Abstract
In this paper, we introduce a new class of weights, the \(A_{\lambda, \infty}\) weights, which contains the classical \(A_{\infty}\) weights. We prove a mixed \(A_{p,q}\)–\(A_{\lambda,\infty}\) type estimate for fractional integral operators.
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1 Introduction and the main results
Fractional integral operators and the associated maximal functions are very useful tools in harmonic analysis and PDE, especially in the study of differentiability or smoothness properties of functions. Recall that, for \(0<\lambda <n\), the fractional integral operator \(I_{\lambda }\) of a locally integrable function f defined on \(\mathbb{R}^{n}\) is given by
And the fractional maximal function \(M_{\lambda }\) is defined by
where the supremum is taken over all cubes Q in \(\mathbb{R}^{n}\) with sides parallel to the axes. We refer to [1,2,3,4,5] for more results on fractional integral operators.
For \(1< p,q<\infty \), we call a locally integrable positive function \(w(x)\) defined on \(\mathbb{R}^{n}\) a weight belongs to \(A_{p,q}( \mathbb{R}^{n})\) if
In [6], Muckenhoupt and Wheeden showed that, for \(1< p< n/(n-\lambda )\) and \(1/q+1/p'=\lambda /n\), the fractional integral operator \(I_{\lambda }\) is bounded from \(L^{p}(w^{p})\) to \(L^{q}(w^{q})\) if and only if w belongs to \(A_{p,q}\). They also proved that the fractional maximal function \(M_{\lambda }\) is bounded from \(L^{p}(w^{p})\) to \(L^{q}(w^{q})\) under the same conditions on the weights. Lacey, Moen, Pérez and Torres [7] proved the sharp weighted bound for fractional integral operators. Specifically,
And the sharp weighted bound for the fractional maximal function was proved by Pradolini and Salinas [8], i.e.,
Hytönen and Lacey [9] introduced a different approach to improving the sharp \(A_{p}\) estimates for Calderón–Zygmund operators using a mixed \(A_{p}\)–\(A_{\infty }\) condition. Cruz-Uribe and Moen [4] studied the corresponding problem for fractional integral operators. Recall that w is said to be a weight in \(A_{\infty }\) if
where M is the Hardy–Littlewood maximal function and \(w(Q):=\int _{Q}w(x)\,dx\). There are several equivalent definitions of the \(A_{\infty }\) weights. For example \(w\in A_{\infty }'\) if
In [10], Fujii proved that \(w\in A_{\infty }\) if and only if \(w\in A_{\infty }'\). It is well known that \(w\in A_{\infty }\) if and only if \(w\in A_{p}\) for some \(p>1\), here \(A_{p}\) denotes the class of Muckenhoupt weights for which
Sbordone and Wik [11] showed that
Hytönen and Pérez [12] showed that \([w]_{A_{\infty }}\lesssim [w]_{A_{\infty }'}\), and in fact \([w]_{A_{\infty }}\) can be substantially smaller.
In this paper, we introduce a new class of weights, called the \(A_{\lambda ,\infty }\) weights, which is defined with the fractional maximal function.
Definition 1.1
Given \(0<\lambda <n\), \(A_{\lambda ,\infty }\) consists of all locally integrable functions \(w(x)\) on \(\mathbb{R}^{n}\) for which
We show that \(A_{\infty }\) is a subset of \(A_{\lambda ,\infty }\). Specifically, we have the following results.
Theorem 1.2
For any \(0<\lambda <n\)and \(w\in A_{\infty }\), we have \(w\in A_{ \lambda ,\infty }\)and
With \(A_{\lambda ,\infty }\) weights, we give a mixed two-weight estimate of fractional integral operators.
Theorem 1.3
Letλ, pandqbe constants such that \(0<\lambda <n\)and \(1/q+1/p'=\lambda /n\). For any \(w\in A_{p,q}\), set \(\mu = w^{q}\)and \(\sigma =w^{-p'}\). Then
The above theorem suggests us to generalize the \(A_{p,q}\) condition for a pair of weights. Given \(1< p,q<\infty \), we say that a pair of weights \((\mu ,\sigma )\) is in the class \(A_{p,q}\) if
With this notation, we can generalize Theorem 1.3 as follows.
Theorem 1.4
Letλ, pandqbe constants such that \(0<\lambda <n\)and \(1/q+1/p'=\lambda /n\). For any pair of weights \((\mu ,\sigma )\in A _{p,q}\)withμ, \(\sigma \in A_{\lambda ,\infty }\), we have
The paper is organized as follows. In Sect. 2, we collect some preliminary results. And in Sect. 3, we give proofs for the main results.
2 Preliminaries
In this section, we introduce some preliminary results.
