Advertisement

Collectanea Mathematica

, Volume 68, Issue 1, pp 129–144 | Cite as

Two weight inequalities for bilinear forms

  • Kangwei Li
Article

Abstract

Let \(1\le p_0<p,q <q_0\le \infty \). Given a pair of weights \((w,\sigma )\) and a sparse family \({\mathcal {S}}\), we study the two weight inequality for the following bi-sublinear form
$$\begin{aligned} B(f, g)= \sum _{Q\in {\mathcal {S}}}\langle |f|^{p_0}\rangle _Q^{\frac{1}{p_0}} \langle |g|^{q_0'}\rangle _Q^{\frac{1}{q_0'}}\lambda _Q\le \mathcal N\Vert f\Vert _{L^{p}(w)}\Vert g\Vert _{L^{q'}(\sigma )}. \end{aligned}$$
When \(\lambda _Q=|Q|\) and \(p=q\), Bernicot, Frey and Petermichl showed that B(fg) dominates \(\langle Tf, g\rangle \) for a large class of singular non-kernel operators. We give a characterization for the above inequality and then obtain the mixed \(A_p-A_\infty \) estimates and the corresponding entropy bounds when \(\lambda _Q=|Q|\) and \(p=q\). We also propose a new conjecture which implies both the one supremum conjecture and the separated bump conjecture.

Keywords

\(A_p-A_\infty \) estimates Two weight theorem One supremum estimate Separated bump conjecture 

Mathematics Subject Classification

42B25 

1 Introduction and main results

The weighted theory for Calderón-Zygmund operators has achieved several advances in the last decades. The quantitative relation between the weighted bound of the operator and the \(A_p\) characteristics has attracted many authors’ interest. The climax of this topic is the settle of the \(A_2\) conjecture, which was due to Hytönen [9]. Lerner [17] also gave a simple proof for it by reducing the problem to study the so-called sparse operators. We refer the readers to [9, 17] and the reference therein for an overview of this topic.

Later on, Hytönen and Lacey [11] extended the \(A_2\) theorem to the mixed \(A_p-A_\infty \) estimate, and proposed the famous one supremum conjecture (which will be recalled below), which is in the theme of “finding the minimal sufficient condition such that the two weight inequality holds”. As far as we know, this conjecture is still open. Another problem in the same theme is the so-called separated bump conjecture (which will be recalled in Sect. 5), which arised from work of Cruz-Uribe and Pérez [4, 5], and Cruz-Uribe et al. [6]. The separated bump conjecture was just verified for the log-bumps [1, 6]. In general, it is still unknown, see also in [14] for more details.

Recently, Bernicot et al. [2] studied the weighted theory beyond Calderón-Zygmund theory. They gave the sharp weighted estimates for a large class of singular non-kernel operators by proving a domination theorem. To be precise, they showed that, if T is bounded on \(L^2\) and satisfies some cancellation property and Cotlar type inequality, then for fg supported in \(5Q_0\) for some cube \(Q_0\), there exists some sparse family \({\mathcal {S}}_0\) such that
$$\begin{aligned} \left| \int _{Q_0} Tf \cdot g d\mu \right| \le C \sum _{P\in {\mathcal {S}}_0}\mu (P) \langle |f|^{p_0}\rangle _{5P}^{1/{p_0}}\langle |g|^{q_0'}\rangle _{5P}^{1/{q_0'}}, \end{aligned}$$
(1.1)
where
$$\begin{aligned} \langle h\rangle _Q=\frac{1}{\mu (Q)}\int _Q h(x) d\mu , \end{aligned}$$
and \(1\le p_0<q_0\le \infty \). Recall that we say \({\mathcal {S}}\) is sparse if for any \(Q\in {\mathcal {S}}\),
$$\begin{aligned} \mu \left( \bigcup _{ Q' \in {\mathcal {S}}, Q'\subsetneq Q }Q' \right) \le \frac{1}{2} \mu (Q). \end{aligned}$$
The above result is in the setting of locally compact separable metric space equipped with a doubling Borel measure \(\mu \), finite on compact sets and strictly positive on any nonempty open set. In the following, we simply use |Q| stands for \(\mu (Q)\).
It is well-known that there exists a finite collection of adjacent dyadic systems \(\mathcal D_k\), \(k=1,\ldots , K\), such that for any cube Q, there exists some \(1\le k_0\le K\) such that \(5Q\subset \tilde{Q}\in \mathcal D_{k_0}\) and \(|\tilde{Q}|\eqsim |Q|\) (e.g. see [10]). Then RHS(1.1) can be dominated by
$$\begin{aligned} \int _{\mathbb {R}^n} \sum _{k=1}^K\sum _{Q\in \mathcal D_k} \langle |f|^{p_0}\rangle _{ Q}^{1/{p_0}}\langle |g|^{q_0'}\rangle _{Q}^{1/{q_0'}}\sum _{\begin{array}{c} P\in {\mathcal {S}}_0\\ \tilde{P}=Q \end{array}} \mathbf 1_{P}(x) dx. \end{aligned}$$
Then following similar arguments as that in [7], it suffices to consider the following bi-sublinear form
$$\begin{aligned} B(f, g)= \sum _{Q\in {\mathcal {S}}}\langle |f|^{p_0}\rangle _Q^{\frac{1}{p_0}} \langle |g|^{q_0'}\rangle _Q^{\frac{1}{q_0'}}\lambda _Q, \end{aligned}$$
where \({\mathcal {S}}\) is a sparse family (here we generalize |Q| to general sequence \(\lambda _Q\)). This is the main object in this paper.

Our first result concern the characterization of two weight norm inequality for the bilinear form.

Theorem 1.1

Let \((w,\sigma )\) be a pair of weights and \(p_0<p,q<q_0\). Suppose that \(\mathcal N\) is the best constant such that the following two weight inequality holds
$$\begin{aligned} B(f,g)\le \mathcal N \Vert f\Vert _{L^p(w)}\Vert g\Vert _{L^{q'}(\sigma )}. \end{aligned}$$
(1.2)
Denote
$$\begin{aligned} \tau _Q= \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{-\frac{1}{q_0}} \frac{\lambda _Q}{|Q|},\,\, u:= w^{\frac{p_0}{p_0-p}},\,\, v:=\sigma ^{\frac{q_0'}{q_0'-q'}}, \end{aligned}$$
and
$$\begin{aligned} T_F(f)=\sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset F \end{array}} \tau _Q \langle f \rangle _Q \mathbf 1_Q. \end{aligned}$$
Then we have
$$\begin{aligned} \mathcal N \eqsim \mathfrak {T}_s+\mathfrak {T}_s^*, \end{aligned}$$
where \(s\in (1,\infty ]\) is determined by
$$\begin{aligned} \frac{1}{s}=\left( \frac{1}{q}-\frac{1}{p}\right) _+ :=\max \left\{ \frac{1}{q}-\frac{1}{p},0 \right\} , \end{aligned}$$
(1.3)
and
$$\begin{aligned} \begin{aligned} \mathfrak {T}_s&:=\sup _{\mathcal {F}}\left\| \left\{ \frac{\Vert T_{F}(u)\Vert _{L^{q}(v)}}{u(F)^{1/p}} \right\} _{F\in {\mathcal {F}}} \right\| _{\ell ^s}, \\ \mathfrak {T}_s^*&:=\sup _{\mathcal {G}} \left\| \left\{ \frac{\Vert T_{G}(v)\Vert _{L^{p'}(u)}}{v(G)^{1/{q'}}} \right\} _{G\in \mathcal G} \right\| _{\ell ^s}, \end{aligned} \end{aligned}$$
(1.4)
and the supremums are taken over all subcollections \({\mathcal {F}}\) and \(\mathcal {G}\) of \(\mathcal {S}\) that are sparse with respect to u and v, respectively.

