Michael T. Lacey

The solution of the Kato square root problem for second order elliptic operators on Rn

We prove the Kato conjecture for elliptic operators on Jfin. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L =-div (AV) with bounded measurable coefficients in IEtn iS the

A note on the almost sure central limit theorem

We give a new proof of the almost sure central limit theorem. It is based on an almost sure invariance principle and extends to weakly dependent random variables.

On Calderón's conjecture.

with constants Cfi;p1;p2 depending only on fi;p1;p2 and p := p1p2 p1+p2 hold. The flrst result of this type is proved in [4], and the purpose of the current paper is to extend the range of exponents p1 and p2 for which (2) is known. In particular, the case p1 =2 ,p2 = 1 is solved to the a‐rmative. This was

A characterization of product BMO by commutators

In this paper we establish a commutator estimate which allows one to concretely identify the product BMO space, BMO(R2+ x R2+), of A. Chang and R. Fefferman, as an operator space on L2(R2). The one-parameter analogue of this result is a well-known theorem of Nehari [8]. The novelty of this paper is that we discuss a situation governed by a twoparameter family of dilations, and so the spaces H 1 and BMO have a more complicated structure. Here R2+ denotes the upper half-plane and BMO(R2+ • R2+) is defined to be the dual of the real-variable Hardy space H 1 on the product domain R2+ x R2+. There are several equivalent ways to define this latter space, and the reader is referred to [5] for the various characterizations. We will be more interested in the biholomorphic analogue of H 1, which can be defined in terms of the boundary values of biholomorphic functions on R 2 • R2+ and will be denoted throughout by Hi(R2+ • cf. [10]. In one variable, the space L2(R) decomposes as the direct sum H 2 ( R ) | where H2(R) is defined as the boundary values of functions in H2(R2+) and H2(R) denotes the space of complex conjugate of functions in H2(R). The space L2(R2), therefore, decomposes as the direct sum of the four spaces H 2 ( R ) | H2(R)@H2(R) , H 2 ( R ) | and H 2 ( R ) | where the tensor products are the Hilbert space tensor products. Let P~-,• denote the orthogonal projection of L2(R 2) onto the holomorphic/anti-holomorphic subspaces, in the first and second variables, respectively, and let Hj denote the one-dimensional Hilbert transform in the j t h variable, j -1 , 2. In terms of the projections P+,• HI=P+,++P+,--P-,+-P-,and H2=P+,++P-,+-P+,--P_,_.

An elementary proof of the A 2 bound

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A2 bounds in that setting.

Two-weight inequality for the Hilbert transform: A real variable characterization, II

Let σ and w be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a Poisson A2 condition, and satisfy the testing conditions below, for the Hilbert transform H, ∫ I H(σ1I) 2 dw . σ(I), ∫ I H(w1I) 2 dσ . w(I), with constants independent of the choice of interval I. Then H(σ ·) maps L(σ) to L(w), verifying a conjecture of Nazarov–Treil–Volberg. The proof uses basic tools of non-homogeneous analysis with two components particular to the Hilbert transform. The first is a global to local reduction, a consequence of prior work of Lacey-Sawyer-Shen-Uriarte-Tuero. The second, an analysis of the local part, is the contribution of this paper.

Sharp weighted bounds for the q‐variation of singular integrals

Any Calderon-Zygmund operator T is pointwise dominated by a convergent sum of positive dyadic operators. We give an elementary self-contained proof of this fact, which is simpler than the probabilistic arguments used for all previous results in this direction. Our argument also applies to the q-variation of certain Calderon-Zygmund operators, a stronger nonlinearity than the maximal truncations. As an application, we obtain new sharp weighted inequalities.

The bilinear maximal functions map into Lp for 2/3 < p < 1

The bilinear maximal operator defined below maps LP x Lq into Lr provided 1 0 2t t In particular Mfg is integrableif f and g are square integrable, answering a conjecture posed by Alberto Calder6n.

An elementary proof of the

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A2 bounds in that setting.

A_2

It is shown that product BMO of S.-Y. A. Chang and R. Fefferman, defined on the space