Abstract
We generalize the Hardy inequality to Hardy–Morrey spaces.
Funding source: Hong Kong Institute of Education
Award Identifier / Grant number: RG21/14-15R
Funding statement: The author is partially supported by HKIEd Internal Research Grant RG21/14-15R.
Acknowledgements
The author would like to thank the referee for his/her valuable suggestions for improving the content of this paper.
References
[1] P. Butzer and F. Fehér, Generalized Hardy and Hardy–Littlewood inequalities in rearrangement-invariant spaces, Comment. Math. Special Issue 1 (1978), 41–64. Search in Google Scholar
[2] D. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Monogr. Math., Springer, Berlin, 2004. 10.1007/978-3-662-07731-3Search in Google Scholar
[3] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland Publishing, Amsterdam, 1985. Search in Google Scholar
[4] G. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), no. 3–4, 314–317. 10.1007/BF01199965Search in Google Scholar
[5] G. Hardy, J. Littlewood and G. Pólya, Inequalities, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1988. Search in Google Scholar
[6] K.-P. Ho, Vector-valued singular integral operators on Morrey type spaces and variable Triebel–Lizorkin–Morrey spaces, Ann. Acad. Sci. Fenn. Math. 37 (2012), no. 2, 375–406. 10.5186/aasfm.2012.3746Search in Google Scholar
[7] K.-P. Ho, Atomic decompositions of weighted Hardy–Morrey spaces, Hokkaido Math. J. 42 (2013), no. 1, 131–157. 10.14492/hokmj/1362406643Search in Google Scholar
[8] K.-P. Ho, Atomic decomposition of Hardy–Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 31–62. 10.5186/aasfm.2015.4002Search in Google Scholar
[9] K.-P. Ho, Hardy’s inequality and Hausdorff operators on rearrangement-invariant Morrey spaces, Publ. Math. Debrecen 88 (2016), no. 1–2, 201–215. 10.5486/PMD.2016.7357Search in Google Scholar
[10] K.-P. Ho, Hardy’s inequalities on Hardy spaces, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 10, 125–130. 10.3792/pjaa.92.125Search in Google Scholar
[11] K.-P. Ho, Atomic decompositions and Hardy’s inequality on weak Hardy–Morrey spaces, Sci. China Math. 60 (2017), no. 3, 449–468. 10.1007/s11425-016-0229-1Search in Google Scholar
[12] K.-P. Ho, Hardy’s inequality on Hardy–Morrey spaces with variable exponents, Mediterr. J. Math. 14 (2017), no. 2, Article ID 79. 10.1007/s00009-016-0811-8Search in Google Scholar
[13] H. Jia and H. Wang, Decomposition of Hardy–Morrey spaces, J. Math. Anal. Appl. 354 (2009), no. 1, 99–110. 10.1016/j.jmaa.2008.12.051Search in Google Scholar
[14] L. Maligranda, Generalized Hardy inequalities in rearrangement invariant spaces, J. Math. Pures Appl. (9) 59 (1980), no. 4, 405–415. Search in Google Scholar
[15] C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. 10.1090/S0002-9947-1938-1501936-8Search in Google Scholar
[16] E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95–103. 10.1002/mana.19941660108Search in Google Scholar
[17] B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser. 219, Longman Scientific & Technical, Harlow, 1990. Search in Google Scholar
[18] Y. Sawano, A note on Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 8, 1223–1242. 10.1007/s10114-009-8247-8Search in Google Scholar
[19] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993. 10.1515/9781400883929Search in Google Scholar
[20] W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Math. 2005, Springer, Berlin, 2010. 10.1007/978-3-642-14606-0Search in Google Scholar
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