For the case of \(p_0=1, q_0=\infty \), this result is already known in [15] for \(p\le q\) and [21] for \(p>q\), see also in [8] for an unified approach for both cases. In this sense, our result extends from the special case \(p_0=1, q_0=\infty \) to general cases. Next, we are concerned with the mixed \(A_p-A_\infty \) type estimate.

Theorem 1.2

Let \((w,\sigma )\) be a pair of weights, \(\lambda _Q=|Q|\), \(p=q\) and \(\mathcal N\) be the best constant such that (1.2) holds. Then
$$\begin{aligned} \mathcal N\lesssim [v]_{A_r}^{\frac{1}{q_0'}-\frac{1}{p'}} \left( [u]_{A_\infty }^{\frac{1}{p}}+ [v]_{A_\infty }^{\frac{1}{p'}} \right) , \end{aligned}$$
where
$$\begin{aligned} r=\left( \frac{q_0}{p} \right) ' \left( \frac{p}{p_0}-1 \right) +1. \end{aligned}$$
We point out Theorem 1.2 improves the main result in [2]. Indeed, notice that Bernicot, Frey and Petermichl showed that, if \(\sigma =w^{1-p'}\), i.e., \(u=v^{1-r'}\), then
$$\begin{aligned} \mathcal N\lesssim [v]_{A_r}^{\max \{\frac{1}{q_0'}, \frac{1}{p_0(r-1)}\}}. \end{aligned}$$
Since
$$\begin{aligned}{}[u]_{A_\infty }^{\frac{1}{p}}\le [u]_{A_{r'}}^{\frac{1}{p}}=[v]_{A_r}^{\frac{1}{p(r-1)}},\quad \,\,\text{ and }\quad \,\, [v]_{A_\infty }^{\frac{1}{p'}}\le [v]_{A_r}^{\frac{1}{p'}}, \end{aligned}$$
we have
$$\begin{aligned}{}[v]_{A_r}^{\frac{1}{q_0'}-\frac{1}{p'}} \left( [u]_{A_\infty }^{\frac{1}{p}}+ [v]_{A_\infty }^{\frac{1}{p'}} \right) \le [v]_{A_r}^{\frac{1}{p_0(r-1)}}+ [v]_{A_r}^{\frac{1}{q_0'}}\le 2[v]_{A_r}^{\max \left\{ \frac{1}{q_0'}, \frac{1}{p_0(r-1)} \right\} }. \end{aligned}$$
Finally we study the one supremum estimate. In [11], Hytönen and Lacey proposed the following one supremum conjecture, which is an extension of the \(A_p-A_\infty \) estimate.

Conjecture 1.1

Let T be a Calderón-Zygmund operator, \(w,\sigma \) be weights and \(1<p<\infty \). Then
$$\begin{aligned} \Vert T(\cdot \sigma )\Vert _{L^p(\sigma )\rightarrow L^p(w)}\lesssim \sup _Q \langle w\rangle _Q^{\frac{1}{p}}\langle \sigma \rangle _Q^{\frac{1}{p'}}(A_\infty (w,Q)^{\frac{1}{p'}}+A_\infty (\sigma ,Q)^{\frac{1}{p}}), \end{aligned}$$
where for any weight v,
$$\begin{aligned} A_\infty (v,Q):= \frac{1}{v(Q)}\int _Q M(v\chi _Q)(x) dx. \end{aligned}$$
Here we give some partial answer to this conjecture for the general indices \(1\le p_0<q_0\le \infty \). Define
$$\begin{aligned} A_r(v,u,Q)= \langle v\rangle _Q^{\frac{1}{p}-\frac{1}{q_0}}\langle u\rangle _Q^{ \left( \frac{1}{p}-\frac{1}{q_0} \right) (r-1)}, \end{aligned}$$
and
$$\begin{aligned}{}[v,u]_{(A_r) (\phi (A_\infty )) }&:= \sup _Q A_r(v,u,Q)(A_\infty (v,Q))^{\frac{1}{p'}} \phi (A_\infty (v,Q)); \\ [u, v]_{(A_{r'}) (\psi (A_\infty )) }&:= \sup _Q A_{r'}(u,v,Q)(A_\infty (u,Q))^{\frac{1}{p}} \psi (A_\infty (u,Q)). \end{aligned}$$
We have the following one supremum estimate

Theorem 1.3

Let \((w,\sigma )\) be a pair of weights, \(\lambda _Q=|Q|\), \(p=q\) and \(\mathcal N\) be the best constant such that (1.2) holds. Let \(\phi ,\psi \) be increasing functions such that
$$\begin{aligned} \int _{1/2}^\infty \left( \frac{1}{ \phi (t) } + \frac{1}{ \psi (t)}\right) \frac{dt}{t}<\infty . \end{aligned}$$
Then
$$\begin{aligned} \mathcal N\lesssim [v,u]_{(A_r) (\phi (A_\infty )) }+ [u,v]_{(A_{r'}) (\psi (A_\infty )) }, \end{aligned}$$
For the case of \(p_0=1\) and \(q_0=\infty \), this was shown by Lacey and Spencer in [16], which was referred to as the separated entropy bounds. Now consider another definition of \(A_\infty \) weights, namely,
$$\begin{aligned}{}[w]_{A_\infty }^{exp }:= \sup _{Q} \langle w\rangle _Q \exp ( \langle \log w^{-1}\rangle _Q ). \end{aligned}$$
It is showed in [13] that
$$\begin{aligned}{}[w]_{A_\infty }\le c_n [w]_{A_\infty }^{exp }. \end{aligned}$$
We shall show that, if we replace \(A_{\infty }\) with \(A_{\infty }^{exp }\), then we can relax the decay condition of \(\phi \) and \(\psi \) slightly. To be precise, define
$$\begin{aligned} A_\infty ^{exp } (v,Q):= \langle v\rangle _Q \exp ( \langle \log v^{-1}\rangle _Q ) \end{aligned}$$
and
$$\begin{aligned}{}[v,u]_{(A_r) (\Phi (A_\infty ^{exp })) }&:= \sup _Q A_r(v,u,Q) (A_\infty ^{exp }(v,Q))^{\frac{1}{p'}} \Phi (A_\infty ^{exp } (v,Q)); \\ [u, v]_{(A_{r'}) (\Psi (A_\infty ^{exp })) }&:= \sup _Q A_{r'}(u,v,Q) (A_\infty ^{exp }(u,Q))^{\frac{1}{p}} \Psi (A_\infty ^{exp } (u,Q)). \end{aligned}$$
We have the following result

Theorem 1.4

Let \((w,\sigma )\) be a pair of weights, \(\lambda _Q=|Q|\), \(p=q\) and \(\mathcal N\) be the best constant such that (1.2) holds. Let \(\Phi ,\Psi \) be increasing functions such that
$$\begin{aligned} \int _{1/2}^\infty \left( \frac{1}{\Phi (t)^{ p }} + \frac{1}{\Psi (t)^{ p' }} \right) \frac{dt}{t}<\infty . \end{aligned}$$
Then
$$\begin{aligned} \mathcal N\lesssim [v,u]_{(A_r) (\Phi (A_\infty ^{exp })) }+ [u,v]_{(A_{r'}) (\psi (A_\infty ^{exp })) }, \end{aligned}$$

The organization of the paper will be as follows. In Sect. 2, we present the proof of Theorem 1.1. In Sect. 3, we prove Theorem 1.2. We study the one supremum estimate in Sect. 4, which contains the proof of Theorems 1.3 and 1.4. And we end this manuscript with a generalized question in Sect. 5.

