Norm estimates for the Kakeya maximal function with respect to general measures.
Let \(0 < \delta \ll 1\) and let \(f\) be a locally integrable real-valued function on \(R^2\). For each direction \(e \in S^1\), let \(K_\delta f(e)\) be the quantity \[ K_\delta f(e) := \sup_T \frac{1}{| T| } \int_T | f| \] where \(T\) ranges over all \(1 \times \delta\) tubes oriented in the direction \(e\). The author addresses the question of establishing sharp weighted \(L^p\) bounds of the form \[ \| K_\delta f \| _{L^p(d\mu)} \leq A \| f \| _{L^p(R^2)} \] for various measures \(\mu\) on the unit circle and \(p \geq 2\). In particular, he establishes the bound \[ A \leq C_p \left(1 + \int_1^{1/\delta} \phi(1/r)\;dr\right)^{1/p} \] where \(C_p\) depends only on \(p\) and \(\phi(r) = \sup_{x \in S^1} \mu(B(x,r))\). This bound is then shown to be sharp in the case of Ahlfors-regular measures \(\mu\). For \(\mu\) equal to arclength measure, these results are due to Bourgain. The author also extends these results from regular measures to measures with finite \(s\)-energy for some \(0 < s < 1\), but with an additional logarithmic loss in the constants, though the author conjectures that in some cases this logarithmic loss can be removed. As an application he introduces the notion of an \(A\)-Kakeya set \(E\) for subsets \(A\) of the unit circle (sets which contain a unit line segment in the directions of \(A\)), and obtains the Hausdorff dimension bound \(\dim(E) \geq \dim(A) + 1\).
scientific article; zbMATH DE number 2083992