论文标题
LP空间和UMD Lattices上的最大Hörmander官能演算
Maximal Hörmander Functional Calculus on Lp Spaces and UMD Lattices
论文作者
论文摘要
令$ a $为具有h {Ö} rmander functional conculus的分析半群的生成器，$ x = l^p（ω，y）$，其中$ y $是umd lattice。使用Banach空间几何学的方法与功能性微积分有关，我们表明，对于H {Ö} rmander Spectral乘数在$ \ infty $处的快速衰减非常快，则具有最大估计$ \ | \ sup_ {t \ geq 0} | m（ta）f | \，\ | _ {l^p（ω，y）} \ sillssim \ | f \ | _ _ {l^p（ω，y）} $。我们还显示Square函数估计$ \ left \ | \ left（\ sum_k \ sup _ {t \ geq 0} | m_k（ta）f_k |^2 \ right）^{\ frac12} \ right \ | | _ {l^p（ω，y）} \ left（\ sum _k | f_k |^2 \ right）^{\ frac12} \ right \ | _ {l^p（ω，y）} $适用于频谱乘数$ m_k $的适当家族，这是Euclidean laplacian在Scalar laplacian on Scalar on Scalar in Scalar in Callued $ l^p（p p p p p（p p p p p p）的新事物。作为推论，我们获得了波传播器和Bochner-riesz手段的最大估计值。最后，我们通过给出了几个运算符$ a $的示例来说明结果，这些$ a $接纳了一些$ l^p（ω，y）$上的h {Ö} rmander functional conculus，并讨论了晶格$ y $和非换档操作员$ $ $ a $ a $ a $ a $ a $ a的示例。
Let $A$ be a generator of an analytic semigroup having a H{ö}rmander functional calculus on $X = L^p(Ω,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for H{ö}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\, \|_{L^p(Ω,Y)} \lesssim \|f\|_{L^p(Ω,Y)}$. We also show square function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2 \right)^{\frac12} \right\|_{L^p(Ω,Y)} \lesssim \left\| \left( \sum _k |f_k|^2 \right)^{\frac12} \right\|_{L^p(Ω,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\mathbb{R}^d)$. As corollaries, we obtain maximal estimates for wave propagators and Bochner--Riesz means. Finally, we illustrate the results by giving several examples of operators $A$ that admit a H{ö}rmander functional calculus on some $L^p(Ω,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.
