TY - JOUR

T1 - On the interpolation constants for variable Lebesgue spaces

AU - Karlovych, Oleksiy

AU - Shargorodsky, Eugene

N1 - info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT#
Publisher Copyright:
© 2023 Wiley-VCH GmbH.

PY - 2023/7

Y1 - 2023/7

N2 - For (Formula presented.) and variable exponents (Formula presented.) and (Formula presented.) with values in [1, ∞], let the variable exponents (Formula presented.) be defined by (Formula presented.) The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space (Formula presented.) to the variable Lebesgue space (Formula presented.) for (Formula presented.), then (Formula presented.) where C is an interpolation constant independent of T. We consider two different modulars (Formula presented.) and (Formula presented.) generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that (Formula presented.) and (Formula presented.), as well as, lead to sufficient conditions for (Formula presented.) and (Formula presented.). We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that (Formula presented.), (Formula presented.) are Lipschitz continuous and bounded away from one and infinity (in this case, (Formula presented.)).

AB - For (Formula presented.) and variable exponents (Formula presented.) and (Formula presented.) with values in [1, ∞], let the variable exponents (Formula presented.) be defined by (Formula presented.) The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space (Formula presented.) to the variable Lebesgue space (Formula presented.) for (Formula presented.), then (Formula presented.) where C is an interpolation constant independent of T. We consider two different modulars (Formula presented.) and (Formula presented.) generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that (Formula presented.) and (Formula presented.), as well as, lead to sufficient conditions for (Formula presented.) and (Formula presented.). We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that (Formula presented.), (Formula presented.) are Lipschitz continuous and bounded away from one and infinity (in this case, (Formula presented.)).

KW - Calderón product

KW - complex method of interpolation

KW - interpolation constant

KW - Riesz–Thorin interpolation theorem

KW - variable Lebesgue space

UR - http://www.scopus.com/inward/record.url?scp=85152251816&partnerID=8YFLogxK

U2 - 10.1002/mana.202100549

DO - 10.1002/mana.202100549

M3 - Article

AN - SCOPUS:85152251816

SN - 0025-584X

VL - 296

SP - 2877

EP - 2902

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

IS - 7

ER -