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 -