In this article we give an overview of the problem of finding sharp constants in matrix weighted norm inequalities for singular integrals, the so-called matrix A 2 conjecture. We begin by reviewing the history of the problem in the scalar case, including a sketch of the proof of the scalar A 2 conjecture. We then discuss the original, qualitative results for singular integrals with matrix weights and the best known quantitative estimates. We give an overview of new results by the author and Bownik, who developed a theory of harmonic analysis on convex set-valued functions. This led to the proof the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix weights, two longstanding problems. Rubio de Francia extrapolation is expected to be a major tool in the proof of the matrix A 2 conjecture, and we discuss some ideas which may lead to a complete solution.

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 $T$.Comment: We improve different parts of the first version, in particular we show the sharpness of our theorem for the vector-valued maximal functio

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We develop the theory of variable exponent Hardy spaces H p(·) . Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that H p(·) functions have an atomic decomposition including a "finite" decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Strömberg and Torchinsky [28] than the classical atomic decomposition. As an application of the atomic decomposition we show that singular integral operators are bounded on H p(·) with minimal regularity assumptions on the exponent p(·).Date: November 15, 2012. 2010 Mathematics Subject Classification. 42B25, 42B30, 42B35.

Abstract. We give a sufficient condition for singular integral operators and, more generally, Calderón-Zygmund operators to satisfy the weak (p, p) inequality u({x ∈ R