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Title: Differentiation along rectangles
Abstract:Lebesgue鈥檚 differentiation theorem states that, when $f$ is a locally integrable function in Euclidean space, its average on the ball $B(x,r)$ centered at $x$ with radius $r$, converges to $f(x)$ for almost every $x$, when $r$ approaches zero. Many questions arise when the family of balls $\{B(x,r)\}$ is replaced by a聽differentiation basis $\mathcal{B}=\bigcup_x \mathcal{B}_x$ (where, for each $x$, $\mathcal{B}_x$ is, roughly speaking, a collection of sets shrinking to the point $x$). In this case, one looks for conditions on $\mathcal{B}$ such that the average of $f$ on sets belonging to $\mathcal{B}_x$ are known to converge to $f(x)$ for a.e. $x$, when those sets shrink to the point $x$. Many interesting phenomena happen when sets in $\mathcal{B}$ have a聽rectangular shape (Lebesgue鈥檚 theorem may or may not hold in this case, depending on the geometrical properties of sets in $\mathcal{B}$). In this talk, we shall discuss some of the history around this problem, as well as recent results obtained with E. D鈥橝niello and J. Rosenblatt in the planar case, when the rectangles in $\mathcal{B}$ are only allowed to lie along a fixed sequence of directions.