Beltrami operators in the plane

Abstract

For a general Calderón-Zygmund operator T on R N , it is shown thatfor all Muckenhoupt weights w ∈ A2. This optimal estimate was known as the A2 conjecture. A recent result of Pérez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper.The proof consists of the following elements: (i) a variant of the Nazarov-Treil-Volberg method of random dyadic systems with just one random system and completely without "bad" parts; (ii) a resulting representation of a general Calderón-Zygmund operator as an average of "dyadic shifts;" and (iii) improvements of the Lacey-Petermichl-Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

Chord-arc curves and the Beurling transform

“…One has to be more careful in the case of chord-arc curves as Bishop showed in [3]. He constructed a quasiconformal map ρ of the disk to itself such that the quasiconformal mapping corresponding to the dilatation 1 2 μ ρ , maps the circle to a curve of Hausdorff dimension >1. The characterization of chord-arc curves given in Theorem 2 provides a new approach to the connectivity problem by translating it into a question regarding the spectrum of a singular operator in a weighted L 2 space.…”

“…In this case (I − μS) −1 : L p (C) → L p (C) (see [1,12]), therefore h ∈ L p (C), where p > 2. Define…”

“…By applying the quasiconformal change of variables u = H • ρ −1 as in [1], we get that H = u • ρ and…”

“…We know that (I − μS) −1 : L p (C) → L p (C) when 2 ≤ p ≤ 1 + 1 μ ∞ (see [1,12]). Therefore, if f ∈ F then ∂ H = (I − μS) −1 (μC f ) ∈ L p 0 (C).…”

On coercivity and irregularity for some nonlinear degenerate elliptic systems

“…Due to the fact that bounds for such L p estimate are very difficult to be optimal, our sufficient condition Theorem 1 is only a qualitative result. In the special case when n = N = 2, we can give a more precise description of p-coercivity for p close to 2 by applying the theory of quasiregular mappings and Beltrami equations in the plane [2,4,21]. When p > 2 we can obtain some pseudo p-coercivity results based on more precise geometric conditions for E in the case p = 2 by applying the Plancherel's identity to rank-one convex quadratic forms.…”