} x\right)

where B(t)=tlog⁡(e+t)B(t)=t \log (\mathrm{e}+t) and Ψ(t)=[tlog⁡(e+tα/n)]n/(n−α)\Psi(t)=\left[t \log \left(\mathrm{e}+t^{\alpha / n}\right)\right]^{n /(n-\alpha)}.

We initially conjectured that the corresponding weighted inequality was

w({x∈Rn:∣[b,Iα]f(x)∣>t})⩽CΨ(∫RnB(∥b∥BMO∣f(x)∣t)w(x)1/q dx)w\left(\left\{x \in \mathbb{R}^{n}:\left|\left[b, I_{\alpha}\right] f(x)\right|>t\right\}\right) \leqslant C \Psi\left(\int_{\mathbb{R}^{n}} B\left(\|b\|_{\mathrm{BMO}} \frac{|f(x)|}{t}\right) w(x)^{1 / q} \mathrm{~d} x\right)

where w∈A1w \in A_{1} and q=n/(n−α)q=n /(n-\alpha). However, this conjecture proved to be false, and we gave a counterexample [3, Example 1.8].

At the time, we were not able to make a new conjecture. One difficulty we had was that locally, our original conjecture appeared to be true; the counterexample works because it exploits the decay of the weight at infinity. Careful consideration of this behavior yielded two results. First, we found the correct weighted endpoint inequality for the commutator on Rn\mathbb{R}^{n}. Second, we showed that if we restrict ourselves to a bounded domain, then there is a sharper endpoint inequality which is even simpler than our original conjecture.

Theorem 1.1. Given α,0<α<n\alpha, 0<\alpha<n, and a function b∈BMOb \in \mathrm{BMO}, let B(t)=B(t)= tlog⁡(e+t),Ψ(t)=[tlog⁡(e+tα/n)]n/(n−α),Θ(t)=t1−α/nlog⁡(e+t−α/n)t \log (\mathrm{e}+t), \Psi(t)=\left[t \log \left(\mathrm{e}+t^{\alpha / n}\right)\right]^{n /(n-\alpha)}, \Theta(t)=t^{1-\alpha / n} \log \left(\mathrm{e}+t^{-\alpha / n}\right), and q=q= n/(n−α)n /(n-\alpha). Then for each weight w∈A1w \in A_{1}, there exists a constant CC such that

w({x∈Rn:∣[b,Iα]f(x)∣>t})⩽CΨ(∫RnB(∥b∥BMO∣f(x)∣t)Θ(w(x))dx)w\left(\left\{x \in \mathbb{R}^{n}:\left|\left[b, I_{\alpha}\right] f(x)\right|>t\right\}\right) \leqslant C \Psi\left(\int_{\mathbb{R}^{n}} B\left(\|b\|_{\mathrm{BMO}} \frac{|f(x)|}{t}\right) \Theta(w(x)) \mathrm{d} x\right)

But, given any bounded domain Ω⊂Rn\Omega \subset \mathbb{R}^{n},

w({x∈Ω:∣[b,Iα]f(x)∣>t})⩽C(∫ΩB(∥b∥BMO∣f(x)∣t)w(x)1/q dx)qw\left(\left\{x \in \Omega:\left|\left[b, I_{\alpha}\right] f(x)\right|>t\right\}\right) \leqslant C\left(\int_{\Omega} B\left(\|b\|_{\mathrm{BMO}} \frac{|f(x)|}{t}\right) w(x)^{1 / q} \mathrm{~d} x\right)^{q}

Remark 1.2. If we replace ff by fχΩ,Ωf \chi_{\Omega}, \Omega unbounded, (1.1) yields a nominally more general result for unbounded domains. However, in both (1.1) and (1.2), the A1A_{1} weights are still defined on all of Rn\mathbb{R}^{n}.

Theorem 1.1 is best understood by comparing inequalities (1.1) and (1.2) to the weighted endpoint inequality for the fractional integral operators due to Muckenhoupt and Wheeden [7]. They showed that if w∈A1w \in A_{1}, then

w({x∈Rn:∣Iαf(x)∣>t})⩽C(1t∫Rn∣f(x)∣w(x)1/q dx)qw\left(\left\{x \in \mathbb{R}^{n}:\left|I_{\alpha} f(x)\right|>t\right\}\right) \leqslant C\left(\frac{1}{t} \int_{\mathbb{R}^{n}}|f(x)| w(x)^{1 / q} \mathrm{~d} x\right)^{q}

Intuitively, Theorem 1.1 shows that the commutator [b,Iα]\left[b, I_{\alpha}\right] is more singular than IαI_{\alpha} itself, and that its singularity is worse at infinity.

