Stefanie Petermichl

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

We establish borderline regularity for solutions of the Beltrami equation f z−μ fz̄ = 0 on the plane, where μ is a bounded measurable function, ‖μ‖∞ = k < 1. What is the minimal requirement of the type f ∈ W 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 loc suffices ifε > 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].

Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol

Abstract We are going to show that the commutator of the Hilbert transform with matrix multiplication by a BMO matrix of size n×n is bounded by a multiple of logn times the BMO-norm of the matrix.

Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

We obtain sharp weighted L p estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 r it is bounded on L p (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on L p (v) by the same increasing function of the r 1 p 1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.

The sharp weighted bound for the Riesz transforms

We establish the best possible bound on the norm of the Riesz transforms as operators in the weighted space L p Rn (ω) for 1 < p < ∞ in terms of the classical Ap characteristic of the weight.

An estimate for weighted Hilbert transform via square functions

We show that the norm of the Hilbert transform as an operator on the weighted space L 2 (w) is bounded by a constant multiple of the 3/2 2 power of the A 2 constant of w, in other words by c sup I ( r I ω -1 I ) 3/2 . We also give a short proof for sharp upper and lower bounds for the dyadic square function.

Weighted little bmo and two-weight inequalities for Journé commutators

We characterize the boundedness of the commutators

Sharp Lp estimates for discrete second order Riesz transforms

[b, T]

Multi-parameter Div–Curl lemmas

with biparameter Journe operators

A version of Burkholder's theorem for operator-weighted spaces

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Various Sharp Estimates for Semi-discrete Riesz Transforms of the Second Order

in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little