2 A characterization of the two weight norm inequalities for bilinear forms

In this section, we give a proof for Theorem 1.1. Recall that \(u= w^{\frac{p_0}{p_0-p}}\) and \(v=\sigma ^{\frac{q_0'}{q_0'-q'}}\). We can write the two weight inequality as follows
$$\begin{aligned} B(f, g)= \sum _{Q\in {\mathcal {S}}}\langle |f|^{p_0}\rangle _Q^{\frac{1}{p_0}} \langle |g|^{q_0'}\rangle _Q^{\frac{1}{q_0'}}\lambda _Q\le C \Vert |f|^{p_0} u^{-1}\Vert _{L^{p/{p_0}}(u)}^{1/{p_0}}\Vert |g|^{q_0'} v^{-1}\Vert _{L^{q'/{q_0'}}(v)}^{1/{q_0'}}, \end{aligned}$$
which is equivalent to
$$\begin{aligned} \sum _{Q\in {\mathcal {S}}}(\langle |f|^{p_0}\rangle _Q^u)^{\frac{1}{p_0}} (\langle |g|^{q_0'}\rangle _Q^v)^{\frac{1}{q_0'}}\lambda _Q \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}} \le C \Vert |f|^{p_0} \Vert _{L^{p/{p_0}}(u)}^{1/{p_0}}\Vert |g|^{q_0'} \Vert _{L^{q'/{q_0'}}(v)}^{1/{q_0'}}, \end{aligned}$$
(2.1)
where for any function h and weight w,
$$\begin{aligned} \langle h\rangle _Q^w:= \frac{1}{w(Q)}\int _Q h d w. \end{aligned}$$
Follow the spirit in [12], we claim that (2.1) is equivalent to
$$\begin{aligned} \sum _{Q\in {\mathcal {S}}} \langle f\rangle _Q^u \langle g\rangle _Q^v \lambda _Q \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}} \le C \Vert f \Vert _{L^{p }(u)} \Vert g \Vert _{L^{q' }(v)} . \end{aligned}$$
(2.2)
In fact, if (2.1) holds, then (2.2) follows immediately from Hölder’s inequality. On the other hand, if (2.2) holds, then we have
$$\begin{aligned}&\sum _{Q\in {\mathcal {S}}}(\langle |f|^{p_0}\rangle _Q^u)^{\frac{1}{p_0}} (\langle |g|^{q_0'}\rangle _Q^v)^{\frac{1}{q_0'}}\lambda _Q \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}}\\&\quad \le \sum _{Q\in {\mathcal {S}}} \langle M_{p_0, u}^{{\mathcal {S}}}(f)\rangle _Q^u \langle M_{q_0', v}^{{\mathcal {S}}}(g)\rangle _Q^v\lambda _Q \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}}\\&\quad \le C \Vert M_{p_0, u}^{{\mathcal {S}}}(f) \Vert _{L^{p }(u)} \Vert M_{q_0', v}^{{\mathcal {S}}}(g) \Vert _{L^{q' }(v)} \\&\quad \le C_{p_0,p,q_0,n} C \Vert f \Vert _{L^{p }(u)} \Vert g \Vert _{L^{q' }(v)}, \end{aligned}$$
where
$$\begin{aligned} M_{p,u}^{{\mathcal {S}}}(f):=\sup _{Q\in {\mathcal {S}}} (\langle |f|^{p}\rangle _Q^u)^{\frac{1}{p}}. \end{aligned}$$
Then by duality, it suffices to give a characterization for the following two weight norm inequality
$$\begin{aligned} \Vert T_\tau (fu)\Vert _{L^{q}(v)}\le C\Vert f\Vert _{L^p(u)}, \end{aligned}$$
(2.3)
where
$$\begin{aligned} T_\tau (f )= \sum _{Q\in {\mathcal {S}}} \tau _Q \langle f \rangle _Q \mathbf 1_Q \end{aligned}$$
and recall that
$$\begin{aligned} \tau _Q= \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{-\frac{1}{q_0}} \frac{\lambda _Q}{|Q|}. \end{aligned}$$
The case of \(p\le q\) was due to Lacey et al. [15], and the case of \(p>q\) was given by Tanaka [21]. Here we follow the unified testing for both cases given by Hänninen, Hytönen et al. [8].

Proposition 2.1

[8, Theorem 1.5] Let \(p,q\in (1,\infty )\) and \(w,\sigma \) be two measures. Let
$$\begin{aligned} \begin{aligned} T(f\sigma )&:=\sum _{Q\in \mathcal {D}}\lambda _Q\int _Q f d\sigma \cdot 1_Q,\qquad \lambda _Q\ge 0,\\ T_{Q}(f\sigma )&:=\sum _{\begin{array}{c} Q'\in \mathcal {D}\\ Q'\subseteq Q \end{array}}\lambda _{Q'}\int _{Q'} f d\sigma \cdot 1_{Q'}. \end{aligned} \end{aligned}$$
(2.4)
We have
$$\begin{aligned} \Vert T (\cdot \sigma )\Vert _{L^p(\sigma )\rightarrow L^q(w)} \eqsim \mathfrak {T}_s+\mathfrak {T}_s^*, \end{aligned}$$
where \(s\in (1,\infty ]\) is determined by
$$\begin{aligned} \frac{1}{s}= \left( \frac{1}{q}-\frac{1}{p} \right) _+ :=\max \left\{ \frac{1}{q}-\frac{1}{p},0 \right\} . \end{aligned}$$
(2.5)
and
$$\begin{aligned} \begin{aligned} \mathfrak {T}_s&:=\sup _{\mathcal {F}} \left\| \left\{ \frac{\Vert T_{F}(\sigma )\Vert _{L^{q}(w)}}{\sigma (F)^{1/p}} \right\} _{F\in {\mathcal {F}}} \right\| _{\ell ^s}, \\ \mathfrak {T}_s^*&:=\sup _{\mathcal {G}} \left\| \left\{ \frac{\Vert T_{G}(w)\Vert _{L^{p'}(\sigma )}}{w(G)^{1/{q'}}} \right\} _{G\in \mathcal G} \right\| _{\ell ^s}, \end{aligned} \end{aligned}$$
(2.6)
where the supremums are taken over all subcollections \({\mathcal {F}}\) and \(\mathcal {G}\) of \(\mathcal {S}\) that are sparse with respect to \(\sigma \) and \(\omega \), respectively.

Now combining (2.3) and Proposition 2.1, Theorem 1.1 follows immediately.

