Three weight Koopman semigroups on Lebesgue spaces

Miana, Pedro J. (Universidad de Zaragoza) ; Poblete, Verónica

Resumen: In this paper, we consider three different semiflows (φt)t≥0, (ψt)t≥0 and (ϕt)t≥0 on the real half-line given by φt(r) := e−tr + 1 − e−t, ψt(r) := etr / 1 + r(et − 1), ϕt(r) := (1 + et)r − 1 + et / (−1 + et)r + 1 + et , for r, t ≥ 0. These semiflows induce three weight Koopman semigroups, (Tγt,p)t>0, (Sγt,p)t>0 and (Rγt,p)t>0 on the fractional Lebesgue spaces T (α)p (tα), closed subspaces of L p(R+) for some α and γ ≥ 0. We describe spectrum sets, point spectrums and resolvent operators of their infinitesimal generators. Three Cesàro-like operators, defined using the Chen fractional integral, Cμ,ν f (r) := 1|r − 1 μ+ν−1 / 1,r |s − 1| μ−1|r − s| ν−1 f (s)ds, r > 0, Cγ μ,ν f (r) := rμ /|r − 1| 1,r|s − 1|μ+γ −1 sμ+ν |r − s|ν−1 f (s)ds, r > 0, Cγμ,ν f (r) := 2ν |r + 1|μ−γ / |r − 1|μ+ν−1 1,r |s − 1|μ−1 /|s + 1|μ+ν−γ |r − s| ν−1 f (s)ds, r > 0, (for certain μ, ν, γ ∈ R and 1,r := (1,r) when r > 1 and 1,r := (r, 1) in the case 0 < r < 1) are subordinated to these C0-semigroups. These representations allow to obtain their norms and spectrum sets.

Idioma: Inglés

DOI: 10.1007/s00605-024-02041-2

Año: 2025

Publicado en: MONATSHEFTE FUR MATHEMATIK 206 (2025), 629-664
ISSN: 0026-9255

Financiación: info:eu-repo/grantAgreement/ES/DGA/E48-20R
Financiación: info:eu-repo/grantAgreement/ES/MCINN/PID2022-137294NB-I00

Tipo y forma: Article (Published version)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)