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This is the first part of a series of three articles. In this paper, we obtain weighted norm inequalities for di erent conical square functions associated with the Heat and the Poisson semigroups generated by a second order divergence form elliptic operator with bounded complex coe cients. We find classes of Muckenhoupt weights where the square functions are comparable and/or bounded. These classes are natural from the point of view of the ranges where the unweighted estimates hold. In doing that, we obtain sharp weighted change of angle formulas which allow us to compare conical square functions with di erent cone apertures in weighted Lebesgue spaces. A key ingredient in our proofs is a generalization of the Carleson measure condition which is more natural when estimating the square functions below p = 2., The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The first author was supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. The second author was supported in part by ICMAT Severo Ochoa project SEV-2011-0087.

Let T be a Calderon-Zygmund operator in a "non-homogeneous" space (X,d,µ), where, in particular, the measure µ may be non- doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil and A. Vol- berg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T?, which is the supremum of the trun- cated operators associated with T. Specifically, for 1 < p < 1, we obtain sucient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L p weighted inequality holds for T?. The main tool to deal with this problem is the theory of vector-valued inequalities for T? and some re- lated operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderon-Zygmund operators in non-homogeneous spaces, developed in our previous pa- per (GM). For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C? to characterize the existence of principal values for functions in weighted L p .

We give a characterization of weighted Hardy spaces H p (w), valid for a rather large collection of wavelets, 0

Recently, F. Nazarov, S. Treil and A. Volberg (and independently X. Tolsa) have extended the classical theory of Calderón-Zygmund operators to the context of a "non-homogeneous" space $(\mathbb {X},d,\mu )$, where, in particular, the measure $\mu $ may be non-doubling. In the present work we study weighted inequalities for these operators. Specifically, for $1 < p < \infty $, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted $L^p$ inequality holds. We deal with this problem by developing a vector-valued theory for Calderón-Zygmund operators on non-homogeneous spaces which is interesting in its own right. For the case of the Cauchy integral operator, which is the most important example, we even prove that the conditions for the weights are also necessary.

We show that if Ω ⊂ ℝ, n ≥ 1, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior corkscrew and Harnack chain conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior corkscrew conditions, and an interior Harnack chain condition. We discuss some implications of this result for theorems of F. and M. Riesz type, and for certain free boundary problems., The first author was partially supported by NSF RTG grant 0838212. The second author was supported by NSF grants DMS-1101244 and DMS-1361701. The third author was supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. He also acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The fourth author was partly supported by the Swedish research council VR. The last author was partially supported by the Robert R. & Elaine F. Phelps Professorship in Mathematics.

Let $E\subset \ree$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the ``Riesz Transform bound> $$\sup_{\eps>0}\int_E\left|\int_{\{y\in E:|x-y|>\eps\}}\frac{x-y}{|x-y|^{n+1}} \,f(y)\, dH^n(y)\right|^2 dH^n(x) \,\leq \,C \int_E|f|^2 dH^n\,.$$ Assume further that $E$ is the boundary of a domain $\Omega\subset\ree$ satisfying the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition. Then $E$ is uniformly rectifiable.

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1

Calderón-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for p¿2, even for kernels with good size/smoothness properties. In this paper, we obtain weak-type estimates for perfect dyadic CZO¿s and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. General CZO¿s satisfy analogous H1¿L1 type estimates. In conjunction with type estimates, we get similar row/column Lp estimates. Our approach also applies to paraproducts/martingale transforms with noncommuting symbols., Guixiang Hong was supported in part by an ANR Grant 2011 BS01 008 01 (France). Luis Daniel López-Sánchez, José María Martell and JavierParcet were supported in part by MEC Grant MTM-2010-16518 (Spain) and by the ERC Grant StG-256997 (European Union)

We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n ≥ 2, for an Ahlfors-David regular domain Ω ⊂ ℝn+1 which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of ∂Ω., The first author was supported by NSF grants DMS-0801079 and DMS-1101244. The second author was supported by MICINN Grant MTM2010-16518, and by CSIC PIE 200850I015. The third author was partially supported by grants DMS-0901524, CAREER DMS- 1056965 (US NSF), Sloan Research Foundation, and MTM2010-16232, MTM2009-14694-C02-01 (Spain).