3 Mixed \(A_p-A_\infty \) type estimate for bilinear forms

In this section, we focus on the sharp constant for the case \(p=q\). In this case, the testing condition degenerate to the Sawyer type testing condition. Namely,
$$\begin{aligned} \mathcal N\eqsim \sup _{R\in {\mathcal {S}}} \frac{ \Vert T_R (v) \Vert _{L^{p'}(u)}}{v(R)^{ 1/{p'} }} +\sup _{R\in {\mathcal {S}}} \frac{\Vert T_R(u)\Vert _{L^p(v)}}{u(R)^{ 1/p}}. \end{aligned}$$
(3.1)
Before further estimates, we introduce the following proposition.

Proposition 3.1

Let \({\mathcal {S}}\) be a sparse family and \(0\le \gamma , \eta <1\) satisfying \(\gamma +\eta <1\). Then
$$\begin{aligned} \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}}\langle u \rangle _Q^\gamma \langle v\rangle _Q^\eta |Q| \lesssim \langle u \rangle _R^\gamma \langle v\rangle _R^\eta |R|. \end{aligned}$$

Proof

Indeed, set \(1/r:=\gamma +\eta \), \(1/s:=\gamma +(1-1/r)/2\) and \(1/{s'}:=1-1/{s}\). Denote
$$\begin{aligned} E_Q:=Q\setminus \bigcup _{Q'\in {\mathcal {S}}, Q'\subsetneq Q} Q'. \end{aligned}$$
By sparseness and Kolmogorov’s inequality, we have
$$\begin{aligned} \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}}\langle u \rangle _Q^\gamma \langle v\rangle _Q^\eta |Q|&\le 2 \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}}\langle u \rangle _Q^\gamma \langle v\rangle _Q^\eta |E_Q|\\&\le 2 \int _{R} M(u\mathbf 1_R)^{\gamma } M(v\mathbf 1_R)^{\eta }dx\\&\le 2 \left( \int _R M(u\mathbf 1_R)^{s\gamma } \right) ^{1/s}\left( \int _R M(v\mathbf 1_R)^{s'\eta } \right) ^{1/{s'}}\\&\lesssim \langle u\rangle _R^\gamma |R|^{1/s} \langle v\rangle _R^{\eta }|R|^{1/{s'}}\\&=\langle u \rangle _R^\gamma \langle v\rangle _R^\eta |R|. \end{aligned}$$
\(\square \)

We also need the following result

Proposition 3.2

[3, Proposition 2.2] Let \(1<s<\infty \), \(\sigma \) be a positive Borel measure and
$$\begin{aligned} \phi =\sum _{Q\in \mathcal D} \alpha _Q \mathbf 1_Q,\qquad \phi _Q=\sum _{Q'\subset Q}\alpha _{Q'} \mathbf 1_{Q'}. \end{aligned}$$
Then
$$\begin{aligned} \Vert \phi \Vert _{L^s(\sigma )}\eqsim \left( \sum _{Q\in \mathcal D} \alpha _Q (\langle \phi _Q\rangle _Q^\sigma )^{s-1}\sigma (Q) \right) ^{1/s}. \end{aligned}$$

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2

As that in [2], we denote by \(\rho \) the critical index \(1+ p_0/{q_0'}\). First, we consider the case \(p\ge \rho \). set
$$\begin{aligned} \alpha = \frac{1}{p_0(r-1)}\min \{p-p_0, 1\}. \end{aligned}$$
We can check that
$$\begin{aligned} \frac{1}{p_0}- (r-1)\alpha \ge 0,\,\, \frac{1}{q_0'}-\alpha \ge 0,\,\,\text{ and }\,\, \frac{1}{p_0}- (r-1)\alpha +\frac{1}{q_0'}-\alpha <1. \end{aligned}$$
By Propositions 3.2 and 3.1, we have
$$\begin{aligned}&\Vert T_R(v) \Vert _{L^{p'}(u)}=\left\| \sum _{\begin{array}{c} Q\in \mathcal S\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} \mathbf 1_Q \right\| _{L^{p'}(u)} \\&\quad \eqsim \left( \sum _{\begin{array}{c} Q\in \mathcal S\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)}\sum _{Q'\subset Q} \langle u\rangle _{Q'}^{-\frac{1}{p_0'}} \langle v\rangle _{Q'}^{\frac{1}{q_0'}} u(Q') \right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad \le [v, u]_{A_r}^{\frac{\alpha }{p} } \left( \sum _{\begin{array}{c} Q\in \mathcal S\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)}\sum _{Q'\subset Q} \langle u\rangle _{Q'}^{\frac{1}{p_0}-(r-1)\alpha } \langle v\rangle _{Q'}^{\frac{1}{q_0'}-\alpha } |Q'|\right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad \lesssim [v, u]_{A_r}^{\frac{\alpha }{p } } \left( \sum _{\begin{array}{c} Q\in \mathcal S\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)} \langle u\rangle _Q^{\frac{1}{p_0}-(r-1)\alpha } \langle v\rangle _{Q }^{\frac{1}{q_0'}-\alpha } |Q | \right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad = [v, u]_{A_r}^{\frac{\alpha }{p } } \left( \sum _{\begin{array}{c} Q\in \mathcal S\\ Q\subset R \end{array}} \langle u\rangle _Q^{\frac{1}{p_0}-(p'-1)((r-1)\alpha +\frac{1}{p_0'})} \langle v\rangle _Q^{\frac{1}{q_0'}+(\frac{1}{q_0'}-\alpha )(p'-1)} |Q| \right) ^{\frac{1}{p'}}\\&\quad \le [v, u]_{A_r}^{\frac{\alpha }{p }+\frac{1}{p'p_0(r-1)}- (\frac{\alpha }{p}+\frac{1}{pp_0'(r-1)}) } \left( \sum _{\begin{array}{c} Q\in \mathcal S\\ Q\subset R \end{array}}v(Q) \right) ^{\frac{1}{p'}} \\&\quad \lesssim [v, u]_{A_r}^{\frac{1}{q_0'}-\frac{1}{p'}}[v]_{A_\infty }^{\frac{1}{p'}} v(R)^{\frac{1}{p'}}, \end{aligned}$$
For the case \(p<\rho \), set
$$\begin{aligned} \tilde{\alpha }= \left( \frac{1}{q_0'}-\frac{1}{p'} \right) p. \end{aligned}$$
Again, we can check that
$$\begin{aligned} \frac{1}{q_0'}-\tilde{\alpha }= & {} \frac{1}{q_0'}(p'-1)(p-1)-\frac{1}{q_0'}(p'-q_0')(p-1)\ge 0,\\&\frac{1}{p_0}-\tilde{\alpha }(r-1)\ge 0, \end{aligned}$$
and
$$\begin{aligned} \frac{1}{p_0}-\tilde{\alpha }(r-1)+ \frac{1}{q_0'}-\tilde{\alpha }= 1-(p-1) \left( \frac{1}{p_0}-\frac{1}{q_0} \right) <1. \end{aligned}$$
By Propositions 3.2 and 3.1 again, we have
$$\begin{aligned}&\Vert T_R(v) \Vert _{L^{p'}(u)}\eqsim \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)}\sum _{Q'\subset Q} \langle u\rangle _{Q'}^{\frac{1}{p_0}} \langle v\rangle _{Q'}^{\frac{1}{q_0'}} |Q'|\right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad \le [v,u]_{A_r}^{\frac{\tilde{\alpha }}{p}} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)}\sum _{Q'\subset Q} \langle u\rangle _{Q'}^{\frac{1}{p_0}-\tilde{\alpha }(r-1)} \langle v\rangle _{Q'}^{\frac{1}{q_0'}-\tilde{\alpha }} |Q'| \right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad \lesssim [v,u]_{A_r}^{\frac{\tilde{\alpha }}{p}} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)} \langle u\rangle _{Q }^{\frac{1}{p_0}-\tilde{\alpha }(r-1)} \langle v\rangle _{Q }^{\frac{1}{q_0'}-\tilde{\alpha }} |Q |\right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad = [v,u]_{A_r}^{\frac{\tilde{\alpha }}{p}} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}} \langle u\rangle _Q^{\frac{1}{p_0}- (p'-1)(\frac{1}{p_0'}+\tilde{\alpha }(r-1))} \langle v\rangle _Q^{\frac{1}{q_0'}+(p'-1)(\frac{1}{q_0'}-\tilde{\alpha })} |Q| \right) ^{\frac{1}{p'}}\\&\quad =[v,u]_{A_r}^{\frac{1}{q_0'}-\frac{1}{p'}} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}\\ Q\subset R \end{array}}v(Q) \right) ^{\frac{1}{p'}}\\&\quad \lesssim [v, u]_{A_r}^{\frac{1}{q_0'}-\frac{1}{p'}}[v]_{A_\infty }^{\frac{1}{p'}} v(R)^{\frac{1}{p'}}. \end{aligned}$$
By symmetry, we have
$$\begin{aligned} \Vert T_R(u) \Vert _{L^{p }(v)}&\lesssim [u, v]_{A_{r'}}^{\frac{1}{p_0}-\frac{1}{p}}[u]_{A_\infty }^{\frac{1}{p}} u(R)^{\frac{1}{p}}\\&= [v, u]_{A_r}^{\frac{1}{q_0'}-\frac{1}{p'}}[u]_{A_\infty }^{\frac{1}{p}} u(R)^{\frac{1}{p}}. \end{aligned}$$
Then the desired estimate follows from (3.1) immediately. \(\square \)