2.1 General dyadic grids
Let \(\mathscr {D}\) be a set consisting of cubes in \(\mathbb{R}^{n}\). Recall that \(\mathscr {D}\) is said to be a general dyadic grid if it satisfies the following three conditions:
- 1.
for any \(Q\in \mathscr {D}\), its side length \(l(Q)\) is of the form \(2^{k}\) for some \(k\in \mathbb{Z}\);
- 2.
\(Q_{1}\cap Q_{2}\in \{Q_{1},Q_{2},\emptyset \}\) for any \(Q_{1},Q _{2} \in \mathscr {D}\);
- 3.
the cubes of a fixed side length \(2^{k}\) form a partition of \(\mathbb{R}^{n}\).
Given a general dyadic grid \(\mathscr {D}\), we call a subset \(\mathcal{S} \subset \mathscr {D}\) a sparse family in \(\mathscr {D}\) if it satisfies
For any \(Q\in \mathcal{S}\), denote
We see from the definition of the sparse family that \(|E(Q)|\geq \frac{1}{2}|Q|\) for any \(Q\in \mathcal{S}\).
Below we will make extensive use of the dyadic grids
Hytönen, Lacey and Pérez [13] proved the following result.
Lemma 2.1
(Three-lattice lemma)
For any cube \(Q\subset \mathbb{R}^{n}\), there exists a shifted dyadic cube
for some \(\alpha \in \{0,\frac{1}{3},\frac{2}{3}\}^{n}\), such that \(Q\subseteq R\)and \(\ell (R)\leq 6\ell (Q)\).
2.2 Dyadic lattice
Let Q be any cube in \(\mathbb{R}^{n}\). A dyadic child of Q is any of the \(2^{n}\) cubes obtained by partitioning Q by n “median hyperplanes” (i.e., the hyperplanes parallel to the faces of Q and dividing each edge into two equal parts).
Passing from Q to its children, then to the children of the children, etc., we obtain a standard dyadic lattice \(\mathcal{D}(Q)\) of subcubes of Q.
We refer to Lerner and Nazarov [14] for more properties of the dyadic lattice.
2.3 Dyadic fractional integral operators
Given \(0<\lambda <n\) and a general dyadic grid \(\mathscr {D}\) in \(\mathbb{R} ^{n}\), we define the dyadic fractional integral operator \(I^{\mathscr {D}}_{ \lambda }\) by
For a sparse family \(\mathcal{S}\subseteq \mathscr {D}\), the sparse dyadic fractional integral operators \(I^{\mathcal{S}}_{\lambda }\) are defined similarly. Cruz-Uribe and Moen [4] proved the following two propositions.
Proposition 2.2
Given \(0<\lambda <n\)and a nonnegative functionf, then \(I_{\lambda }f\)is pointwise equivalent to a linear combination of dyadic fractional integral operators, i.e.,
Proposition 2.3
Given a bounded, nonnegative functionfwith compact support and a dyadic grid \(\mathscr {D}\), there exists a sparse family \(\mathcal {S}\)such that, for allλwith \(0<\lambda <n\), we have
2.4 Testing condition
Let λ, p and q be constants such that \(0<\lambda <n\) and \(1< p\leq q<\infty \). Lacey, Sawyer and Uriarte-Tuero [15] reduced the proof of the boundedness for \(I_{\lambda }^{\mathcal {S}}(\cdot \sigma )\) from \(L^{p}(\sigma )\) to \(L^{q}(\mu )\) to the boundedness of the testing condition
where the operator \(I^{\mathcal{S}(R)}_{\lambda }\) is defined by
Proposition 2.4
([15, Theorem 1.11])
Suppose thatλ, pandqare constants such that \(0<\lambda <n\)and \(1< p\leq q<\infty \). Let \(\mathscr {D}\)be a dyadic grid and let \(\mathcal {S}\)be a sparse subset of \(\mathscr {D}\). For any pair of weights \((\mu ,\sigma )\), we have
3 Proofs of the main results
First, we show that a weight in \(A_{p,q}\) is associated with a weight in some \(A_{\lambda ,\infty }\).
Theorem 3.1
Suppose thatλ, pandqare constants such that \(0<\lambda <n\)and \(1/q+1/p'=\lambda /n\). Let \(w\in A_{p,q}\)and set \(\mu = w^{q}\). Then \(\mu \in A_{\lambda ,\infty }\)and
Proof
Since \(w\in A_{p,q}\), we have \(w^{-1}\in A_{q',p'}\) and
Using (1.1), this gives us
Fix some cube \(Q\in \mathbb{R}^{n}\). Since \(\frac{1}{p'}+\frac{1}{q}=\frac{ \lambda }{n}\), we see from Hölder’s inequality that
Hence
This completes the proof. □
Next we show that \(A_{\infty }\) is contained in \(A_{\lambda ,\infty }\) for any \(0<\lambda <n\).
