Bounds for Kakeya-type maximal operators associated with \(k\)-planes (Q2372674)

Let \(G(d,k)\) denote the Grassmannian manifold of \(k\)-dimensional linear subspaces of \(\mathbb R^d\). For \(L\in G(d,k)\) one defines \(\mathcal N^k[f](L)=\sup_{x\in \mathbb R^d}\int_{x+L}f(y)\,dy\) where \(f\) is supported on the unit ball \(B(0,1)\subset \mathbb R^d\). For \(L\in G(d,k)\), \(a\in \mathbb R^d\) and \(\delta>0\) define the \(\delta\) plate centered at \(a\), \(L_\delta (a)\) be the \(\delta\) neighbourhood in \(\mathbb R^d\) of the intersection of \(B(a,\frac12)\) with \(L+a\). The Kakeya-type maximal operator is defined by \(\mathcal M_\delta^k[f](L)=\sup_{a\in \mathbb R^d}| L_\delta(a)| ^{-1} \int_{L_\delta(a)}f(y)\,dy\). Let \(k_{cr}\) be the solution of the equation \(d=\frac732^{k_{cr}-2}+k_{cr}\). The main result is: Suppose \(4\leq k<d\), \(k>k_{cr}\) and \(p\geq (d-1)/2\). Then \[ \| \mathcal N^kf\| _{L^p(G(d,k))}\leq \| f\| _{L^p(\mathbb R^d)}. \] Another result is: including the case \(k<k_{cr}\), under some conditions \[ \| M_\delta^kf\| _{L^p(G(d,k))}\leq C\delta^{-\alpha/p}\| f\| _{L^p(\mathbb R^d)}, \] for some \(\alpha>0\). These results are concerned with one of Bourgain's: every \((d,k)\) set has positive Lebesgue measure, where a \((d,k)\) set is a subset of \(\mathbb R^d\) containing a translate of every \(k\)-dimensional plane [\textit{J. Bourgain}, Geom. Funct. Anal. 1, No. 2, 147--187 (1991; Zbl 0756.42014)].