4 Mixed \(A_p-A_\infty \) type estimates with one supremum

In this section, we study the one supremum estimate. And this could be done by just slightly modify the arguments in the previous section. We first prove Theorem 1.3.

Proof of Theorem 1.3

By symmetry, we only need to estimate \(\Vert T_R(v) \Vert _{L^{p'}(u)}\). Denote
$$\begin{aligned} {\mathcal {S}}_a=\{Q\in {\mathcal {S}}: 2^a\le A_\infty (v,Q)< 2^{a+1}\}. \end{aligned}$$
We will abuse of using the notations T and \( T_R(v)\) slightly, which is now understood as summation over \({\mathcal {S}}_a\) instead of \({\mathcal {S}}\). We still consider the case \(p\ge \rho \) first, we have
$$\begin{aligned}&\Vert T_R(v) \Vert _{L^{p'}(u)} \eqsim \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}_a\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)}\sum _{Q'\subset Q} \langle u\rangle _{Q'}^{\frac{1}{p_0}} \langle v\rangle _{Q'}^{\frac{1}{q_0'}} |Q'|\right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad \le \left( \frac{[v,u]_{(A_r) (\phi (A_\infty )) }}{2^{\frac{a}{p'}}\phi (2^a)}\right) ^{\frac{\alpha q_0}{q_0-p}} \\&\quad \quad \times \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}_a\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q) \left( \frac{1}{u(Q)}\times \sum _{Q'\subset Q} \langle u\rangle _{Q'}^{\frac{1}{p_0}-(r-1)\alpha } \langle v\rangle _{Q'}^{\frac{1}{q_0'}-\alpha } |Q'|\right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad \lesssim \left( \frac{[v,u]_{(A_r) (\phi (A_\infty )) }}{2^{\frac{a}{p'}}\phi (2^a)}\right) ^{\frac{\alpha q_0}{q_0-p}}\\&\quad \quad \times \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}_a\\ Q\subset R \end{array}} \langle u\rangle _Q^{-\frac{1}{p_0'}} \langle v\rangle _Q^{\frac{1}{q_0'}} u(Q)\times \left( \frac{1}{u(Q)} \langle u\rangle _Q^{\frac{1}{p_0}-(r-1)\alpha } \langle v\rangle _{Q }^{\frac{1}{q_0'}-\alpha } |Q | \right) ^{p'-1} \right) ^{\frac{1}{p'}}\\&\quad = \left( \frac{[v,u]_{(A_r) (\phi (A_\infty )) }}{2^{\frac{a}{p'}}\phi (2^a)} \right) ^{\frac{\alpha q_0}{q_0-p}} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}_a\\ Q\subset R \end{array}} \langle u\rangle _Q^{\frac{1}{p_0}-(p'-1)((r-1)\alpha +\frac{1}{p_0'})} \langle v\rangle _Q^{\frac{1}{q_0'}+(\frac{1}{q_0'}-\alpha )(p'-1)} |Q| \right) ^{\frac{1}{p'}}\\&\quad \le \frac{[v,u]_{(A_r) (\phi (A_\infty )) }}{2^{\frac{a}{p'}}\phi (2^a)} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}_a\\ Q\subset R \end{array}}v(Q) \right) ^{\frac{1}{p'}} \\&\quad \lesssim \frac{1}{\phi (2^a) } [v,u]_{(A_r) (\phi (A_\infty )) } v(R)^{\frac{1}{p'}}. \end{aligned}$$
For the case \(p<\rho \), similar arguments show that
$$\begin{aligned} \Vert T_R(v) \Vert _{L^{p'}(u)}\lesssim \frac{1}{\phi (2^a) } [v,u]_{(A_r) (\phi (A_\infty )) } v(R)^{\frac{1}{p'}}. \end{aligned}$$
It remains to sum over a, by definition, \(a \ge 0\), we have
$$\begin{aligned} \sup _{R\in {\mathcal {S}}} \frac{ \Vert T_R (v) \Vert _{L^{p'}(u)}}{v(R)^{ 1/{p'} }}&\lesssim \sum _{a\ge 0 }\frac{1}{\phi (2^a) } [v,u]_{(A_r) (\phi (A_\infty )) }\\&\lesssim \sum _{a\ge 0}\int _{2^{a-1}}^{2^a}\frac{1}{\phi (t) }\frac{dt}{t}[v,u]_{(A_r) (\phi (A_\infty )) }\\&= \int _{1/2}^{\infty } \frac{1}{\phi (t) }\frac{dt}{t}[v,u]_{(A_r) (\phi (A_\infty )) }. \end{aligned}$$
This completes the proof. \(\square \)

Next we prove Theorem 1.4.

