Weighted weak-type inequalities for maximal operators and singular integrals

Abstract

We prove quantitative, one-weight, weak-type estimates for maximal operators, singular integrals, fractional maximal operators and fractional integral operators. We consider a kind of weak-type inequality that was first studied by Muckenhoupt and Wheeden (Indiana Univ. Math. J. 26(5):801–816, 1977) and later in Cruz-Uribe et al. (Int. Math. Res. Not. 30:1849–1871, 2005). We obtain quantitative estimates for these operators in both the scalar and matrix weighted setting using sparse domination techniques. Our results extend those obtained by Cruz-Uribe et al. (Rev. Mat. Iberoam. 37(4):1513–1538, 2021) for singular integrals and maximal operators when $p=1$.

Proof of Theorem A.1

In this section we provide a simple proof of a lower bound for the constant in Theorem 1.3. This bound is worse than the sharp result found in [24], but we believe it provides some insight into the behavior of weights in multiplier weak-type inequalities. The authors would like to thank an anonymous referee for suggesting this example, which is better than the example we originally sketched in [10].

Theorem A.1

When $n=1$, given $w\in A_1$, for the maximal operator we must have that

$$\begin{aligned} \Vert wM(\cdot \, w^{-1}) \Vert _{L^1 \rightarrow L^{1,\infty }} > rsim [w]_{A_1}. \end{aligned}$$

To prove this result we construct an explicit weight $w\in A_1$ and function f such that

$$\begin{aligned} \sup _{\lambda> 0}\,\lambda |\{x\in (0,\infty ) : w(x)|Mf(x)|>\lambda \}| \le C_0 \int _\mathbb {R}|f(x)|w(x)\,dx, \end{aligned}$$

(A.1)

where the constant must satisfy $C_0 > rsim [w]_{A_1}$. For $N > 0$, consider the weight $w(x)=\upchi _{(-\infty , 0)} + N\upchi _{(0,\infty )}$. Then, given any $a< 0 < b$, we have

$$\begin{aligned} \frac{1}{b-a}\int _a^b w(x)\,dx = \frac{|a| + Nb}{b - a} = \frac{Nb + |a|}{b + |a|} \le N. \end{aligned}$$

Hence, $Mw(x)\rightarrow N$ as $x\rightarrow 0^-$. It follows that $[w]_{A_1} = N$. Define $f(x) = \upchi _{(-2,-1)}$. Then $\int _{\mathbb {R}} |f(x)|w(x)\,dx = 1$ and for $x \in (0, \infty )$ we have $Mf(x) \ge \frac{1}{x + 2}$. Therefore, for $\lambda = \frac{N}{3}$,

$$\begin{aligned}{} & {} |\{x \in \mathbb {R}: w(x)Mf(x)> \lambda \}| \ge \left| \left\{ x \in \mathbb {R}: \frac{w(x)}{x + 2}>\frac{N}{3}\right\} \right| \\{} & {} \quad \ge \left| \left\{ x\in (0,\infty ): \frac{N}{x + 2}> \frac{N}{3}\right\} \right| \ge |\{x\in (0,\infty ): 3 > x + 2\}| = 1. \end{aligned}$$

Consequently,

$$\begin{aligned} \frac{N}{3} \le \sup _{\lambda> 0}\,\lambda |\{x\in \mathbb {R}: w(x)|Mf(x)|>\lambda \}| \le C_0 \int _\mathbb {R}|f(x)|w(x)\,dx = C_0, \end{aligned}$$

and so $C_0 > rsim [w]_{A_1}$, as desired.