Theorem 1.1 should also be compared to the analogous result for commutators of singular integral operators (formally corresponding to the case α=0\alpha=0 ) due to Pérez [9]. If TT is a singular integral operator, b∈BMOb \in \mathrm{BMO} and w∈A1w \in A_{1}, then

w({x∈Rn:∣[b,T]f(x)∣>t})⩽C∫RnB(∥b∥BMO∣f(x)∣t)w(x)dxw\left(\left\{x \in \mathbb{R}^{n}:|[b, T] f(x)|>t\right\}\right) \leqslant C \int_{\mathbb{R}^{n}} B\left(\|b\|_{\mathrm{BMO}} \frac{|f(x)|}{t}\right) w(x) \mathrm{d} x

The proof of Theorem 1.1 follows the same outline as the proof of the unweighted result on Rn\mathbb{R}^{n} [3, Theorem 1.1]. The first step is to prove that the associated Orlicz fractional maximal operator,

Mα,Bf(x)=sup⁡Q∋x∣Q∣α/n∥f∥B,QM_{\alpha, B} f(x)=\sup _{Q \ni x}|Q|^{\alpha / n}\|f\|_{B, Q}

satisfies inequalities analogous to (1.1) and (1.2).

Theorem 1.3. With the same notation and hypotheses as in Theorem 1.1, we have that

w({x∈Rn:Mα,Bf(x)>t})⩽CΨ(∫RnB(∣f(x)∣t)Θ(w(x))dx)w\left(\left\{x \in \mathbb{R}^{n}: M_{\alpha, B} f(x)>t\right\}\right) \leqslant C \Psi\left(\int_{\mathbb{R}^{n}} B\left(\frac{|f(x)|}{t}\right) \Theta(w(x)) \mathrm{d} x\right)

and

w({x∈Ω:Mα,Bf(x)>t})⩽C(∫ΩB(∣f(x)∣t)w(x)1/q dx)qw\left(\left\{x \in \Omega: M_{\alpha, B} f(x)>t\right\}\right) \leqslant C\left(\int_{\Omega} B\left(\frac{|f(x)|}{t}\right) w(x)^{1 / q} \mathrm{~d} x\right)^{q}

We prove Theorem 1.3 in Section 2 below; since we can do so with essentially no more work, we prove a generalization that holds for a large class of Young functions.

The proof of Theorem 1.1 now proceeds as in the unweighted case. Here we sketch the main steps, and we refer the reader to [3] for details. On the left-hand side of (1.1) (or (1.2)) we replace ∣[b,Iα]f(x)∣\left|\left[b, I_{\alpha}\right] f(x)\right| by Md([b,Iα]f)(x)M^{d}\left(\left[b, I_{\alpha}\right] f\right)(x), where MdM^{d} is the dyadic maximal operator. We then use the good- λ\lambda inequality relating MdM^{d} and the sharp maximal operator M#M^{\#} (Lemma 6.1 in [3], which remains true in the weighted case since w∈A1)\left.w \in A_{1}\right) to replace this with M#([b,Iα]f)(x)M^{\#}\left(\left[b, I_{\alpha}\right] f\right)(x). Next, we apply the inequality

M#([b,Iα]f)(x)⩽C∥b∥BMO[Iαf(x)+Mα,Bf(x)]M^{\#}\left(\left[b, I_{\alpha}\right] f\right)(x) \leqslant C\|b\|_{\mathrm{BMO}}\left[I_{\alpha} f(x)+M_{\alpha, B} f(x)\right]

(Theorem 1.3 in [3]), which then reduces the estimate to endpoint inequalities for IαI_{\alpha} and Mα,BM_{\alpha, B}. To complete the proof we apply inequality (1.3) and Theorem 1.3, and use the fact that w1/q⩽Θ(w)w^{1 / q} \leqslant \Theta(w).