Proof of Theorem 1.4

We follow the same strategy as that in the proof of Theorem 1.3. Set
$$\begin{aligned} {\mathcal {S}}_a=\{Q\in {\mathcal {S}}: 2^a\le A_\infty ^{exp }(v,Q)< 2^{a+1}\}. \end{aligned}$$
We haveThis completes the proof. \(\square \)

5 Further discussions

In this section, we propose a new conjecture which implies both the one supremum conjecture (with \(A_\infty \) replaced by \(A_\infty ^{exp }\)) and the separated bump conjecture. On the other hand, it shares many similar results as the later two problems. To be precise, define
$$\begin{aligned}{}[u,v]_{A, q_0,p,r}&= \sup _{Q}\langle v\rangle _Q^{ \frac{1}{p}-\frac{1}{q_0} } \langle u\rangle _Q^{ \left( \frac{1}{p}-\frac{1}{q_0} \right) (r-1)} \frac{\langle u\rangle _Q^{\frac{1}{p}}}{\langle u^{\frac{1}{p}}\rangle _{A, Q}},\\ [v,u]_{B, p_0',p',r'}&= \sup _{Q}\langle u\rangle _Q^{\frac{1}{p'}-\frac{1}{p_0'}} \langle v\rangle _Q^{ \left( \frac{1}{p'}-\frac{1}{p_0'} \right) (r'-1)} \frac{\langle v\rangle _Q^{\frac{1}{p'}}}{\langle v^{\frac{1}{p'}}\rangle _{B, Q}}, \end{aligned}$$
where \(A\in B_p\) and \(B\in B_{p'}\). Recall that we say a Young function A belongs to \(B_p\) if
$$\begin{aligned} \int _{1/2}^\infty \frac{A(t)}{t^p}\frac{dt}{t}<\infty , \end{aligned}$$
and the Luxembourg norm \(\langle f\rangle _{A,Q}\) is defined by
$$\begin{aligned} \langle f\rangle _{A,Q}:=\inf \{\lambda >0: \langle A(f/\lambda )\rangle _{Q}\le 1 \}. \end{aligned}$$
We propose the following conjecture.

Conjecture 5.1

Let \((w,\sigma )\) be a pair of weights, \(\lambda _Q=|Q|\), \(p=q\) and \(\mathcal N\) be the best constant such that (1.2) holds. Let \(A\in B_p\) and \(B\in B_{p'}\). Then there holds
$$\begin{aligned} \mathcal N \le C ([u,v]_{A, q_0,p,r}+[v,u]_{B, p_0', p',r'}), \end{aligned}$$
where the constant \(C>0\) is independent of w and \(\sigma \).

Now we shall see that this conjecture implies both the one supremum conjecture and also the separated bump conjecture. For general indices \(p_0\) and \(q_0\), the separated bump conjecture can be stated as follows

Conjecture 5.2

Let \((w,\sigma )\) be a pair of weights, \(\lambda _Q=|Q|\), \(p=q\) and \(\mathcal N\) be the best constant such that (1.2) holds. Let \(A\in B_p\) and \(B\in B_{p'}\). Then there holds
$$\begin{aligned} \mathcal N \le C ((u,v)_{A, q_0,p,r}+(v,u)_{B, p_0', p',r'}), \end{aligned}$$
where the constant \(C>0\) is independent of w and \(\sigma \) and
$$\begin{aligned} (u,v)_{A, q_0,p,r}&= \sup _{Q}\langle v\rangle _Q^{ \frac{1}{p}-\frac{1}{q_0} } \langle u\rangle _Q^{ \left( \frac{1}{p}-\frac{1}{q_0} \right) (r-1)} \frac{\langle u^{\frac{1}{p'}}\rangle _{\bar{A}, Q}}{\langle u\rangle _Q^{\frac{1}{p'}}},\\ (v,u)_{B, p_0',p',r'}&= \sup _{Q}\langle u\rangle _Q^{\frac{1}{p'}-\frac{1}{p_0'}} \langle v\rangle _Q^{ \left( \frac{1}{p'}-\frac{1}{p_0'} \right) (r'-1)} \frac{\langle v^{\frac{1}{p}}\rangle _{\bar{B}, Q}}{\langle v\rangle _Q^{\frac{1}{p}}}. \end{aligned}$$
Indeed, by general Hölder’s inequality,
$$\begin{aligned} \frac{\langle u\rangle _Q^{\frac{1}{p}}}{\langle u^{\frac{1}{p}}\rangle _{A, Q}}\le \frac{ \langle u^{\frac{1}{p'}}\rangle _{\bar{A}, Q}}{\langle u \rangle _{ Q}^{\frac{1}{p'}}},\,\, \frac{\langle v\rangle _Q^{\frac{1}{p'}}}{\langle v^{\frac{1}{p'}}\rangle _{B, Q}}\le \frac{ \langle v^{\frac{1}{p}}\rangle _{\bar{B}, Q}}{\langle v \rangle _{ Q}^{\frac{1}{p}}}, \end{aligned}$$
which means Conjecture 5.1 implies separated bump conjecture. On the other hand, by Jensen’s inequality,Then
$$\begin{aligned} \frac{\langle u\rangle _Q^{\frac{1}{p}}}{\langle u^{\frac{1}{p}}\rangle _{A, Q}}\le A_\infty ^{exp }(u,Q)^{\frac{1}{p}}, \end{aligned}$$
which means Conjecture 5.1 implies also the one supremum conjecture.
Although Conjecture 5.1 is stronger than both separated bump conjecture and one supremum conjecture, it still contain the essential property as the separated bump conjecture and one supremum conjecture. Indeed, we are free to write
$$\begin{aligned} \langle \sigma ^{\frac{1}{p}}\rangle _{A, Q}=\langle \sigma \rangle _{\tilde{A}, Q}^{\frac{1}{p}}, \end{aligned}$$
where \(\tilde{A}(t)= A(t^{1/p})\). Since \(M_{A}^{\mathcal D}\) is bounded on \(L^p\) (see [20]), we have \(M_{\tilde{A}}^{\mathcal D}\) is bounded on \(L^1\), and this is the key point. We shall see this by showing the following result.

Theorem 5.1

Let \((w,\sigma )\) be a pair of weights, \(\lambda _Q=|Q|\), \(p=q\) and \(\mathcal N\) be the best constant such that (1.2) holds. Let \(A\in B_p\), \(B\in B_{p'}\) and \(\phi ,\psi \) be increasing functions such that
$$\begin{aligned} \int _{1/2}^\infty \left( \frac{1}{\phi (t)^p } + \frac{1}{\psi (t)^{p'}}\right) \frac{dt}{t}<\infty . \end{aligned}$$
Then
$$\begin{aligned} \mathcal N\lesssim [u,v]_{A, q_0,p,r,\psi }+ [v,u]_{B, p_0',p',r',\phi }, \end{aligned}$$
where
$$\begin{aligned}{}[u,v]_{A,q_0,p,r,\psi }&= \sup _{Q}\langle v\rangle _Q^{ \frac{1}{p}-\frac{1}{q_0} } \langle u\rangle _Q^{ \left( \frac{1}{p}-\frac{1}{q_0} \right) (r-1)} \frac{\langle u\rangle _Q^{\frac{1}{p}}}{\langle u^{\frac{1}{p}}\rangle _{A, Q}}\psi \left( \frac{\langle u\rangle _Q^{\frac{1}{p}}}{\langle u^{\frac{1}{p}}\rangle _{A, Q}} \right) ,\\ [v,u]_{B, p_0',p',r',\phi }&= \sup _{Q}\langle u\rangle _Q^{\frac{1}{p'}-\frac{1}{p_0'}} \langle v\rangle _Q^{ \left( \frac{1}{p'}-\frac{1}{p_0'} \right) (r'-1)} \frac{\langle v\rangle _Q^{\frac{1}{p'}}}{\langle v^{\frac{1}{p'}}\rangle _{B, Q}}\phi \left( \frac{\langle v\rangle _Q^{\frac{1}{p'}}}{\langle v^{\frac{1}{p'}}\rangle _{B, Q}} \right) . \end{aligned}$$