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions and paraproducts, and the vector-valued maximal operator of C. Fefferman-Stein. Also we can derive a very sharp two-weight bump type condition for $T$., We improve different parts of the first version, in particular we show the sharpness of our theorem for the vector-valued maximal function

The extrapolation theorem of Rubio de Francia is one of the deepest results in the study of weighted norm inequalities in harmonic analysis: it is simple to state but has profound and diverse applications. The goal of this book is to give a systematic development of the theory of extrapolation, one which unifies known results and expands them in new directions. In addition, we want to show how extrapolation theory, broadly defined, can be applied to the theory of weighted norm inequalities. We describe new and simpler proofs of known results, and then prove new results and show how these lead to additional open questions.

In this chapter we prove our fundamental generalization of the Rubio de Francia extrapolation theorem. Our result combines the two most important generalizations we discussed in Chapter 2: generalized maximal operators and the elimination of the operator. We then consider the results we get by rescaling, particularly A ∞ extrapolation. Next, we prove four variants of our main result: extrapolation with sharp constants, off-diagonal extrapolation, extrapolation with pairs of positive operators in place of the maximal operator (which we apply to one-sided A p weights), and limited range extrapolation. Finally, we survey some of the many possible applications of our results. Extrapolation in the context of Banach function spaces and modular spaces will be discussed in Chapter 4.

In this chapter we give our new proof of the Rubio de Francia extrapolation theorem, Theorem 1.4, and discuss how our proof allows a number of powerful generalizations. For the convenience of the reader we restate it here.

In this chapter and the next we apply the two-weight extrapolation theorems in Chapters 7 and 8 to the theory of two-weight norm inequalities.

In this chapter we consider further variations of the two-weight extrapolation theorem proved in Chapter 7.

In this chapter we give our generalization of the Rubio de Francia extrapolation theorem to the two-weight setting.

In this chapter we extend our theory of extrapolation to get norm inequalities on Banach function spaces starting from inequalities in weighted L p.

Our primary goal in this chapter is to define the appropriate analogs of A 1 weights and prove a “reverse factorization” property for pairs of weights that satisfy the A p bump conditions defined in Chapter 5. As we discussed in Chapter 2, reverse factorization is an essential tool in the proof of the Rubio de Francia extrapolation theorem; in the two-weight case these ideas play a similar but less direct role, as we will discuss in Chapter 7.

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators., Comment: To appear in the Electronic Research Announcements in Mathematical Sciences

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds., Comment: 12 pages. Fourth of 4 papers. Important revision: improvement of main result by eliminating use of Poincar\'e inequalities replaced by the weaker Gaussian keat kernel bounds

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-$\lambda$ inequality with two-parameters and the other uses Calder\'on-Zygmund decomposition. These results apply well to singular 'non-integral' operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, 'non-integral' that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all $L^p$ spaces for $1 < p < \infty$. Pointwise estimates are then replaced by appropriate localized $L^p-L^q$ estimates. We obtain weighted $L^p$ estimates for a range of $p$ that is different from $(1,\infty)$ and isolate the right class of weights. In particular, we prove an extrapolation theorem ' \`a la Rubio de Francia' for such a class and thus vector-valued estimates., Comment: 43 pages. Series of 4 papers

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces. Stewart Faculty Development Fund (Trinity College) Ministerio de Educación y Ciencia

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant $L^p-L^q$ estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators., Comment: 40 pages. Second of 4 papers. Can be read independently

We present an extrapolation theory that allows us to obtain, from weighted Lp inequalities on pairs of functions for p fixed and all A∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A∞ weights and also modular inequalities with A∞ weights. Vector-valued inequalities are obtained automatically, without the need of a Banachvalued theory. This provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators. In particular, we obtain weighted, and vector-valued, extensions of the classical theorems of Boyd and Lorentz-Shimogaki. The key is to develop appropriate versions of Rubio de Francia’s algorithm. Ministerio de Ciencia y Tecnología Ministerio de Educación y Ciencia

We present a general framework to deal with commutators of singular integral operators with BMO functions. Hörmander type conditions associated with Young functions are assumed on the kernels. Coifman type estimates, weighted norm inequalities and two-weight estimates are considered. We give applications to homogeneous singular integrals, Fourier multipliers and one-sided operators.

We give an overview of the generalized Calder\'on-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigroup and its gradient. Such applications are summarized., Comment: survey for the El Escorial 2008 proceedings