Abstract. We prove a pointwise estimate for positive dyadic shifts of complexity m which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.
For any Calderón–Zygmund operator T, any weight w, and α>1, the operator T is bounded as a map from L1false(ML0.166667emlog0.166667emlog0.166667emLfalse(log0.166667emlog0.166667emlog0.166667emLfalse)αwfalse) into weak‐L1false(wfalse). The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman–Fefferman, Pérez, and Hytönen–Pérez, on the L(logL)ϵ scale. Also, for square functions Sf, and weights w∈Ap, the norm of S from Lpfalse(wfalse) to weak‐Lpfalse(wfalse), 2⩽p<∞, is bounded by [w]Ap1/2(1+log[w]A∞)1/2, which is a sharp estimate.
We study the problem of dominating the dyadic strong maximal function by (1, 1)-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance.
We give a quantitative embedding of the Muckenhoupt class
A
1
A_1
into
A
∞
A_\infty
. In particular, we show how
ϵ
\epsilon
depends on
[
w
]
A
1
[w]_{A_1}
in the inequality which characterizes
A
∞
A_\infty
weights:
\[
w
(
E
)
w
(
Q
)
≤
(
|
E
|
|
Q
|
)
ϵ
,
\frac {w(E)}{w(Q)} \leq \biggl ( \frac {|E|}{|Q|} \biggr )^\epsilon ,
\]
where
Q
Q
is any dyadic cube and
E
E
is any subset of
Q
Q
. This embedding yields a sharp reverse-Hölder inequality as an easy corollary.
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