51 research outputs found
Uniform rectifiability and harmonic measure II: Poisson kernels in imply uniform rectifiability
We present the converse to a higher dimensional, scale-invariant version of a
classical theorem of F. and M. Riesz. More precisely, for , for an ADR
domain \Omega\subset \re^{n+1} which satisfies the Harnack Chain condition
plus an interior (but not exterior) Corkscrew condition, we show that absolute
continuity of harmonic measure with respect to surface measure on
, with scale invariant higher integrability of the Poisson
kernel, is sufficient to imply uniformly rectifiable of
A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis
Subject to a range of side conditions, the two weight inequality for the
Hilbert transform is characterized in terms of (1) a Poisson A_2 condition on
the weights (2) A forward testing condition, in which the two weight inequality
is tested on intervals (3) and a backwards testing condition, dual to (2). A
critical new concept in the proof is an Energy Condition, which incorporates
information about the distribution of the weights in question inside intervals.
This condition is a consequence of the three conditions above. The Side
Conditions are termed 'Energy Hypotheses'. At one endpoint they are necessary
for the two weight inequality, and at the other, they are the Pivotal
Conditions of Nazarov-Treil-Volberg. This new concept is combined with a known
proof strategy devised by Nazarov-Treil-Volberg. A counterexample shows that
the Pivotal Condition are not necessary for the two weight inequality.Comment: 60 pages, 1 figure. v3. An important revision: The Energy Condition
is reformulated, a key concept of the proof, is reformulated. The main result
is unchanged. v4. important display corrected. v6: The earlier versions
incorrectly claimed a characterization, as was pointed out to us by S. Treil
v7. Corrections in Section
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