1,300 research outputs found
The Perfect Local Theorem and Twisted Martingale Transforms
A local Tb Theorem provides a flexible framework for proving the boundedness
of a Calder\'on-Zygmund operator T. One needs only boundedness of the operator
T on systems of locally pseudo-accretive functions \{b_Q\}, indexed by cubes.
We give a new proof of this Theorem in the setting of perfect (dyadic) models
of Calder\'on-Zygmund operators, imposing integrability conditions on the b_Q
functions that are the weakest possible. The proof is a simple direct argument,
based upon a new inequality for transforms of so-called twisted martingale
differences.Comment: 12 pages v2: typos corrected in section 5 v3: The twisted martingale
transform inequalities appeared in Auscher Routin (arXiv:1011.1747), which is
now reflected in the tex
The local non-homogeneous Tb theorem for vector-valued functions
We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg
to the setting of singular integrals with operator-valued kernel that act on
vector-valued functions. Here, `vector-valued' means `taking values in a
function lattice with the UMD (unconditional martingale differences) property'.
A similar extension (but for general UMD spaces rather than UMD lattices) of
Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier
by the first author, and it has found applications in the work of Mayboroda and
Volberg on square-functions and rectifiability. Our local version requires
several elaborations of the previous techniques, and raises new questions about
the limits of the vector-valued theory
Self-improvement of pointwise Hardy inequality
We prove the self-improvement of a pointwise -Hardy inequality. The proof
relies on maximal function techniques and a characterization of the inequality
by curves
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