Differentiating along rectangles in lacunary directions

Abstract

We show that, given some lacunary sequence of angles $\theta =({\theta }_j{)}_{j\in \mathbb{N}}$ not converging too fast to zero, it is possible to build a rare differentiation basis $\mathcal{B}$ of rectangles parallel to the axes that differentiates $L^1({\mathbb{R}}^2)$ while the basis ${\mathcal{B}}_{\theta }$ obtained from $\mathcal{B}$ by allowing its elements to rotate around their lower left vertex by the angles ${\theta }_j$, $j\in \mathbb{N}$, fails to differentiate all Orlicz spaces lying between $L^1({\mathbb{R}}^2)$ and $L\log L({\mathbb{R}}^2)$.

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• Differentiating along rectangles with fixed shapes in a set of directions