Vector Valued Transference
| Autor: | Berkson, E., Gillespie, T. A., Torrea, J. L. |
|---|---|
| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Our principal result is the following. Let $X$ and $Y$ be Banach spaces, let $G$ be a locally compact abelian group, and let $K$ be an operator valued kernel defined on $G$ with values in the space of bounded linear operators from $X$ to $Y$. Suppose that $R$ and $\tilde{R}$ are representations of $G$ on $X$ and $Y$ respectively that intertwine the values of $K$. Then, under suitable boundedness conditions on $R, \tilde{R}$ and $K$, the formula $$T_Kx = \int_GK(u)R_{-u}xdu $$ defines a bounded linear operator $T_K$ from $X$ to $Y$ with norm controlled by norm of convolution by $K$ as a mapping from $L^p_X(G)$ into $L^p_Y(G)$, (for all values of $p$ in the range $1\le p < \infty$.) A number of applications to the geometry of Banach spaces are given. Several results are proved in the setting of abstract commutative harmonic analysis. We outline the proof of the affirmative resolution of a conjecture of Rubio de Francia. This technique of transference is used to obtain dimension free estimates for certain operators in an $\Bbb{R}^n$ setting. Comment: This survey is the core of a talk given by the third author at the International Conference and 13th Academic Symposium of China on Functional Analysis and Applications, Wuhan 2003. It was published by Research Information Ltd, UK |
| Databáze: | arXiv |
| Externí odkaz: |
|