Constructive Approximation ( IF 2.3 ) Pub Date : 2021-03-08 , DOI: 10.1007/s00365-021-09526-5 Bae Jun Park
We study multiplier theorems on a vector-valued function space, which is a generalization of the results of Calderón and Torchinsky, and Grafakos, He, Honzík, and Nguyen, and an improvement of the result of Triebel. For \(0<p<\infty \) and \(0<q\le \infty \) we obtain that if \(r>\frac{d}{s-(d/\min {(1,p,q)}-d)}\), then
$$\begin{aligned} \big \Vert \big \{\big ( m_k \widehat{f_k}\big )^{\vee }\big \}_{k\in {\mathbb {Z}}}\big \Vert _{L^p(\ell ^q)}\lesssim _{p,q} \sup _{l\in {\mathbb {Z}}}{\big \Vert m_l(2^l\cdot )\big \Vert _{L_s^r({\mathbb {R}}^d)}} \big \Vert \big \{f_k\big \}_{k\in {\mathbb {Z}}}\big \Vert _{L^p(\ell ^q)}, ~~f_k\in {\mathcal {E}}(A2^k), \end{aligned}$$under the condition \(\max {(|d/p-d/2|,|d/q-d/2|)}<s<d/\min {(1,p,q)}\). An extension to \(p=\infty \) will be additionally considered in the scale of Triebel–Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by Sobolev spaces \(L_s^r\) with \(r\le \frac{d}{s-(d/\min {(1,p,q)}-d)}\).
中文翻译：
向量值函数空间上的傅立叶乘法器
我们在向量值函数空间上研究乘子定理，这是对Calderón和Torchinsky以及Grafakos，He，Honzík和Nguyen的结果的推广，并且对Triebel的结果进行了改进。对于\（0 <p <\ infty \）和\（0 <q \ le \ infty \），我们得出，如果\（r> \ frac {d} {s-（d / \ min {（1，p， q）}-d）} \），然后
$$ \ begin {aligned} \ big \ Vert \ big \ {\ big（m_k \ widehat {f_k} \ big）^ {\ vee} \ big \} _ {k \ in {\ mathbb {Z}}} \\大\ Vert _ {L ^ p（\ ell ^ q）} \ lesssim _ {p，q} \ sup _ {l \ in {\ mathbb {Z}}} {\大\ Vert m_l（2 ^ l \ cdot ）\ big \ Vert _ {L_s ^ r（{\ mathbb {R}} ^ d）}}} \ big \ Vert \ big \ {f_k \ big \} _ {k \ in {\ mathbb {Z}}} \\大\ Vert _ {L ^ p（\ ell ^ q）}，~~ f_k \ in {\ mathcal {E}}（A2 ^ k），\ end {aligned} $$在条件\（\ max {（| d / pd / 2 |，| d / qd / 2 |）} <s <d / \ min {（1，p，q）} \）的情况下。\（p = \ infty \）的扩展将在Triebel–Lizorkin空间的范围内另外考虑。我们的结果是在这个意义上，在上述估计的Sobolev空间不能被Sobolev空间来代替尖锐\（L_S ^ R \）与\（R \文件\压裂{d} {S-（d / \分钟{（1 ，p，q）}-d）} \）。
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