2. Endpoint results for Orlicz fractional maximal operators

In this section we state and prove two endpoint inequalities for the Orlicz fractional maximal operator. Theorem 1.3 will be an immediate consequence of these results. Hereafter we will assume that the reader is familiar with the basic facts about Orlicz spaces, maximal operators and Muckenhoupt ApA_{p} weights, and we refer the reader to [4],[5],[6],[10][4],[5],[6],[10] for further information. Also, we will draw heavily on our work [3], and we urge the reader to consult that paper.

Remark 2.1. The conclusions of our main results in this section, Theorems 2.3 and 2.5, remain true if α=0\alpha=0. (See Pérez [8].) However, to avoid technical difficulties, we restrict ourselves to α>0\alpha>0.

To state our results, we need one definition.
Definition 2.2. Given an increasing function φ\varphi, define a function hφh_{\varphi} by

hφ(s)=sup⁡t>0φ(st)φ(t),0⩽s<∞h_{\varphi}(s)=\sup _{t>0} \frac{\varphi(st)}{\varphi(t)}, \quad 0 \leqslant s<\infty

The function hφh_{\varphi} could be infinite if s>1s>1, but if φ\varphi is doubling, then it is finite for all 0<s<∞0<s<\infty. (See Maligranda [6, Theorem 11.7].) If φ\varphi is submultiplicative, then hφ≈φh_{\varphi} \approx \varphi. Also, for all s,t>0,φ(st)⩽hφ(s)φ(t)s, t>0, \varphi(s t) \leqslant h_{\varphi}(s) \varphi(t).

Theorem 2.3. Given α,0<α<n\alpha, 0<\alpha<n, let BB be a Young function such that B(t)/tn/αB(t) / t^{n / \alpha} is decreasing for all t>0t>0, and let w∈A1w \in A_{1}. Then there exists a constant CC depending only on BB and the A1A_{1} constant of ww such that for all t>0,Mα,Bt>0, M_{\alpha, B} satisfies the modular weak-type inequality

Φ(w({x∈Rn:Mα,Bf(x)>t}))⩽C∫RnB(f(x)t)hΦ(w(x))dx\Phi\left(w\left(\left\{x \in \mathbb{R}^{n}: M_{\alpha, B} f(x)>t\right\}\right)\right) \leqslant C \int_{\mathbb{R}^{n}} B\left(\frac{f(x)}{t}\right) h_{\Phi}(w(x)) \mathrm{d} x

for all non-negative f∈LB(Rn)f \in L_{B}\left(\mathbb{R}^{n}\right), where

Φ(s)={0 if s=0shB(sα/n) if s>0\Phi(s)= \begin{cases}0 & \text { if } s=0 \\ \frac{s}{h_{B}\left(s^{\alpha / n}\right)} & \text { if } s>0\end{cases}

Inequality (1.4) follows easily from Theorem 2.3. Since BB is submultiplicative, hB≈Bh_{B} \approx B, so

Φ(t)≈t1−α/nlog⁡(e+tα/n)\Phi(t) \approx \frac{t^{1-\alpha / n}}{\log \left(\mathrm{e}+t^{\alpha / n}\right)}

The function Φ\Phi is invertible with

Φ−1(t)≈Ψ(t)=[tlog⁡(e+tα/n)]n/(n−α)\Phi^{-1}(t) \approx \Psi(t)=\left[t \log \left(\mathrm{e}+t^{\alpha / n}\right)\right]^{n /(n-\alpha)}

Thus, inequality (2.1) yields (1.4), provided that

hΦ(t)⩽CΘ(t)=Ct1−α/nlog⁡(e+t−α/n)h_{\Phi}(t) \leqslant C \Theta(t)=C t^{1-\alpha / n} \log \left(\mathrm{e}+t^{-\alpha / n}\right)

However, this follows from the definition: since hBh_{B} is submultiplicative and hB≈Bh_{B} \approx B,

hΦ(s)=sup⁡t>0Φ(st)Φ(t)=ssup⁡t>0hB(tα/n)hB((st)α/n)⩽shB(s−α/n)⩽CsB(s−α/n)=CΘ(s)h_{\Phi}(s)=\sup _{t>0} \frac{\Phi(s t)}{\Phi(t)}=s \sup _{t>0} \frac{h_{B}\left(t^{\alpha / n}\right)}{h_{B}\left((s t)^{\alpha / n}\right)} \leqslant s h_{B}\left(s^{-\alpha / n}\right) \leqslant C s B\left(s^{-\alpha / n}\right)=C \Theta(s)

Remark 2.4. Note that if we let t=s−1t=s^{-1} and use the fact that hB(t)⩾cB(t)h_{B}(t) \geqslant c B(t), we get hΦ(s)⩾cΘ(s)h_{\Phi}(s) \geqslant c \Theta(s). Hence, Θ\Theta is the best possible function we can get by this means.

