The boundedness of Riesz 𝑠-transforms of measures in ℝⁿ

Let μ \mu be a finite nonzero Borel measure in R n \mathbb {R}^{n} satisfying 0 > c − 1 r s ≤ μ B ( x , r ) ≤ c r s > ∞ 0 >c^{-1}r^{s}\le \mu B(x,r)\le cr^{s} >\infty for all x ∈ spt ⁡ μ x\in \operatorname {spt}\mu and 0 > r ≤ 1 0 > r\le 1 and some c > 0 c >0 . If the Riesz s s -transform C s , μ ( x ) = ∫ y − x | y − x | s + 1 d μ y \begin{equation*}{\mathcal {C}}_{s,\mu }(x)=\int \frac {y-x}{|y-x|^{s+ 1}}\, d\mu y \end{equation*} is essentially bounded, then s s is an integer. We also give a related result on the L 2 L^{2} -boundedness.

Lyhyen matematiikan ylioppilaskoetehtävät vuosina 1970-2009