论文标题
Dirichlet系列的Riesz投影和有限的平均振荡
Riesz projection and bounded mean oscillation for Dirichlet series
论文作者
论文摘要
我们证明，Riesz投影的规范是从$ l^\ infty（\ bbb {t}^n）$到$ l^p（\ bbb {t}^n）$是所有$ n \ ge 1 $ $ 1 $ $ 1 $，仅当$ p \ le 2 $时，就解决了Marzo和eip in 2011。 $ h^p（\ bbb {t}^{\ infty}）$不包含$ h^1的双空间（\ bbb {t}^{\ infty}）$对于任何$ p> 2 $。然后，我们注意到，$ h^1（\ bbb {t}^{\ infty}）$的双重二通过bohr lift，$ \ operatatorname {bmoa} $ of右半平面的dirichlet系列的空间。我们提供了几个条件，显示了此$ \ operatotorname {bmoa} $ Space与Dirichlet系列的其他空间相关联。 Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.
We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.
