Differentiating along rectangles in lacunary directions

TL;DR

This paper constructs a special basis of rectangles that differentiates L^1 but fails for certain rotated bases, revealing limits of differentiation in relation to lacunary angle sequences.

Contribution

It introduces a differentiation basis that differentiates L^1 but not certain rotated bases, highlighting the impact of lacunary angle sequences on differentiation properties.

Findings

Differentiation basis differentiates L^1( R^2).

Rotated basis fails to differentiate certain Orlicz spaces.

Results depend on lacunary sequence properties.

Abstract

We show that, given some lacunary sequence of angles $\mathbf{\theta}=(\theta_j)_{j\in\N}$ 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 $\mathcal{B}_{\mathbf{\theta}}$ 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 $L\log L(\mathbb{R}^2)$.