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 -