An estimate for the Kakeya maximal operator on functions of square radial type (Q1970882)

The Kakeya (or Nikodým) maximal operator is essentially defined by the formula \[ f^{**}_\delta(x) = \sup_{x \in T} |T|^{-1} \int_T f \] where \(\delta > 0\) is a small parameter and \(T\) ranges over all \(1 \times \delta\) tubes in \(R^n\) centered at \(x\). The Kakeya maximal conjecture asserts that this operator is bounded on \(L^n\) with only a logarithmic loss in the parameter \(\delta\). This conjecture has been verified when \(n=2\), or when the function \(f\) is radial or a tensor product of one-dimensional functions, but is open in general. In this paper the author proves the conjecture assuming that \(f\) is ``square-radial'', or, more precisely, that \(f\) is constant on the boundary of all cubes which are centered at the origin.