论文标题
$ s^1 $结合的傅立叶乘数在$ h^1（{\ mathbb r}）$和Semigroups的功能计算
$S^1$-bounded Fourier multipliers on $H^1({\mathbb R})$ and functional calculus for semigroups
论文作者
论文摘要
令$ t \ colon h^1（{\ Mathbb r}）\ to H^1（{\ Mathbb r}）$是分析性强大空间上有界的傅立叶乘数$ h^1（{\ Mathbb r}）\ subset l^1（ r} _+）$是它的象征，即，$ \ wideHat {t（h）} = m \ wideHat {h} $ for H^1 in H^1（{\ Mathbb r}）$。令$ s^1 $成为所有跟踪班级操作员的Banach banach banach banach oon $ \ ell^ell^2 $。我们表明，$ t $承认一个有界的张量扩展名$ t \ edlline {\ otimes} i_ {s_1} \ colon h^1（{\ mathbb r}; s^1）\ to H^1（{\ m m if Mathb r}; s^1）$ if和natif $ n hilbert Space $ hilbert Space $ hilbert Space $ l^\ infty（{\ Mathbb r} _+; {\ Mathcal H}）$，使得$ m（s+t）= \langleα（t），β（s）\ rangle _ {\ rangle _ {\ nathcal h} $几乎每个$（s，s，s，s，s，t）\ in {这样的傅立叶乘数均可列出$ s^1 $结合，我们让$ {\ mathcal m} _ {s^1}（h^1}（h^1（{\ mathbb r}））$表示所有$ s^1 $ bunded fourier multiperiers的Banach空间。接下来，我们将此结果分为两个步骤，将此结果应用于功能性演算估计。首先，我们介绍一个新的Banach代数$ {\ Mathcal a} _ {0，s^1}（{\ Mathbb c} _+）$ to $ {\ MathBB C} _+ = \ bigl \ big big big = \ big big {z \ in { re}（z）> 0 \ bigr \} $，并显示其双空间与$ {\ Mathcal M} _ {s^1}（h^1（{\ Mathbb r}））$一致。其次，给定任何有限的$ C_0 $ -Semigroup $（t_t）_ {t \ geq 0} $在Hilbert Space上，任何$ B \ in L^1（{\ Mathbb r} _+）$，我们建立了一个估计$ \ big \ big \ big \ big big big \ vert \ fert_0^\ int_0^\ infty fty b（big） \ vert l_b \ vert _ {{{\ Mathcal a} _ {0，s^1}（{\ Mathbb r}）} $，其中$ l_b $表示$ b $的laplace变换。这改善了前两位作者最近获得的先前功能分积估计。
Let $T\colon H^1({\mathbb R})\to H^1({\mathbb R})$ be a bounded Fourier multiplier on the analytic Hardy space $H^1({\mathbb R})\subset L^1({\mathbb R})$ and let $m\in L^\infty({\mathbb R}_+)$ be its symbol, that is, $\widehat{T(h)}=m\widehat{h}$ for all $h\in H^1({\mathbb R})$.Let $S^1$ be the Banach space of all trace class operators on $\ell^2$. We show that $T$ admits a bounded tensor extension $T\overline{\otimes} I_{S_1}\colon H^1({\mathbb R};S^1) \to H^1({\mathbb R};S^1)$ if and only if there exist a Hilbert space $\mathcal H$ and two functions $α, β\in L^\infty({\mathbb R}_+;{\mathcal H})$ such that $m(s+t) = \langleα(t),β(s)\rangle_{\mathcal H}$ for almost every $(s,t)\in{\mathbb R}_+^2$. Such Fourier multipliers arecalled $S^1$-bounded and we let ${\mathcal M}_{S^1}(H^1({\mathbb R}))$ denote the Banach space of all $S^1$-bounded Fourier multipliers. Next we apply this result to functional calculus estimates, in two steps. First we introduce a new Banach algebra ${\mathcal A}_{0,S^1}({\mathbb C}_+)$ of bounded analytic functions on ${\mathbb C}_+ =\bigl\{z\in{\mathbb C}\, :\, {\rm Re}(z)>0\bigr\}$ and show that its dual space coincides with ${\mathcal M}_{S^1}(H^1({\mathbb R}))$. Second, given any bounded $C_0$-semigroup $(T_t)_{t\geq 0}$ on Hilbert space, and any $b\in L^1({\mathbb R}_+)$, we establish an estimate $\bigl\Vert\int_0^\infty b(t) T_t\, dt\bigr\Vert\lesssim \Vert L_b\Vert_{{\mathcal A}_{0,S^1}({\mathbb R})}$, where $L_b$ denotes the Laplace transform of $b$. This improves previous functional calculus estimates recently obtained by the first two authors.
