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Heisenberg Hausdorff Dimension of Besicovitch Sets

100%

Venieri L.

nr 1

We consider (bounded) Besicovitch sets in the Heisenberg group and prove that Lp estimates for the Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.

Some remarks on Bochner-Riesz means

75%

Thangavelu S.

nr 2

217-230

We study $L^p$ norm convergence of Bochner-Riesz means $S_R^δ f$ associated with certain non-negative differential operators. When the kernel $S_R^m(x,y)$ satisfies a weak estimate for large values of m we prove $L^p$ norm convergence of $S_R^δ f$ for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.

Blow-up of regular submanifolds in Heisenberg groups and applications

75%

Magnani V.

nr 1

82-109

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.