Joan Orobitg

BMO for nondoubling measures

1. Introduction. The Calderón-Zygmund theory of singular integrals has been traditionally considered with respect to a measure satisfying a doubling condition. Recently, Tolsa [T] and, independently, Nazarov, Treil, and Volberg [NTV] have shown that this standard doubling condition was not really necessary. Likewise, in the homogeneous spaces setting, functions of bounded mean oscillation, BMO, and its predual H 1 , the atomic Hardy space, play an important role in the theory of singular integrals. This note is an attempt to find good substitutes for the spaces BMO and H 1 when the underlying measure is nondoubling. Our hope was that we would have been able to prove some results of Tolsa, Nazarov, Treil, and Volberg, via BMO-H 1 interpolation, but in this respect we were unsuccessful. Let µ be a nonnegative Radon measure on R n. A function f ∈ L 1 loc (µ) is said to belong to BMO(µ) if the inequality

Ap weights for nondoubling measures in Rn and applications

We study an analogue of the classical theory of Ap(µ) weights in R n without assuming that the underlying measure µ is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderon-Zygmund operators (in the sense of (NTV1)). We also consider commutators of those Calderon-Zygmund operators with bounded mean oscillation functions (BMO), extending the main result from (CRW). Finally, we study self-improving properties of Poincare-B.M.O. type inequalities within this context, more precisely we show that if f is a locally integrable function satisfying 1 µ(Q) R Q |f fQ|dµ a(Q) for all cubes Q, then it is possible to deduce higher L p integrability result of f assuming certain simple geometric condition on the functional a.

Choquet Integrals, Hausdorff Content and the Hardy–Littlewood Maximal Operator

Using the BMO-H1 duality (among other things), D. R. Adams proved in [1] the strong type inequality where C is some positive constant independent of f. Here M is the Hardy–Littlewood maximal operator in Rn, Hα is the α-dimensional Hausdorff content, and the integrals are taken in the Choquet sense. The Choquet integral of φ ⩾ 0 with respect to a set function C is defined by Precise definitions of M and Hα will be given below. For an application of (1) to the Sobolev space W1, 1 (Rn), see [1, p. 114]. The purpose of this note is to provide a self-contained, direct proof of a result more general than (1). 1991 Mathematics Subject Classification 28A12, 28A25, 42B25.

Beltrami Equation with Coefficient in Sobolev and Besov Spaces

Our goal in this work is to present some function spaces on the complex plane

Beltrami equations with coefficient in the Sobolev space W1,p

\C

Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

,

Well-posedness for the continuity equation for vector fields with suitable modulus of continuity

X(\C)

A note on transport equation in quasiconformally invariant spaces

, for which the quasiregular solutions of the Beltrami equation,

Extra cancellation of even Calderón–Zygmund operators and quasiconformal mappings

\bar\partial f (z) = \mu(z) \partial f (z)

Extra cancellation of even CaldernZygmund operators and quasiconformal mappings

, have first derivatives locally in