Weights of reasonable growth and their application
Abstract
In the paper, we introduce the concept of weight with reasonable growth on a locally compact group $G$. We verify that these weights form a natural class to work with, by examining the most common examples. We proceed with the discussion of the $L^p$conjecture. With the use of RieszThorinSteinWeiss interpolation, we establish that \mbox{$L^p_{\omega}(G)\star L^p_{\omega}(G)\subset L^p_{\omega}(G)$}, $p>1$ implies that $L^p_{\omega}(G)\star L^q_{\omega}(G)\subset L^q_{\omega}(G)$ for $q$ which lies between $p$ and $p'$. At last, we confirm the $L^p_{\omega}$conjecture for weights of $(p,q)$reasonable growth.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 DOI:
 10.48550/arXiv.1709.00380
 arXiv:
 arXiv:1709.00380
 Bibcode:
 2017arXiv170900380K
 Keywords:

 Mathematics  Functional Analysis
 EPrint:
 Error in the proof