arXiv subjects
Publications and source records attributed to Nikita Dobronravov.
We prove that Besicovitch's covering theorem holds for the parabolic metric $d_p((x_1,t_1),(x_2,t_2))=(|x_1-x_2|^{p}+|t_1-t_2|^{\frac{p}{2}})^{\frac{1}{p}}$ on $\mathbb{R}^n\times\mathbb{R}$ if and only if $p\geqslant 2$. If $p=2$, this answers a question posed by P. Mattila in the affirmative.
A version of the Uncertainty Principle says: There does not exist a non zero function in $L_p(\mathbb{R}^d)$ if its Fourier transform is supported by a set of finite $α$-Hausdorff measure with $α<2d/p$. This UP does not hold at the endpoint $α=2d/p$. We find the sharp form of the UP in the limit case. We prove that there exists a non-zero function in the Lorentz space $L_{p,q}(\mathbb{R}^d)$ such that its Fourier transform is supported by a set of zero $(\frac{2d}{p},β)$-Netrusov--Hausdorff capacity if and only if $β>\frac{q}{2(q-1)}$.