Integral conditions for Hardy-type operators involving suprema

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• 作者：Martin Křepela
• 关键词：Operators with suprema ; Hardy ; type inequalities ; Weights
• 刊名：Collectanea Mathematica
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：68
• 期：1
• 页码：21-50
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Applications of Mathematics; Algebra; Geometry; Analysis;
• 出版者：Springer Milan
• ISSN：2038-4815
• 卷排序：68

文摘

We characterize the validity of the weighted inequality $$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) \int _s^\infty g(x)\,\mathrm {d}x\Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty g^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$for all nonnegative functions g on \((0,\infty )\), with exponents in the range \(1\le p<\infty \) and \(0<q<\infty \).Moreover, we give an integral characterization of the inequality $$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) f(s) \Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty f^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$being satisfied for all nonnegative nonincreasing functions f on \((0,\infty )\) in the case \(0<q<p<\infty \), for which an integral condition was previously unknown.KeywordsOperators with supremaHardy-type inequalitiesWeightsMathematics Subject Classification47G1026D15References1.Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press, Boston (1988)MATHGoogle Scholar2.Gogatishvili, A., Křepela, M., Pick, L., Soudský, F.: Embeddings of classical Lorentz spaces involving weighted integral means. (2016)3.Gogatishvili, A., Kufner, A., Persson, L.-E.: Some new scales of weight characterizations of the class \(B_p\). Acta Math. Hung. 123, 365–377 (2009)MathSciNetCrossRefMATHGoogle Scholar4.Gogatishvili, A., Mustafayev R.: Weighted iterated Hardy-type inequalities. (2015)5.Gogatishvili, A., Opic, B., Pick, L.: Weighted inequalities for Hardy-type operators involving suprema. Collect. Math. 57, 227–255 (2006)MathSciNetMATHGoogle Scholar6.Gogatishvili, A., Persson, L.-E., Stepanov, V.D., Wall, P.: On scales of equivalent conditions that characterize the weighted Stieltjes inequality. Dokl. Math. 86, 738–739 (2012)7.Gogatishvili, A., Persson, L.-E., Stepanov, V.D., Wall, P.: Some scales of equivalent conditions to characterize the Stieltjes inequality: the case \(q<p\). Math. Nachr. 287, 242–253 (2014)8.Gogatishvili, A., Pick, L.: Duality principles and reduction theorems. Math. Inequal. Appl. 43, 539–558 (2000)9.Gogatishvili, A., Pick, L.: A reduction theorem for supremum operators. J. Comput. Appl. Math. 208, 270–279 (2007)10.Gogatishviliand, A., Stepanov, V.D.: Reduction theorems for operators on the cones of monotone functions. J. Math. Anal. Appl. 405, 156–172 (2013)MathSciNetCrossRefGoogle Scholar11.Goldman, M.L., Heinig, H.P., Stepanov, V.D.: On the principle of duality in Lorentz spaces. Can. J. Math. 48, 959–979 (1996)12.Grosse-Erdmann K.-G.: The Blocking Technique, Weighted Mean Operators and Hardy’s Inequality, Lecture Notes in Mathematics, 1679. Springer-Verlag, Berlin (1998)13.Sinnamon, G.: Transferring monotonicity in weighted norm inequalities. Collect. Math. 54, 181–216 (2003)MathSciNetMATHGoogle ScholarCopyright information© Universitat de Barcelona 2016Authors and AffiliationsMartin Křepela12Email author1.Department of Mathematics and Computer Science, Faculty of Health, Science and TechnologyKarlstad UniversityKarlstadSweden2.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic About this article CrossMark Publisher Name Springer Milan Print ISSN 0010-0757 Online ISSN 2038-4815 About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips