Hardy type inequalities with kernels: The current status and some new results

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• 作者：Alois Kufner ; Lars-Erik Persson and Natasha Samko
• 刊名：Mathematische Nachrichten
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：290
• 期：1
• 页码：57-65
• 全文大小：171K
• ISSN：1522-2616

文摘

We consider the general Hardy type operator brary/content/render" xmlns="http://www.w3.org/1998/Math/MathML">b>Hkb>f(x)=bsup>∫0xbsup>k(x,t)f(t)dt, where brary/content/render" xmlns="http://www.w3.org/1998/Math/MathML">k(x,t) is a positive and measurable kernel. To characterize the weights u and v so that brary/content/render" xmlns="http://www.w3.org/1998/Math/MathML">b>Hkb>:b>Lpb>(u)→b>Lqb>(v) is still an open problem for any parameters p and q. However, for special cases the solution is known for some parameters p and q. In this paper the current status of this problem is described and discussed mainly for the case brary/content/render" xmlns="http://www.w3.org/1998/Math/MathML">1<p≤q<∞. In particular, some new scales of characterizations in classical situations are described, some new proofs and results are given and open questions are raised.