TY - JOUR
T1 - On interpolation of reflexive variable Lebesgue spaces on which the Hardy-Littlewood maximal operator is bounded
AU - Diening, Lars
AU - Karlovych, Oleksiy
AU - Shargorodsky, Eugene
N1 - info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT#
Lars Diening was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1283/2 2021 – 317210226
Publisher Copyright:
© 2022 Walter de Gruyter GmbH, Berlin/Boston.
PY - 2022/6/1
Y1 - 2022/6/1
N2 - We show that if the Hardy-Littewood maximal operator M is bounded on a reflexive variable exponent space Lp(·) (ℝd), then for every q ϵ (1, ∞), the exponent p(·) admits, for all sufficiently small θ > 0, the representation 1/p(x) = θ/q + 1 - θ/ r(x), x ϵ ℝd, such that the operator M is bounded on the variable Lebesgue space Lr(·) (ℝd). This result can be applied for transferring properties like compactness of linear operators from standard Lebesgue spaces to variable Lebesgue spaces by using interpolation techniques.
AB - We show that if the Hardy-Littewood maximal operator M is bounded on a reflexive variable exponent space Lp(·) (ℝd), then for every q ϵ (1, ∞), the exponent p(·) admits, for all sufficiently small θ > 0, the representation 1/p(x) = θ/q + 1 - θ/ r(x), x ϵ ℝd, such that the operator M is bounded on the variable Lebesgue space Lr(·) (ℝd). This result can be applied for transferring properties like compactness of linear operators from standard Lebesgue spaces to variable Lebesgue spaces by using interpolation techniques.
KW - Hardy-Littlewood maximal operator
KW - interpolation
KW - Variable Lebesgue space
UR - http://www.scopus.com/inward/record.url?scp=85127871075&partnerID=8YFLogxK
U2 - 10.1515/gmj-2022-2152
DO - 10.1515/gmj-2022-2152
M3 - Article
AN - SCOPUS:85127871075
SN - 1072-947X
VL - 29
SP - 347
EP - 352
JO - Georgian Mathematical Journal
JF - Georgian Mathematical Journal
IS - 3
ER -