A NULLSTELLENSATZ FOR AMOEBAS

552 ASTALA, CLOP, MATEU, OROBITG, and URIARTE-TUERO
and consequently, we can write
f = 1 z ∗ µ = R(I 1 ∗ µ) = I 1−α ∗ R(I α ∗ µ),
where R is a Calderón-Zygmund operator and ‖f ‖ Ẇ 1−α,q =‖R(I α ∗µ)‖ q ‖I α ∗µ‖ q .
For the converse, let f = I 1−α ∗ g be an admissible function for γ 1−α,q .Wehave
that, up to a multiplicative constant, T = ∂f is an admissible distribution for Ċ α,p
because
I α ∗ T = R t (g),
where R t is the transpose of R. Thus Ċ α,p (E) 1/p ≥|〈T,1〉| = |f ′ (∞)|, and the proof
is complete.
We end up with new quasiconformal invariants built on the Riesz capacities.
THEOREM 2.10
Let φ : C → C be a principal K-quasiconformal mapping of the plane which is
conformal on C \ E.Let1