Theorem 2.5. Given a bounded domain Ω⊂Rn\Omega \subset \mathbb{R}^{n} and given α,0<α<n\alpha, 0<\alpha<n, let BB be a Young function such that B(t)/tn/αB(t) / t^{n / \alpha} is decreasing for all t>0t>0. Suppose further that there exists r,1⩽r⩽n/αr, 1 \leqslant r \leqslant n / \alpha, such that hB(t)⩽Ctrh_{B}(t) \leqslant C t^{r} for all t⩽6diam⁡(Ω)αt \leqslant 6 \operatorname{diam}(\Omega)^{\alpha}. Then for each w∈A1w \in A_{1} there exists a constant CC, depending on B,diam⁡(Ω)B, \operatorname{diam}(\Omega) and the A1A_{1} constant of ww, such that for all non-negative f∈LB(Ω)f \in L_{B}(\Omega),

w({x∈Ω:Mα,Bf(x)>t})1−rα/n⩽C∫ΩB(f(x)t)w(x)1−rα/n dxw\left(\left\{x \in \Omega: M_{\alpha, B} f(x)>t\right\}\right)^{1-r \alpha / n} \leqslant C \int_{\Omega} B\left(\frac{f(x)}{t}\right) w(x)^{1-r \alpha / n} \mathrm{~d} x

Inequality (1.5) is an immediate consequence of Theorem 2.5. Since B(t)=B(t)= tlog⁡(e+t),hB≈Bt \log (\mathrm{e}+t), h_{B} \approx B, so if t⩽6diam⁡(Ω)t \leqslant 6 \operatorname{diam}(\Omega), then hB(t)⩽Clog⁡(e+6diam⁡(Ω))th_{B}(t) \leqslant C \log (\mathrm{e}+6 \operatorname{diam}(\Omega)) t. Thus inequality (2.3) holds with r=1r=1, and this yields (1.5).

Proof of Theorem 2.3. Our proof is very similar to the proof of Theorem 3.3 in [3], and we refer the reader there for many lemmas and technical details.

Fix a non-negative function ff and t>0t>0. Define Et={x∈Rn:Mα,Bf(x)>t}E_{t}=\left\{x \in \mathbb{R}^{n}: M_{\alpha, B} f(x)>t\right\}.
For each x∈Etx \in E_{t} there exists a cube Qx∋xQ_{x} \ni x such that

∣Qx∣α/n∥f∥B,Qx>t\left|Q_{x}\right|^{\alpha / n}\|f\|_{B, Q_{x}}>t

The collection {Qx}x∈Et\left\{Q_{x}\right\}_{x \in E_{t}} covers EtE_{t}. Thus, by Lemma 3.14 in [3], there exists β>0\beta>0 and a collection of disjoint dyadic cubes {Pj}\left\{P_{j}\right\} such that Et⊂⋃j3PjE_{t} \subset \bigcup_{j} 3 P_{j} and

∣Pj∣α/n∥f∥B,Pj>βt\left|P_{j}\right|^{\alpha / n}\|f\|_{B, P_{j}}>\beta t

By the properties of the Luxemburg norm on Orlicz spaces and by Definition 2.2,

1<1∣Pj∣∫PjB(∣Pj∣α/nf(x)βt)dx⩽ChB(∣3Pj∣α/n)∣3Pj∣∫PjB(f(x)t)dx=CΦ(∣3Pj∣)∫PjB(f(x)t)dx\begin{aligned} 1 & <\frac{1}{\left|P_{j}\right|} \int_{P_{j}} B\left(\frac{\left|P_{j}\right|^{\alpha / n} f(x)}{\beta t}\right) \mathrm{d} x \\ & \leqslant \frac{C h_{B}\left(\left|3 P_{j}\right|^{\alpha / n}\right)}{\left|3 P_{j}\right|} \int_{P_{j}} B\left(\frac{f(x)}{t}\right) \mathrm{d} x \\ & =\frac{C}{\Phi\left(\left|3 P_{j}\right|\right)} \int_{P_{j}} B\left(\frac{f(x)}{t}\right) \mathrm{d} x \end{aligned}