Proof

The proof is quite similar to the proof of Theorem 1.4. Set
$$\begin{aligned} {\mathcal {S}}_a:=\{Q\in {\mathcal {S}}: 2^a\le \frac{\langle v\rangle _Q^{\frac{1}{p'}}}{\langle v^{\frac{1}{p'}}\rangle _{B, Q}}< 2^{a+1}\}. \end{aligned}$$
Then we have
$$\begin{aligned} \Vert T_R(v) \Vert _{L^{p'}(u)}\lesssim & {} \sum _{a\ge 0} \frac{[v,u]_{B, p_0',p',r',\phi }}{2^{ a }\phi (2^a)} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}_a\\ Q\subset R \end{array}}v(Q) \right) ^{\frac{1}{p'}}\\\le & {} [v,u]_{B, p_0',p',r',\phi }\sum _{a\ge 0} \frac{1}{\phi (2^a)} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}}_a\\ Q\subset R \end{array}}\langle v^{\frac{1}{p'}}\rangle _{B, Q}^{p'} |Q| \right) ^{\frac{1}{p'}}\\\le & {} [v,u]_{B, p_0',p',r',\phi } \left( \sum _{a\ge 0} \frac{1}{\phi (2^a)^p} \right) ^{\frac{1}{p}} \left( \sum _{\begin{array}{c} Q\in {\mathcal {S}} \\ Q\subset R \end{array}}\langle v^{\frac{1}{p'}}\rangle _{B, Q}^{p'} |Q| \right) ^{\frac{1}{p'}}\\\lesssim & {} [v,u]_{B, p_0',p',r',\phi } v(R)^{\frac{1}{p'}} \Vert M_B\Vert _{L^{p'}\rightarrow L^{p'}} \left( \int _{1/2}^\infty \frac{1}{\phi (t)^{ p}}\frac{dt}{t} \right) ^{\frac{1}{p}}. \end{aligned}$$
Then the desired result follows immediately from Theorem 1.1. \(\square \)

Remark 5.1

In [14], Lacey showed the same result with \(\langle u\rangle _Q^{\frac{1}{p}}/\langle u^{\frac{1}{p}}\rangle _{A, Q}\) replaced by \(\langle u^{\frac{1}{p'}}\rangle _{\bar{A}, Q}/\langle u\rangle _Q^{\frac{1}{p'}}\) and analogously for v when \(p_0=1\) and \(q_0=\infty \) (Recall that in this case, \(u=\sigma \) and \(v=w\)). And therefore, our estimate improves Lacey’s bound. On the other hand, it is easy to see that it improves Theorem 1.4 as well.

We also have the following result.

Theorem 5.2

Let \((w,\sigma )\) be a pair of weights, \(\lambda _Q=|Q|\), \(p=q\) and \(\mathcal N\) be the best constant such that (1.2) holds. Let \(A\in B_p\), \(B\in B_{p'}\). Then
$$\begin{aligned} \mathcal N\lesssim [u,v]_{A, B}, \end{aligned}$$
where
$$\begin{aligned}{}[u,v]_{A, B}= \sup _{Q}\langle v\rangle _Q^{ \frac{1}{p}-\frac{1}{q_0} } \langle u\rangle _Q^{ \left( \frac{1}{p}-\frac{1}{q_0} \right) (r-1)} \frac{\langle u\rangle _Q^{\frac{1}{p}}}{\langle u^{\frac{1}{p}}\rangle _{A, Q}}\cdot \frac{\langle v\rangle _Q^{\frac{1}{p'}}}{\langle v^{\frac{1}{p'}}\rangle _{B, Q}}. \end{aligned}$$

Proof

Instead of using Proposition 3.2, we shall use the technique of parallel stopping cubes here. We define the principal cubes \({\mathcal {F}}\) for (fu) as follows
$$\begin{aligned} {\mathcal {F}}:= & {} \bigcup _{k=0}^\infty {\mathcal {F}}_k, \quad {\mathcal {F}}_0:= \{\textit{maximal cubes in }{\mathcal {S}}\}\\ {\mathcal {F}}_{k+1}:= & {} \bigcup _{F\in {\mathcal {F}}_k}\mathrm {ch}_{\mathcal {F}}(F),\quad \mathrm {ch}_{\mathcal {F}}(F):= \{ Q\subsetneq F\, \textit{maximal s.t.} \langle f\rangle _Q^u>2\langle f\rangle _F^u \}, \end{aligned}$$
and analogously define \(\mathcal G\) for (gv). We also denote by \(\pi _{\mathcal {F}} (Q)\) the minimal cube in \({\mathcal {F}}\) which contains Q, and \(\pi (Q)=(F, G)\) if \(\pi _{\mathcal {F}}(Q)=F\) and \(\pi _{\mathcal G}(Q)=G\). By construction, we have
$$\begin{aligned} \sum _{F\in {\mathcal {F}}} (\langle f\rangle _F^u)^p u(F) \lesssim \Vert f\Vert _{L^p(u)}^p. \end{aligned}$$
(5.1)
Now we start our arguments from (2.2). We have
$$\begin{aligned} \sum _{Q\in {\mathcal {S}}} \langle f\rangle _Q^u \langle g\rangle _Q^v \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}}|Q|&= \sum _{F\in {\mathcal {F}}}\sum _{G\in \mathcal G}\sum _{\pi (Q)=(F,G)} \langle f\rangle _Q^u \langle g\rangle _Q^v \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}}|Q|\\&\lesssim \sum _{F\in {\mathcal {F}}} \langle f\rangle _F^u \sum _{\begin{array}{c} G\in \mathcal G\\ G\subset F \end{array}} \langle g\rangle _G^v \sum _{\pi (Q)=(F,G)} \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}}|Q|\\&+ \sum _{G\in \mathcal G} \langle g\rangle _G^v \sum _{\begin{array}{c} F\in {\mathcal {F}}\\ F\subset G \end{array}} \langle f\rangle _F^u\sum _{\pi (Q)=(F,G)} \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}}|Q|\\&:= I+II. \end{aligned}$$
By symmetry, we only need to estimate I. We have
$$\begin{aligned}&\sum _{\pi (Q)=(F,G)} \langle u\rangle _Q^{\frac{1}{p_0}} \langle v\rangle _Q^{\frac{1}{q_0'}}|Q|\\&\quad =\sum _{\pi (Q)=(F,G)} \langle u\rangle _Q^{\frac{1}{p_0}-\frac{1}{p}} \langle v\rangle _Q^{\frac{1}{q_0'}-\frac{1}{p'}}\frac{\langle u\rangle _Q^{\frac{1}{p}}}{\langle u^{\frac{1}{p}}\rangle _{A, Q}}\cdot \frac{\langle v\rangle _Q^{\frac{1}{p'}}}{\langle v^{\frac{1}{p'}}\rangle _{B, Q}}\langle u^{\frac{1}{p}}\rangle _{A, Q}\cdot \langle v^{\frac{1}{p'}}\rangle _{B, Q} |Q|\\&\quad \le [u,v]_{A, B} \left( \sum _{\pi (Q)=(F,G)} \langle u^{\frac{1}{p}}\rangle _{A, Q}^p |Q| \right) ^{\frac{1}{p}} \left( \sum _{\pi (Q)=(F,G)}\langle v^{\frac{1}{p'}}\rangle _{B, Q}^{p'}|Q| \right) ^{\frac{1}{p'}}\\&\quad \lesssim [u,v]_{A, B} \left( \sum _{\pi (Q)=(F,G)} \langle u^{\frac{1}{p}}\rangle _{A, Q}^p |Q| \right) ^{\frac{1}{p}} v(G)^{\frac{1}{p'}}. \end{aligned}$$
Then by Hölder’s inequality,
$$\begin{aligned} I&\le [u,v]_{A, B} \sum _{F\in {\mathcal {F}}} \langle f\rangle _F^u \left( \sum _{\begin{array}{c} G\in \mathcal G\\ G\subset F \end{array}}\sum _{\pi (Q)=(F,G)} \langle u^{\frac{1}{p}}\rangle _{A, Q}^p |Q| \right) ^{\frac{1}{p}} \left( \sum _{\begin{array}{c} G\in \mathcal G\\ \pi _{\mathcal {F}}(G)= F \end{array}} (\langle g\rangle _G^v )^{p'} v(G)\right) ^{\frac{1}{p'}}\\&\lesssim [u,v]_{A, B} \sum _{F\in {\mathcal {F}}} \langle f\rangle _F^u u(F)^{\frac{1}{p}} \left( \sum _{\begin{array}{c} G\in \mathcal G\\ \pi _{\mathcal {F}}(G)= F \end{array}} (\langle g\rangle _G^v )^{p'} v(G) \right) ^{\frac{1}{p'}}\\&\le [u,v]_{A, B} \left( \sum _{F\in {\mathcal {F}}} (\langle f\rangle _F^u)^p u(F) \right) ^{\frac{1}{p}} \left( \sum _{F\in {\mathcal {F}}}\sum _{\begin{array}{c} G\in \mathcal G\\ \pi _{\mathcal {F}}(G)= F \end{array}} (\langle g\rangle _G^v )^{p'} v(G) \right) ^{\frac{1}{p'}}\\&\lesssim [u,v]_{A, B} \Vert f\Vert _{L^p(u)} \Vert g\Vert _{L^{p'}(v)}, \end{aligned}$$
where (5.1) is used in the last step. \(\square \)

