A family of equivalent norms for Lebesgue spaces

Abstract

If $$\psi :[0,\ell ]\rightarrow [0,\infty [$$ ψ : [ 0 , ℓ ] → [ 0 , ∞ [ is absolutely continuous, nondecreasing, and such that $$\psi (\ell )>\psi (0)$$ ψ ( ℓ ) > ψ ( 0 ) , $$\psi (t)>0$$ ψ ( t ) > 0 for $$t>0$$ t > 0 , then for $$f\in L^1(0,\ell )$$ f ∈ L 1 ( 0 , ℓ ) , we have $$\begin{aligned} \Vert f\Vert _{1,\psi ,(0,\ell )}:=\int \limits _0^\ell \frac{\psi '(t)}{\psi (t)^2}\int \limits _0^tf^*(s)\psi (s)dsdt\approx \int \limits _0^\ell |f(x)|dx=:\Vert f\Vert _{L^1(0,\ell )},\quad (*) \end{aligned}$$ ‖ f ‖ 1 , ψ , ( 0 , ℓ ) : = ∫ 0 ℓ ψ ′ ( t ) ψ ( t ) 2 ∫ 0 t f ∗ ( s ) ψ ( s ) d s d t ≈ ∫ 0 ℓ | f ( x ) | d x = : ‖ f ‖ L 1 ( 0 , ℓ ) , ( ∗ ) where the constant in $$ > rsim $$ ≳ depends on $$\psi $$ ψ and $$\ell $$ ℓ . Here by $$f^*$$ f ∗ we denote the decreasing rearrangement of f. When applied with f replaced by $$|f|^p$$ | f | p , $$1<p<\infty $$ 1 < p < ∞ , there exist functions $$\psi $$ ψ so that the inequality $$\Vert |f|^p\Vert _{1,\psi ,(0,\ell )}\le \Vert |f|^p\Vert _{L^1(0,\ell )}$$ ‖ | f | p ‖ 1 , ψ , ( 0 , ℓ ) ≤ ‖ | f | p ‖ L 1 ( 0 , ℓ ) is not rougher than the classical one-dimensional integral Hardy inequality over bounded intervals $$(0,\ell )$$ ( 0 , ℓ ) . We make an analysis on the validity of $$(*)$$ ( ∗ ) under much weaker assumptions on the regularity of $$\psi $$ ψ , and we get a version of Hardy’s inequality which generalizes and/or improves existing results.

Modulars from Nakano onwards

“…3. Recently, in F., Jain [57], it has been shown that if ρ(f ) = f L 1 (0, ) and ψ : [0, ] → [0, ∞[ is absolutely continuous, nondecreasing, and such that ψ( ) > ψ(0), ψ(t) > 0 for t > 0, then…”