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.

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We are going to give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform.1991 Mathematics Subject Classification. 42B20, 42A50, 47B35.

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].

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We prove that if µ is a d-dimensional Ahlfors-David regular measure in R d+1 , then the boundedness of the d-dimensional Riesz transform in L 2 (µ) implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of µ.

We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderón-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 Calderón-Zygmund operators with respect to very bad measures.