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Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\lambda}{\lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles, even rectangles with sides are parallel to the coordinate axes.

But if we restrict things to $L^p$ functions where $1<p<\infty$, then Lebesgue’s differentiation theorem does hold true for rectangles with sides parallel to the coordinate axes, but it does not hold true for arbitrary rectangles. So my question is, is a maximal set of directions known for which Lebesgue’s differentiation theorem holds for $L^p$ functions for the collection of all rectangles oriented in those directions?

Now this journal paper says that a maximal set of directions is not known for tubes in $\mathbb{R}^n$ for which Lebesgue’s differentiation theorem holds (though the paper makes a conjecture about it). And a rectangle is nothing but a tube in $\mathbb{R}^2$. But it’s possible that the answer is known for this special case.

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