Remark 5.2

In [18], Lerner proved the so-called bump conjecture, i.e., the same result with \(\langle u\rangle _Q^{\frac{1}{p}}/\langle u^{\frac{1}{p}}\rangle _{A, Q}\) replaced by \(\langle u^{\frac{1}{p'}}\rangle _{\bar{A}, Q}/\langle u\rangle _Q^{\frac{1}{p'}}\) and analogously for v when \(p_0=1\) and \(q_0=\infty \). Therefore, our result improves the bump theorem. In [19], Lerner and Moen also proved the following estimate, for any Calderón-Zygmund operator T, there holds
$$\begin{aligned} \Vert T\Vert _{L^p(w)}&\le \sup _Q (\langle w\rangle _Q \langle w^{1-p'}\rangle _Q^{p-1})^{ \frac{1}{p-1}} A_\infty ^{exp }(w)(Q)^{1-\frac{1}{p-1}}\\ {}&=\sup _{Q} \langle w\rangle _Q^{\frac{1}{p}}\langle w^{1-p'}\rangle _Q^{\frac{1}{p'}}A_\infty ^{exp }(w,Q)^{\frac{1}{p'}} A_\infty ^{exp }(w^{1-p'},Q)^{\frac{1}{p}}. \end{aligned}$$
Therefore, our result improves the above estimate as well.

Notes

Acknowledgments

The author would like to thank Prof. Tuomas P. Hytönen for suggesting this problem and for many helpful discussions which improve the quality of this paper.

References

  1. 1.
    Anderson, T., Cruz-Uribe, D., Moen, K.: Logarithmic bump conditions for Calderón-Zygmund operators on spaces of homogeneous type. Publ. Mat. 59, 17–43 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernicot, F., Frey, D., Petermichl, S.: Sharp weighted norm estimates beyond Calderón-Zygmund theory. Anal. PDE 9(5), 1079–1113 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cascante, Carme, Ortega, Joaquin M., Verbitsky, Igor E.: Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels. Indiana Univ. Math. J. 53(3), 845–882 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cruz-Uribe, D., Pérez, C.: Sharp two-weight, weak-type norm inequalities for singular integral operators. Math. Res. Lett. 6(3–4), 417–427 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cruz-Uribe, D., Pérez, C.: Two-weight, weak-type norm inequalities for fractional integrals, Calderón- Zygmund operators and commutators. Indiana Univ. Math. J. 49(2), 697–721 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cruz-Uribe, D., Reznikov, A., Volberg, A.: Logarithmic bump conditions and the two-weight boundedness of Calderón-Zygmund operators. Adv. Math. 255, 706–729 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hänninen, T.: Remark on dyadic pointwise domination and median oscillation decomposition. (2015). arXiv:1502.05942
  8. 8.
    Hänninen, T., Hytönen, T., Li, K.: Two-weight \(L^p\)-\(L^q\) bounds for positive dyadic operators: unified approach to \(p\le q\) and \(p>q\). Potential Anal. (2016). doi: 10.1007/s11118-016-9559-9 zbMATHGoogle Scholar
  9. 9.
    Hytönen, T.: The sharp weighted bound for general Calderón-Zygmund operators. Ann. Math. 175, 1473–1506 (2012)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126(1), 1–33 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hytönen, T., Lacey, M.: The \(A_p\)-\(A_\infty \) inequality for general Calderón-Zygmund operators. Indiana Univ. Math. J. 61, 2041–2052 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hytönen, T., Li, K.: Weak and strong \(A_p\)-\(A_\infty \) estimates for square functions and related operators, preprint. (2015). arXiv:1509.00273
  13. 13.
    Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). J. Anal. PDE 6(2013), 777–818 (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Lacey, M.: On the separated bumps conjecture for Calderón-Zygmund operators. Hokkaido Math. J. 45, 223–242 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lacey, M., Sawyer, E., Uriarte-Tuero, I.: Two weight inequalities for discrete positiove operators. (2009). arXiv:0911.3437
  16. 16.
    Lacey, M., Spencer, S.: On entropy bumps for Calderón-Zygmund operators. Concr. Oper. 2, 47–52 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lerner, A.: A simple proof of the \(A_2\) conjecture. Int. Math. Res. Not. (2012). doi: 10.1093/imrn/rns145 Google Scholar
  18. 18.
    Lerner, A.: On an estimate of Calderón-Zygmund operators by dyadic positive operators. J. Anal. Math. 121, 141–161 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lerner, A., Moen, K.: Mixed \(A_p\)-\(A_\infty \) estimates with one supremum. Studia Math. 219, 247–267 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pérez, C.: Weighted norm inequalities for singular integral operators. J. Lond. Math. Soc. 49(1994), 296–308 (1994). (no. 2)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tanaka, H.: A characterization of two-weight trace inequalities for positive dyadic operators in the upper triangle case. Potential Anal. 41(2), 487–499 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b)University of HelsinkiHelsinkiFinland

Personalised recommendations