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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.01420 |
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| _version_ | 1866913996833030144 |
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| author | Kiwerski, Tomasz Tomaszewski, Jakub |
| author_facet | Kiwerski, Tomasz Tomaszewski, Jakub |
| contents | We will provide a complete description of the space $M(X_F,X_G)$ of pointwise multipliers between two Calderón--Lozanovskiĭ spaces $X_F$ and $X_G$ built upon a rearrangement invariant space $X$ and two Young functions $F$ and $G$. Meeting natural expectations, the space $M(X_F,X_G)$ turns out to be another Calderón--Lozanovskiĭ space $X_{G \ominus F}$ with $G \ominus F$ being the appropriately understood generalized Young conjugate of $G$ with respect to $F$. Nevertheless, our argument is not a mere transplantation of existing techniques and requires a rather delicate analysis of the interplay between the space $X$ and functions $F$ and $G$. Furthermore, as an example to illustrate applications, we will solve the factorization problem for Calderón--Lozanovskiĭ spaces. All this not only complements and improves earlier results (basically giving them the final touch), but also confirms the conjecture formulated by Kolwicz, Leśnik and Maligranda in [Pointwise multipliers of Calderón--Lozanovskiĭ spaces, Math. Nachr. 286 (2012), no. 8-9, 876--907]. We will close this work by formulating a number of open questions that outline a promising panorama for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01420 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A few last words on pointwise multipliers of Calderón--Lozanovskiĭ spaces Kiwerski, Tomasz Tomaszewski, Jakub Functional Analysis We will provide a complete description of the space $M(X_F,X_G)$ of pointwise multipliers between two Calderón--Lozanovskiĭ spaces $X_F$ and $X_G$ built upon a rearrangement invariant space $X$ and two Young functions $F$ and $G$. Meeting natural expectations, the space $M(X_F,X_G)$ turns out to be another Calderón--Lozanovskiĭ space $X_{G \ominus F}$ with $G \ominus F$ being the appropriately understood generalized Young conjugate of $G$ with respect to $F$. Nevertheless, our argument is not a mere transplantation of existing techniques and requires a rather delicate analysis of the interplay between the space $X$ and functions $F$ and $G$. Furthermore, as an example to illustrate applications, we will solve the factorization problem for Calderón--Lozanovskiĭ spaces. All this not only complements and improves earlier results (basically giving them the final touch), but also confirms the conjecture formulated by Kolwicz, Leśnik and Maligranda in [Pointwise multipliers of Calderón--Lozanovskiĭ spaces, Math. Nachr. 286 (2012), no. 8-9, 876--907]. We will close this work by formulating a number of open questions that outline a promising panorama for future research. |
| title | A few last words on pointwise multipliers of Calderón--Lozanovskiĭ spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2410.01420 |