The growth conditions assumed on BB imply (see Lemma 3.12 in [3]) that

Φ(w(Et))⩽Φ(∑jw(3Pj))⩽∑jΦ(w(3Pj))\Phi\left(w\left(E_{t}\right)\right) \leqslant \Phi\left(\sum_{j} w\left(3 P_{j}\right)\right) \leqslant \sum_{j} \Phi\left(w\left(3 P_{j}\right)\right)

Hence, if we combine the two inequalities above and apply Definition 2.2, we get

Φ(w(Et))⩽C∑jΦ(w(3Pj))Φ(∣3Pj∣)∫PjB(f(x)t)dx⩽C∑jhΦ(w(3Pj)∣3Pj∣)∫PjB(f(x)t)dx\begin{aligned} \Phi\left(w\left(E_{t}\right)\right) & \leqslant C \sum_{j} \frac{\Phi\left(w\left(3 P_{j}\right)\right)}{\Phi\left(\left|3 P_{j}\right|\right)} \int_{P_{j}} B\left(\frac{f(x)}{t}\right) \mathrm{d} x \\ & \leqslant C \sum_{j} h_{\Phi}\left(\frac{w\left(3 P_{j}\right)}{\left|3 P_{j}\right|}\right) \int_{P_{j}} B\left(\frac{f(x)}{t}\right) \mathrm{d} x \end{aligned}

since w∈A1w \in A_{1} and the PjP_{j} 's are disjoint,

Φ(w(Et))⩽C∑j∫PjB(f(x)t)hΦ(w(x))dx⩽C∫RnB(f(x)t)hΦ(w(x))dx\begin{aligned} \Phi\left(w\left(E_{t}\right)\right) & \leqslant C \sum_{j} \int_{P_{j}} B\left(\frac{f(x)}{t}\right) h_{\Phi}(w(x)) \mathrm{d} x \\ & \leqslant C \int_{\mathbb{R}^{n}} B\left(\frac{f(x)}{t}\right) h_{\Phi}(w(x)) \mathrm{d} x \end{aligned}

This completes the proof.
The proof of Theorem 2.5 requires two lemmas.

Lemma 2.6. Given α,0<α<n\alpha, 0<\alpha<n, let BB be a Young function such that B(t)/tn/αB(t) / t^{n / \alpha} is decreasing for all t>0t>0. Then hB(s)⩽sn/αh_{B}(s) \leqslant s^{n / \alpha} for all s⩾1s \geqslant 1.

Proof. Fix s⩾1s \geqslant 1. Then for all t>0t>0,

B(st)(st)n/α⩽B(t)tn/α, or equivalently, B(st)B(t)⩽sn/α\frac{B(s t)}{(s t)^{n / \alpha}} \leqslant \frac{B(t)}{t^{n / \alpha}}, \quad \text { or equivalently, } \quad \frac{B(s t)}{B(t)} \leqslant s^{n / \alpha}

Taking the supremum over all tt we get the desired inequality.

Lemma 2.7. Given α,0<α<n\alpha, 0<\alpha<n, let BB be a Young function such that B(t)/tn/αB(t) / t^{n / \alpha} is decreasing for all t>0t>0. If QQ and Qˉ\bar{Q} are cubes and ff is a function such that supp⁡(f)⊂Q⊂Qˉ\operatorname{supp}(f) \subset Q \subset \bar{Q}, then

∣Qˉ∣α/n∥f∥B,Qˉ⩽∣Q∣α/n∥f∥B,Q|\bar{Q}|^{\alpha / n}\|f\|_{B, \bar{Q}} \leqslant|Q|^{\alpha / n}\|f\|_{B, Q}

Proof. Let s=∣Qˉ∣/∣Q∣⩾1s=|\bar{Q}| /|Q| \geqslant 1. Then by the definition of the Luxemburg norm and by Lemma 2.6,

