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.

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Abstract. We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [RR]. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain Ω ⊂ R n+1 , n ≥ 2, with a uniformly rectifiable boundary, which satisfies the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. In a companion paper to this one [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant L q bounds, with q > 1, on the Poisson kernel.Résumé. On présente une version invariante paréchelles et en dimension supé-rieureà 3 d'un théorème classique de F. et M. Riesz [RR]. Plus précisément, oń etablit l'absolue continuité de la mesure harmonique par rapportà la mesure de surface, ainsi qu'un gain d'intégrabilité pour le noyau de Poisson, pour un domaine Ω ⊂ R n+1 , n ≥ 2,à bord uniformément rectifiable, vérifiant une condition de chaîne de Harnack et une condition de type "points d'ancrage" ou "corkscrew" intérieure (mais pas extérieure). L'article associé [HMU]établit une réciproque, c'est-à-dire l'uniforme rectifiabilité du bord en supposant des estimées invariantes paréchelle L q pour q > 1 sur le noyau de Poisson.