固定拷贝指数集与Hardy空间间复合算子的严格奇异性
Fixed-Copy Exponent Sets and Strict Singularity of Composition Operators Between Hardy Spaces
AI总结：
该研究定义固定拷贝指数集，确定Hardy空间间有界复合算子的该集合，明确复合算子严格奇异性的等价条件，解决相关公开问题，引入局部化方法完成证明。
AI中文摘要：
针对Banach空间间的有界算子\boldsymbol{T}，我们引入了\boldsymbol{固定拷贝指数集}：\n\\[\n\begin{aligned}\n\operatorname{Fix}_{\ell}(T) &:= \{r\ge1:T\text{ 固定一个 }\ell^r\text{ 的拷贝}\},\n\operatorname{Fix}_{L}(T) &:= \{r\ge1:T\text{ 固定一个 }L^r(0,1)\text{ 的拷贝}\}.\n\end{aligned}\n\\]\n对于\boldsymbol{1\le p,q<\infty}，我们完全确定了每一个有界复合算子\boldsymbol{C_\varphi:H^p\to H^q}对应的上述两个集合。特别地，\boldsymbol{C_\varphi}是严格奇异的当且仅当\boldsymbol{\operatorname{Fix}_{\ell}(C_\varphi)=\varnothing}。该分类还对每一个\boldsymbol{r\ge1}完全刻画了\boldsymbol{\ell^r}-奇异与\boldsymbol{L^r(0,1)}-奇异子类。进一步地，它完全解决了Laitila、Nieminen、Saksman和Tylli提出的问题4.3(1)与4.3(2)。证明中引入了一种新的局部化方法，用于从边界下界估计中生成固定拷贝，其核心是一个与复合算子无关的局部化定理：对每一个满足\boldsymbol{m(E)>0}的可测集\boldsymbol{E\subset\mathbb T}及\boldsymbol{1\le p<r\le2}，它在\boldsymbol{H^p}中构造了一个\boldsymbol{L^r(0,1)}的单拷贝，使得对所有\boldsymbol{1\le s\le p}，同时成立\boldsymbol{L^s(E)}下界估计，且常数仅通过\boldsymbol{m(E)}依赖于\boldsymbol{E}。该分类将此方法与拉回测度准则、已知的固定拷贝结果及经典子空间限制相结合。对于\boldsymbol{r<2}，该局部化定理通过稳定积分与解析提升证明；端点\boldsymbol{r=2}则通过适配\boldsymbol{E}的缺项构造处理。
英文摘要：
For a bounded operator \(T\) between Banach spaces, we introduce the \emph{fixed-copy exponent sets} \[ \begin{aligned} \operatorname{Fix}_{\ell}(T) &:= \{r\ge1:T\text{ fixes a copy of }\ell^r\},\\ \operatorname{Fix}_{L}(T) &:= \{r\ge1:T\text{ fixes a copy of }L^r(0,1)\}. \end{aligned} \] For \(1\le p,q<\infty\), we completely determine both sets for every bounded composition operator \(C_φ:H^p\to H^q\). In particular, \(C_φ\) is strictly singular if and only if \(\operatorname{Fix}_{\ell}(C_φ)=\varnothing\). The classification also gives complete characterizations of the \(\ell^r\)-singular and \(L^r(0,1)\)-singular subclasses for every \(r\ge1\). As a further consequence, it completely resolves Problems~4.3\textup{(1)} and~4.3\textup{(2)} posed by Laitila, Nieminen, Saksman, and Tylli. The proofs introduce a new localization method for producing fixed copies from boundary lower estimates. Its main ingredient is a localization theorem independent of composition operators: for every measurable \(E\subset\mathbb T\) with \(m(E)>0\) and \(1\le p<r\le2\), it constructs a single copy of \(L^r(0,1)\) in \(H^p\) on which lower \(L^s(E)\) estimates hold simultaneously for all \(1\le s\le p\), with constants depending on \(E\) only through \(m(E)\). The classification combines this method with pullback measure criteria, known fixed-copy results, and classical subspace restrictions. For \(r<2\), the localization theorem is proved using stable integrals and analytic lifting; the endpoint \(r=2\) is handled by an \(E\)-adapted lacunary construction.