∣Qˉ∣α/n∥f∥B,Qˉ=∣Q∣α/n∥sα/nf∥B,Qˉ=∣Q∣α/ninf⁡{λ>0:1∣Qˉ∣∫QˉB(sα/n∣f(x)∣λ)dx⩽1}⩽∣Q∣α/ninf⁡{λ>0:1∣Qˉ∣∫QhB(sα/n)B(∣f(x)∣λ)dx⩽1}⩽∣Q∣α/ninf⁡{λ>0:1∣Qˉ∣∫QB(∣f(x)∣λ)dx⩽1}=∣Q∣α/n∥f∥B,Q\begin{aligned} |\bar{Q}|^{\alpha / n}\|f\|_{B, \bar{Q}} & =|Q|^{\alpha / n}\left\|s^{\alpha / n} f\right\|_{B, \bar{Q}} \\ & =|Q|^{\alpha / n} \inf \left\{\lambda>0: \frac{1}{|\bar{Q}|} \int_{\bar{Q}} B\left(\frac{s^{\alpha / n}|f(x)|}{\lambda}\right) \mathrm{d} x \leqslant 1\right\} \\ & \leqslant|Q|^{\alpha / n} \inf \left\{\lambda>0: \frac{1}{|\bar{Q}|} \int_{Q} h_{B}\left(s^{\alpha / n}\right) B\left(\frac{|f(x)|}{\lambda}\right) \mathrm{d} x \leqslant 1\right\} \\ & \leqslant|Q|^{\alpha / n} \inf \left\{\lambda>0: \frac{1}{|\bar{Q}|} \int_{Q} B\left(\frac{|f(x)|}{\lambda}\right) \mathrm{d} x \leqslant 1\right\} \\ & =|Q|^{\alpha / n}\|f\|_{B, Q} \end{aligned}

P r o of of Theorem 2.5. The proof of this result is nearly the same as the proof of Theorem 2.3. The major difference is that we must show that we can restrict the size of the cubes used to compute Mα,BM_{\alpha, B}. Fix x∈Ωx \in \Omega, and let QQ be any cube containing xx. If ℓ(Q)⩾diam⁡(Ω)\ell(Q) \geqslant \operatorname{diam}(\Omega), then we can maximize the quantity ∣Q∣α/n∥f∥B,Q|Q|^{\alpha / n}\|f\|_{B, Q} by taking QQ to be such that Ω⊂Q\Omega \subset Q. Further, by Lemma 2.7 we can increase this quantity by choosing QQ such that ℓ(Q)⩽diam⁡(Ω)\ell(Q) \leqslant \operatorname{diam}(\Omega). Consequently, in computing Mα,BM_{\alpha, B} we can restrict ourselves to cubes whose sidelength is at most the diameter of Ω\Omega.

Thus, in the proof of Theorem 2.3, when we cover the set EtE_{t} by cubes QxQ_{x}, we may assume that each of these cubes has sidelength bounded by diam⁡(Ω)\operatorname{diam}(\Omega). We then use Lemma 3.14 in [3] to replace these cubes by a collection of dyadic cubes PjP_{j}. But from the proof of this lemma (see [2], [8]) we see that the sidelength of each PjP_{j} is bounded by twice the largest sidelength of the QxQ_{x} 's; therefore ℓ(Pj)⩽2diam⁡(Ω)\ell\left(P_{j}\right) \leqslant 2 \operatorname{diam}(\Omega) for all jj.

By assumption, hB(t)⩽Ctrh_{B}(t) \leqslant C t^{r} for t⩽6diam⁡(Ω)αt \leqslant 6 \operatorname{diam}(\Omega)^{\alpha}. Therefore, hB(∣3Pj∣α/n)⩽h_{B}\left(\left|3 P_{j}\right|^{\alpha / n}\right) \leqslant ∣3Pj∣rα/n\left|3 P_{j}\right|^{r \alpha / n} for all jj. Hence, we can modify the argument that yielded (2.4) to get

1<C∣3Pj∣1−rα/n∫PjB(f(x)t)dx1<\frac{C}{\left|3 P_{j}\right|^{1-r \alpha / n}} \int_{P_{j}} B\left(\frac{f(x)}{t}\right) \mathrm{d} x

If we now let Φ(t)=t1−rα/n\Phi(t)=t^{1-r \alpha / n}, then hΦ(t)=Φ(t)h_{\Phi}(t)=\Phi(t) and we can repeat the remainder of the argument in the proof of Theorem 2.3 to get (2.3).

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Authors’ addresses: D. Cruz-Uribe, SFO, Dept. of Mathematics, Trinity College, Hartford, CT 06106-3100, USA, e-mail: david.cruzuribe@mail.trincoll.edu; A. Fiorenza, Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli, Via Monteoliveto, 3, I-80134 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, e-mail: fiorenza@unina.it.