Proof of Theorem 1.2
Fix some cube \(Q\subseteq \mathbb{R}^{n}\). Let
By translations and dilations, we see from Lemma 2.1 that for any cube \(K\subseteq Q\) there exists some \(R\in \mathcal{D}(Q ^{\beta })\) for some \(\beta \in \{-\frac{2}{3},-\frac{1}{3},0, \frac{1}{3},\frac{2}{3}\}^{n}\) such that \(R\in \mathcal{D}(Q^{\beta })\) and \(\ell (R)\leq 6\ell (K)\). Hence
where
So it sufficient to estimate
Among each \(\mathcal{D}(Q^{\beta })\), a subset of principal cubes \(\mathscr {P}^{\beta }=\bigcup_{m=0}^{\infty }\mathscr {P}_{m}^{\beta }\) is constructed as follows: \(\mathscr{P}_{0}^{\beta }=\{Q^{\beta }\}\), and then inductively \(\mathscr{P}_{m+1}^{\beta }\) consists of all maximal \(P'\in \mathcal{D}(Q^{\beta })\) such that
for some \(P\in \mathscr{P}_{m}^{\beta }\) with \(P\supset P'\).
Since \(\lambda /n<1\), we see from the definition of \(\mathscr {P}^{\beta }\) that, for any \(P\in \mathscr{P}_{m}^{\beta }\),
That is,
For any \(P\in \mathscr{P}^{\beta }\), we denote
By (3.1), we have
By the definition of \(Q^{\beta }\) and \(\mathcal{D}(Q^{\beta })\), for any \(R\in \mathcal{D}(Q^{\beta })\) with \(R\cap Q\neq \emptyset \), we have \(|R\cap Q|\geq \frac{1}{3^{n}}|R|\). For any \(P\in \mathscr {P}^{\beta }\), it is easy to see that \(P\mathrel{\cap} Q\neq \emptyset \). Combining with \(|E(P)| \geq (1-\frac{1}{2\cdot 3^{n}})|P|\), we obtain
Hence
Now we get the conclusion as desired. □
Given a locally integrable function f, a Borel measure ν and a cube Q, we denote \(\langle f\rangle _{Q}:=\frac{1}{|Q|}\int _{Q}f(x)\,dx\) and \(\langle f\rangle _{Q}^{\nu }:=\frac{1}{\nu (Q)}\int _{Q}f(x)\,d\nu (x)\). The following result is used in the proof of Theorem 1.4.
Proposition 3.2
([16, Proposition 2.2])
Let \(1< s<\infty \), νbe a positive Borel measure and
Then we have
To prove Theorem 1.4, we also need the following lemma.
Lemma 3.3
([17, Lemma 5.2])
For all \(\gamma \in [0,1)\), we have \(\sum_{L:L\subseteq P}\langle w \rangle _{L}^{\gamma }|L|\lesssim \langle w \rangle _{P}^{\gamma }|P|\).
We are now ready to give a proof of Theorem 1.4.
Proof of Theorem 1.4
By Propositions 2.2, 2.3 and 2.4, it suffices to show that
There are two cases.
Case 1: \(q\geq 2\). By Proposition 3.2, we have
Since \(\frac{p'}{q'}>1\), this give us
Note that \(q\geq 2\). We have
It follows from Lemma 3.3 that
and
Hence
Notice that
This gives us \(\frac{q}{p}>\frac{n}{\lambda }\).
Since \(\mathcal{S}\) is a sparse family and \(|E(Q)|>\frac{1}{2}|Q|\), we have
So
Taking the supreme over all cubes \(R\in \mathscr {D}\), we obtain
Case 2: \(1< q<2\). In this case, we have \(0\leq 1-\frac{q}{p'}<1\). Using Proposition 3.2 and Lemma 3.3, we get
We see from the arguments in Case 1 that
Hence
Taking the supreme over all cubes \(R\in \mathscr {D}\), we obtain
The estimates of \(\varTheta ^{\mathscr{D}}_{\sigma ,\mu }\) can be proved with the symmetry. This completes the proof. □
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Acknowledgements
The authors thank the referees very much for valuable suggestions, which helped to improve the paper.
Funding
This work was partially supported by the National Natural Science Foundation of China (11525104, 11531013, 11761131002 and 11801282).
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Pan, J., Sun, W. Two-weight norm inequalities for fractional integral operators with \(A_{\lambda,\infty}\) weights. J Inequal Appl 2019, 284 (2019). https://doi.org/10.1186/s13660-019-2239-8
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DOI: https://doi.org/10.1186/s13660-019-2239-8