A family of equivalent norms for Lebesgue spaces

摘要

If psi : [0, l] -> [0, infinity[ is absolutely continuous, nondecreasing, and such that psi(l) > psi(0), psi(t) > 0 for t > 0, then for f is an element of L-1(0, l), we have parallel to f parallel to(1,psi,(0,l)) := integral(l)(0)psi'(t)/psi(t)(2) integral(t)(0)f *(s)psi(s)dsdt approximate to integral(l)(0) |f(x)|dx =: parallel to f parallel to(L1( 0,l),) (*) where the constant in greater than or similar to depends on psi and l. Here by f * we denote the decreasing rearrangement of f. When applied with f replaced by |f|(p), 1 < p < infinity, there exist functions psi so that the inequality parallel to|f|(p)parallel to(1,psi,(0,l)) <= parallel to|f|(p)parallel to(L1(0,l)) is not rougher than the classical one-dimensional integral Hardy inequality over bounded intervals (0, l). We make an analysis on the validity of (*) under much weaker assumptions on the regularity of psi, and we get a version of Hardy's inequality which generalizes and/or improves existing results.