A simple expression is presented that is equivalent to the norm of the Lp v → Lq u embedding of the cone of quasi-concave functions in the case 0 < q < p < ∞. The result is extended to more general cones and the case q = 1 is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of related operators on whole spaces. An equivalent norm for the dual of the Lorentz space
Γp(v) = { f: ( ∫0 ∞ (f**)pv )1/p < ∞ }
is also given. The expression is simple and concrete. An application is made to describe the weights for which the Hardy Littlewood Maximal Function is bounded on these Lorentz spaces.
@article{urn:eudml:doc:41463,
title = {Embeddings of concave functions and duals of Lorentz spaces.},
journal = {Publicacions Matem\`atiques},
volume = {46},
year = {2002},
pages = {489-515},
mrnumber = {MR1934367},
zbl = {1043.46026},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41463}
}
Sinnamon, Gord. Embeddings of concave functions and duals of Lorentz spaces.. Publicacions Matemàtiques, Tome 46 (2002) pp. 489-515. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41463/