AbstractThe Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{1,\infty }(uv)} \le C_{u,v} \Vert f \Vert _{L^{1}(u)}, \end{aligned}$$
Mf
v
L
1
,
∞
(
u
v
)
≤
C
u
,
v
‖
f
‖
L
1
(
u
)
,
where $$u\in A_1$$
u
∈
A
1
and $$uv\in A_{\infty }$$
u
v
∈
A
∞
. We prove a novel extension of this result to the general restricted weak type case. That is, for $$p>1$$
p
>
1
, $$u\in A_p^{{\mathcal {R}}}$$
u
∈
A
p
R
, and $$uv^p \in A_\infty $$
u
v
p
∈
A
∞
, $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{p,\infty }(uv^p)} \le C_{u,v} \Vert f \Vert _{L^{p,1}(u)}. \end{aligned}$$
Mf
v
L
p
,
∞
(
u
v
p
)
≤
C
u
,
v
‖
f
‖
L
p
,
1
(
u
)
.
From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the m-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including m-linear Calderón-Zygmund operators, avoiding the $$A_\infty $$
A
∞
extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $$A_p^{{\mathcal {R}}}$$
A
p
R
. Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $${\mathcal {M}}$$
M
, denoted by $$A_{\mathbf {P}}^{{\mathcal {R}}}$$
A
P
R
, establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, $$A_p^{{\mathcal {R}}}$$
A
p
R
and $$A_{\mathbf {P}}^{{\mathcal {R}}}$$
A
P
R
weights, and Lorentz spaces.
On a Weak Type Estimate for Sparse Operators of Strong Type
