Generalized upper and lower Legendre conjugates for weight functions
Abstract
We introduce and study new transformations between two functions satisfying some basic growth properties and generalize the known lower and upper Legendre conjugate (or envelope). We also investigate how these transformations modify recently defined growth indices for weight functions. A special but important and useful situation, to which the knowledge is then applied, is when considering associated weight functions which are expressed in terms of an underlying weight sequence. In this case these transformations precisely correspond to the point-wise product resp. point-wise division of the given sequences. Therefore, the new approach studied in this work illustrates the genuineness and importance and suggests applications for weighted spaces in different directions.
Cite
@article{arxiv.2505.07497,
title = {Generalized upper and lower Legendre conjugates for weight functions},
author = {Gerhard Schindl},
journal= {arXiv preprint arXiv:2505.07497},
year = {2026}
}
Comments
41 pages; several changes according to the comments of the referees have been applied (non-standard cases separated in the new Appendix, technical gaps in Lemma 2.5 and Lemma 4.1 are corrected)