TY - JOUR
T1 - Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline cases
AU - Rafeiro, Humberto
AU - Samko, Stefan
AU - Umarkhadzhiev, Salaudin
N1 - Funding Information:
The research of H. Rafeiro was supported by a Research Start‐up Grant of United Arab Emirates University, Al Ain, United Arab Emirates via Grant No. G00002994. The research of S. Samko was supported by: (a) Russian Foundation for Basic Research under the grant 19‐01‐00223 and (b) TUBITAK and Russian Foundation for Basic research under the grant No 20‐51‐46003. The research of S. Umarkhadzhiev was supported by: (a) Russian Foundation for Basic Research under the grant 18‐01‐00094 and (b) TUBITAK and Russian Foundation for Basic research under the grant No 20‐51‐46003.
Publisher Copyright:
© 2022 Wiley-VCH GmbH.
PY - 2022/5
Y1 - 2022/5
N2 - We define the grand Lebesgue space corresponding to the case (Formula presented.) and similar grand spaces for Morrey and Morrey type spaces, also for (Formula presented.), on open sets in (Formula presented.). We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases (Formula presented.) for Lebesgue spaces and (Formula presented.) for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.
AB - We define the grand Lebesgue space corresponding to the case (Formula presented.) and similar grand spaces for Morrey and Morrey type spaces, also for (Formula presented.), on open sets in (Formula presented.). We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases (Formula presented.) for Lebesgue spaces and (Formula presented.) for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.
KW - BMO
KW - grand Lebesgue spaces
KW - Riesz potential operator
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U2 - 10.1002/mana.202000347
DO - 10.1002/mana.202000347
M3 - Article
AN - SCOPUS:85113990559
VL - 295
SP - 991
EP - 1007
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
SN - 0025-584X
IS - 5
ER -