A family of equivalent norms for Lebesgue spaces

• Published in: Archiv der Mathematik Vol. 116; no. 2; pp. 179 - 192
• Main Authors: Fiorenza, Alberto, Jain, Pankaj
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2021; Springer Nature B.V
• Subjects: Inequality; Mathematics; Mathematics and Statistics; Norms; Monotone functions; 26D10; 46E30; G; Absolutely continuous functions; Lebesgue spaces; spaces; Banach function space norms; Integral inequalities; 26D15; Weighted Lebesgue spaces
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-020-01534-4

If ψ : [ 0 , ℓ ] → [ 0 , ∞ [ is absolutely continuous, nondecreasing, and such that ψ ( ℓ ) > ψ ( 0 ) , ψ ( t ) > 0 for t > 0 , then for f ∈ L 1 ( 0 , ℓ ) , we have ‖ 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 ≳ depends on ψ and ℓ . Here by f ∗ we denote the decreasing rearrangement of f . When applied with f replaced by | f | p , 1 < p < ∞ , there exist functions ψ so that the inequality ‖ | 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 , ℓ ) . We make an analysis on the validity of ( ∗ ) under much weaker assumptions on the regularity of ψ , and we get a version of Hardy’s inequality which generalizes and/or improves existing results.