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  1. Free, publicly-accessible full text available September 1, 2024
  2. Abstract

    We derive sparse bounds for the bilinear spherical maximal function in any dimension . When , this immediately recovers the sharp bound of the operator and implies quantitative weighted norm inequalities with respect to bilinear Muckenhoupt weights, which seems to be the first of their kind for the operator. The key innovation is a group of newly developed continuity improving estimates for the single‐scale bilinear spherical averaging operator.

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  3. Grafakos, Loukas (Ed.)
    We establish the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti- symmetric part. In particular, the coefficients are not necessarily bounded. We prove that for some $p\in (1,\infty)$, the Dirichlet problem for the elliptic equation $Lu= \dv A\nabla u=0$ in the upper half-space $\mathbb{R}^{n+1},\, n\geq 2,$ is uniquely solvable when the boundary data is in $L^p(\mathbb{R}^n,dx)$, provided that the coefficients are independent of the vertical variable. This result is equivalent to saying that the elliptic measure associated to $L$ belongs to the $A_\infty$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity. 
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