Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular

Abstract

We establish borderline regularity for solutions of the Beltrami equation f z −µfz = 0 on the plane, where µ is a bounded measurable function, µ ∞ = k < 1. What is the minimal requirement of the type f ∈ W 1,q loc which guarantees that any solution of the Beltrami equation with any µ ∞ = k < 1 is a continuous function? A deep result of K. Astala says that f ∈ W 1,1+k+ε loc suffices if ε > 0. On the other hand, O. Lehto and T. Iwaniec showed that q < 1 + k is not sufficient. In [2], the following question was asked: What happens for the borderline case q = 1 + k? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia's extrapolation technique and two-weight estimates for the martingale transform from [26].

Weak and Strong Type $$A_1$$–$$A_\infty $$ Estimates for Sparsely Dominated Operators

“…This generalizes the positive result of the well-known -conjecture, stating that for all Calderón–Zygmund operators T one hasIndeed, the result in (1.3) recovers this result since Calderón–Zygmund operators are in the class . Historically, the estimate (1.4) was first proven to be true for the Beurling–Ahlfors transform by Petermichl and Volberg [38], solving an optimal regularity problem for solutions to Beltrami equations. In between this period and the time that (1.4) was established in full generality by Hytönen [24], it was shown by Lerner, Ombrosi, and Pérez [32] that for all Calderón–Zygmund operators T one hasfor all , showing a significantly better exponent of the constant of the weight when considering the smaller class of weights .…”

Beltrami operators in the plane

“…(45)] with implications to Beltrami equations, the case of the Beurling-Ahlfors transform B ∈ L (L 2 (C)) was first settled by Petermichl and Volberg [29], and with an alternative proof by Dragičević and Volberg [7]. Petermichl also obtained the sharp bounds for the Hilbert transform H ∈ L (L 2 (R)) [27] and then for the Riesz transforms R i ∈ L (L 2 (R N )) in arbitrary dimension N ∈ Z + [28].…”

Convex integration and the Lp theory of elliptic equations

“…sym represents the space of 2 × 2 symmetric matrices with real entries. Theorem 1.1 is a combination of results due to F. Leonetti and V. Nesi [24], the first author [1] and S. Petermichl and A. Volberg [34]. The proof is based on the fact [24] that locally any weak solution to (1.2) coincides with the real part of a K-quasiregular mapping, K being the ellipticity constant of (1.1).…”

“…The results in [1] imply that any K -quasiregular mapping in W 1,q belongs actually to the space W 1, p , whenever 2K K +1 < q < p < 2K K −1 . Finally, the end point case q = 2K K +1 was recently covered by S. Petermichl and A. Volberg (see [4,13,34]).…”

“…The key point in the proof is that under the assumptions of the theorem the complex gradient of the solution, ∂ z u = (u x , −u y ), is a K -quasiregular mapping. Therefore the ideas from [1,4,34] apply. In analogy with the result of Piccinini and Spagnolo it was recently shown by A. Baernstein II and L. V. Kovalev ( [5]) that the complex gradients of solutions to equation (1.9) belong to a better Hölder space than general quasiregular mappings.…”