Alexander Volberg

Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.

The Tb-theorem on non-homogeneous spaces

0 Introduction: main objects and results 3 0.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.2 An application of T1-heorem: electric intensity capacity . . . . . . . . . . . . 7 0.3 How to interpret Calderon–Zygmund operator T? . . . . . . . . . . . . . . . 9 0.3.1 Bilinear form is defined on Lipschitz functions . . . . . . . . . . . . . 10 0.3.2 Bilinear form is defined for smooth functions . . . . . . . . . . . . . . 11 0.3.3 Apriori boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.4 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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].

Accretive system Tb-theorems on nonhomogeneous spaces

We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calder ón-Zygmund operator on L 2(μ). We do not assume any kind of doubling condition on the measure μ, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L . Thus we answer positively a question of Christ as to whether the L -assumption can be replaced by a BMO assumption. We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry. In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calder ón-Zygmund operators with respect to very bad measures. 0. Introduction: Main objects and results In what follows, the symbol K (x, y) stands for a Calder ón-Zygmund kernel defined for x, y ∈ Rn of orderd: |K (x, y)| ≤ C1 |x − y|d , |K (x1, y)− K (x2, y)| + |K (y, x1)− K (y, x2)| ≤ C2|x1 − x2| |x1 − y|d+α for a positiveα and any pointsx1, x2, y satisfying|x1 − x2| ≤ (1/2)|x1 − y|. We always assume that μ is adapted toK via the Ahlfors condition: μ ( B(x, r ) ) ≤ C3r d. However, we assume no estimate from below. DUKE MATHEMATICAL JOURNAL Vol. 113, No. 2, c

The Bellman function, the two-weight Hilbert transform, and embeddings of the model spacesKθ

This paper is devoted to embedding theorems for the spaceKθ, where θ is an inner function in the unit disc D. It turns out that the question of embedding ofKθ into L2(Μ) is virtually equivalent to the boundedness of the two-weight Hilbert transform. This makes the embedding question quite difficult (general boundedness criteria of Hunt-Muckenhoupt-Wheeden type for the twoweight Hilbert transform have yet to be found). Here we are not interested in sufficient conditions for the embedding ofKg into L2(Μ) (equivalent to a certain two-weight problem for the Hilbert transform). Rather, we are interested in the fact that a certain natural set of conditions is not sufficient for the embedding ofKθ intoL2 (Μ) (equivalently, a certain set of conditions is not sufficient for the boundedness in a two-weight problem for the Hilbert transform). In particular, we answer (negatively) certain questions of W. Cohn about the embedding ofKθ into L2(Μ). Our technique leads naturally to the conclusion that there can be a uniform embedding of all the reproducing kernels ofKθ but the embedding of the wholeKθ intoL2(Μ) may fail. Moreover, it may happen that the embedding into a potentially larger spaceL2(μ) fails too.

Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball

The purposes of this paper are twofold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil, and Volberg to handle Bergman-type singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for the Besov-Sobolev space of analytic functions

Heating of the Ahlfors–Beurling operator, and estimates of its norm

B^sigma_2

Subordination by conformal martingales in Lp and zeros of Laguerre polynomials

on the complex ball of

The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions

{Bbb{C}}^d

THE POWER LAW FOR THE BUFFON NEEDLE PROBABILITY OF THE FOUR-CORNER CANTOR SET

. In particular, we demonstrate that for any