The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ is unbounded for $1<s<2$

Details The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ is unbounded for $1<s<2$ The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ is unbounded for $1<s<2$

2011-09-10

arxiv.org/abs/1109.2260v3

Mathematics

Analysis of PDEs

23 pages; here is the final version

Scientific paper

In this paper, we prove that for $s\in(1,2)$ there exists no totally lower irregular finite positive Borel measure $\mu$ in $\R^2$ with\break $\mathcal H^s(\supp\mu)<+\infty$ such that $\|R\mu\|\ci{L^\infty(m_2)}<+\infty$, where $R\mu=\mu\ast\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesgue measure in $\R^2$. Combined with known results of Prat and Vihtil\"a, this shows that for any non-integer $s\in(0,2)$ and any finite positive Borel measure in $\R^2$ with $\mathcal H^s(\supp\mu)<+\infty$, we have $\|R\mu\|\ci{L^\infty(m_2)}=\infty$.

Affiliated with

Also associated with

No associations

Rating

The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ is unbounded for $1<s<2$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ is unbounded for $1<s<2$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ is unbounded for $1<s<2$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-19778