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Abstract
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We dominate nonintegral singular operators by adapted sparse operators and
derive optimal norm estimates in weighted spaces. Our assumptions on the
operators are minimal and our result applies to an array of situations, whose
prototypes are Riesz transforms or multipliers, or paraproducts associated with a
second-order elliptic operator. It also applies to such operators whose unweighted
continuity is restricted to Lebesgue spaces with certain ranges of exponents
with
. The norm estimates
obtained are powers
of the characteristic used by Auscher and Martell. The critical exponent in this case is
. We prove
when
and
when
. In particular, we are
able to obtain the sharp
estimates for nonintegral singular operators which do not fit into the class of
Calderón–Zygmund operators. These results are new even in Euclidean space and
are the first ones for operators whose kernel does not satisfy any regularity
estimate.
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Keywords
singular operators, weights
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Mathematical Subject Classification 2010
Primary: 42B20, 58J35
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Milestones
Received: 5 October 2015
Revised: 19 February 2016
Accepted: 30 March 2016
Published: 29 July 2016
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