TY - JOUR

T1 - Remarks on a limiting case of hardy type inequalities

AU - Sano, Megumi

AU - Sobukawa, Takuya

N1 - Funding Information:
the proof is similar. Therefore, we omit the proof in that case. □ Acknowledgement. This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University and was partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). Also, the first author was supported by JSPS KAKENHI Early-Career Scientists, No. JP19K14568 and the second author was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(B), No. JP15H03621.
Publisher Copyright:
© 2020 Element D.O.O.. All rights reserved.

PY - 2020

Y1 - 2020

N2 - The classical Hardy inequality holds in Sobolev spaces W01, p when 1 ≤ p < N . In the limiting case where p = N , it is known that by introducing a logarithmic weight function in the Hardy potential, some inequality which is called the critical Hardy inequality holds in W01,N . In this note, in order to give an explanation of the appearance of the logarithmic function in the potential, we derive the logarithmic function from the classical Hardy inequality with best constant via some limiting procedure as p ↗ N . We show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W02, p with p ∈ (1, N2 ) and the Poincaré inequality.

AB - The classical Hardy inequality holds in Sobolev spaces W01, p when 1 ≤ p < N . In the limiting case where p = N , it is known that by introducing a logarithmic weight function in the Hardy potential, some inequality which is called the critical Hardy inequality holds in W01,N . In this note, in order to give an explanation of the appearance of the logarithmic function in the potential, we derive the logarithmic function from the classical Hardy inequality with best constant via some limiting procedure as p ↗ N . We show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces W02, p with p ∈ (1, N2 ) and the Poincaré inequality.

KW - Extrapolation

KW - Hardy inequality

KW - Limiting case

KW - Pointwise estimate of radial functions

KW - Sobolev embedding

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U2 - 10.7153/MIA-2020-23-102

DO - 10.7153/MIA-2020-23-102

M3 - Article

AN - SCOPUS:85107761349

SN - 1331-4343

VL - 23

SP - 1425

EP - 1440

JO - Mathematical Inequalities and Applications

JF - Mathematical Inequalities and Applications

IS - 4

ER -