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A solution of the bump conjecture for all Calder'on-Zygmund operators: the Bellman function approach | | Fedor Nazarov
; Alexander Reznikov
; Sergei Treil
; Alexander Volberg
; | | Date: |
11 Feb 2012 | | Abstract: | In this paper we prove a long-standing conjecture in the theory of two-weight
norm inequalities. It was believed for quite a while that bumping $A_2$
condition by Orlicz norms one gets a sharp sufficient condition for the
boundedness of all Calder’on-Zygmund operators $T$ between two weighted
spaces, $T: L^2(v)
ightarrow L^2(u)$. T In the present paper we completely
prove the conjecture in $L^2$ metric. Here is the main result:
Let weights $u,v$ satisfy sharp bump $A_2$ assumption. Then any
Calder’on-Zygmund operator is bounded between two weighted spaces, $T:
L^2(v)
ightarrow L^2(u)$.
The proofs will be "automatic", which means that they are given by formulas.
To find these formulas we adopt the Bellman function approach. We first
consider the proof for dyadic singular operators: dyadic shifts of various
complexity and dyadic paraproducts. Then we join this together by a random
geometric construction, which decomposes each Calder’on-Zygmund into such
dyadic operators. Notice that dyadic operators are graded by their complexity.
So our estimates in the dyadic part of the paper should address the right
dependence of the estimates of two-weight dyadic shifts on their complexity (in
the presence of sharp bump $A_2$ assumption.
We also give a new "automatic" proof of the sharp bump results proved by
P’erez for maximal operators. The bump condition here is slightly different,
it is a one-sided bump condition. It is not a new result, but its prove is
exactly the same as for Calder’on-Zygmund operators. The same idea of giving a
formula for a certain function, which, in its turn, "automatically" gives the
proof of the desired inequality is very conspicuous in Section
ef{max}. It is
the last section, but we advise the reader to read it first as a warm up. | | Source: | arXiv, 1202.2406 | | Services: | Forum | Review | PDF | Favorites | |
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