We prove that if p>1 and ψ:]0,p−1]→]0,∞[ is just nondecreasing and differentiable (hence not necessarily Δ2), then for every f Lebesgue measurable function on (0,1) sup0<1Sψ(t)‖f⁎‖Ljavax.xml.bind.JAXBElement@5695bc9b(t,1), where f⁎ denotes the decreasing rearrangement of f and Sψ is defined, for ε∈]0,p−1[, through [Formula presented] where cψ is the normalizing constant chosen so that ν((p−1)−)=1. If ψ is in a class of functions satisfying the Δ2 condition, essentially characterized by the so-called ∇′ condition, then inequality (⁎) is sharp, in the sense that both sides are equivalent. Estimate (⁎) generalizes an inequality of the type obtained by the second author with Farroni and Giova in [6] under the growth condition ψ∈Δ2.
An estimate of the blow-up of Lebesgue norms in the non-tempered case / Di Fratta, G.; Fiorenza, A.; Slastikov, V.. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 493:2(2021), p. 15. [10.1016/j.jmaa.2020.124550]
An estimate of the blow-up of Lebesgue norms in the non-tempered case
Di Fratta G.;Fiorenza A.
;
2021
Abstract
We prove that if p>1 and ψ:]0,p−1]→]0,∞[ is just nondecreasing and differentiable (hence not necessarily Δ2), then for every f Lebesgue measurable function on (0,1) sup0<1Sψ(t)‖f⁎‖Ljavax.xml.bind.JAXBElement@5695bc9b(t,1), where f⁎ denotes the decreasing rearrangement of f and Sψ is defined, for ε∈]0,p−1[, through [Formula presented] where cψ is the normalizing constant chosen so that ν((p−1)−)=1. If ψ is in a class of functions satisfying the Δ2 condition, essentially characterized by the so-called ∇′ condition, then inequality (⁎) is sharp, in the sense that both sides are equivalent. Estimate (⁎) generalizes an inequality of the type obtained by the second author with Farroni and Giova in [6] under the growth condition ψ∈Δ2.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.