Abstract
It is shown that every nonlinear centralizer from \(L_p\) to \(L_q\) is trivial unless \(q=p\). This means that if \(q\ne p\), the only exact sequence of quasi-Banach \(L_\infty \)-modules and homomorphisms \(0\rightarrow L_q\rightarrow Z\rightarrow L_p\rightarrow 0\) is the trivial one where \(Z=L_q\oplus L_p\). From this it follows that the space of centralizers on \(L_p\) is essentially independent on \(p\in (0,\infty )\), which confirms a conjecture by Kalton.
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Research supported in part by MTM2010-20190-C02-01 and Junta de Extremadura GR10113.