Home Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
Article
Licensed
Unlicensed Requires Authentication

Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces

  • Hongliang Li EMAIL logo , Qinxiu Sun and Xiao Yu
Published/Copyright: January 10, 2018

Abstract

Given measurable functions ϕ, ψ on + and a kernel function k(x,y)0 satisfying the Oinarov condition, we study the Hardy operator

Kf(x)=ψ(x)0xk(x,y)ϕ(y)f(y)𝑑y,x>0,

between Orlicz–Lorentz spaces ΛXG(w), where f is a measurable function on +. We obtain sufficient conditions of boundedness of K:Λu0G0(w0)Λu1G1(w1) and K:Λu0G0(w0)Λu1G1,(w1). We also look into boundedness and compactness of K:Λu0p0(w0)Λu1p1,q1(w1) between weighted Lorentz spaces. The function spaces considered here are quasi-Banach spaces rather than Banach spaces. Specializing the weights and the Orlicz functions, we restore the existing results as well as we achieve new results in the new and old settings.

MSC 2010: 46E30; 46B42

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11401530

Award Identifier / Grant number: 11461033

Award Identifier / Grant number: 11561057

Award Identifier / Grant number: LQ13A010018

Award Identifier / Grant number: 20151BAB211002

Funding statement: Supported by National Natural Science Foundation of China (11401530, 11461033, 11561057), Natural Science Foundation of Zhejiang Province of China (LQ13A010018) and Natural Science Foundation of Jiangxi Province of China (20151BAB211002).

Acknowledgements

The first author expresses his deep gratitude to Professor Anna Kamińska for fruitful discussions. The paper was written while the first author was visiting the Department of Mathematics of the University of Memphis. He expresses his thanks to the university for the generous hospitality given to him during his visit.

References

[1] K. F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26. 10.4064/sm-72-1-9-26Search in Google Scholar

[2] M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), no. 2, 727–735. 10.1090/S0002-9947-1990-0989570-0Search in Google Scholar

[3] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988. Search in Google Scholar

[4] M. J. Carro, J. A. Raposo and J. Soria, Recent developments in the theory of Lorentz spaces and weighted inequalities, Mem. Amer. Math. Soc. 187 (2007), no. 877, 1–128. 10.1090/memo/0877Search in Google Scholar

[5] M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112 (1993), no. 2, 480–494. 10.1006/jfan.1993.1042Search in Google Scholar

[6] D. E. Edmunds, P. Gurka and L. Pick, Compactness of Hardy-type integral operators in weighted Banach function spaces, Studia Math. 109 (1994), no. 1, 73–90. Search in Google Scholar

[7] D. E. Edmunds, V. Kokilashvili and A. Meskhi, Boundedness and Compactness Integral Operators, Springer, Cham, 2002. 10.1007/978-94-015-9922-1Search in Google Scholar

[8] E. V. Ferreyra, Weighted Lorentz norm inequalities for integral operators, Studia Math. 96 (1990), no. 2, 125–134. 10.4064/sm-96-2-125-134Search in Google Scholar

[9] Z. Fu, L. Grafakos, S. Lu and F. Zhao, Sharp bounds for m-linear Hardy and Hilbert operators, Houston J. Math. 38 (2012), no. 1, 225–244. Search in Google Scholar

[10] Z. W. Fu, S. L. Gong, S. Z. Lu and W. Yuan, Weighted multilinear Hardy operators and commutators, Forum Math. 27 (2015), no. 5, 2825–2851. 10.1515/forum-2013-0064Search in Google Scholar

[11] A. Gogatishvili and V. D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of monotone functions, Uspekhi Mat. Nauk 68 (2013), no. 4(412), 3–68. 10.1070/RM2013v068n04ABEH004849Search in Google Scholar

[12] M. L. Goldman, Estimates for restrictions of monotone operators on the cone of decreasing functions in Orlicz space, Mat. Zametki 100 (2016), no. 1, 30–46. 10.1134/S0001434616070038Search in Google Scholar

[13] M. L. Goldman, Estimates for the norms of monotone operators on weighted Orlicz–Lorentz classes, Dokl. Akad. Nauk 471 (2016), no. 2, 131–135. 10.1134/S1064562416060065Search in Google Scholar

[14] H. P. Heinig and Q. Lai, Weighted modular inequalities for Hardy-type operators on monotone functions, JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), no. 1, Article ID 10. Search in Google Scholar

[15] R. A. Hunt, On L(p,q) spaces, Enseign. Math. (2) 12 (1966), 249–276. Search in Google Scholar

[16] A. Kamińska, Some remarks on Orlicz–Lorentz spaces, Math. Nachr. 147 (1990), 29–38. 10.1002/mana.19901470104Search in Google Scholar

[17] A. Kamińska, Uniform convexity of generalized Lorentz spaces, Arch. Math. (Basel) 56 (1991), no. 2, 181–188. 10.1007/BF01200349Search in Google Scholar

[18] A. Kamińska and L. Maligranda, On Lorentz spaces Γp,w, Israel J. Math. 140 (2004), 285–318. 10.1007/BF02786637Search in Google Scholar

[19] A. Kamińska and L. Maligranda, Order convexity and concavity of Lorentz spaces Λp,w, 0<p<, Studia Math. 160 (2004), no. 3, 267–286. 10.4064/sm160-3-5Search in Google Scholar

[20] A. Kamińska and Y. Raynaud, Isomorphic copies in the lattice E and its symmetrization E(*) with applications to Orlicz–Lorentz spaces, J. Funct. Anal. 257 (2009), no. 1, 271–331. 10.1016/j.jfa.2009.02.016Search in Google Scholar

[21] L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982. Search in Google Scholar

[22] V. Kokilashvili and A. Meskhi, Two-weighted inequalities for integral operators in Lorentz spaces defined on homogeneous groups, Georgian Math. J. 6 (1999), no. 1, 65–82. 10.1023/A:1022930410227Search in Google Scholar

[23] M. A. Krasnosel’skiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces. Translated from the First Russian Edition by Leo F. Boron, P. Noordhoff, Groningen, 1961. Search in Google Scholar

[24] H. Li, The Riesz convergence property on weighted Lorentz spaces and Orlicz–Lorentz spaces, Quaest. Math. 36 (2013), no. 2, 181–196. 10.2989/16073606.2013.801129Search in Google Scholar

[25] H. Li and A. Kamińska, Boundedness and compactness of Hardy operator on Lorentz-type spaces, Math. Nachr. 290 (2017), no. 5–6, 852–866. 10.1002/mana.201600049Search in Google Scholar

[26] E. Lomakina and V. Stepanov, On the compactness and approximation numbers of Hardy-type integral operators in Lorentz spaces, J. Lond. Math. Soc. (2) 53 (1996), no. 2, 369–382. 10.1112/jlms/53.2.369Search in Google Scholar

[27] E. Lomakina and V. Stepanov, On the Hardy-type integral operators in Banach function spaces, Publ. Mat. 42 (1998), no. 1, 165–194. 10.5565/PUBLMAT_42198_09Search in Google Scholar

[28] G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950), 37–55. 10.2307/1969496Search in Google Scholar

[29] L. Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 1–49. Search in Google Scholar

[30] F. J. Martín-Reyes, P. Ortega Salvador and M. D. Sarrión Gavilán, Boundedness of operators of Hardy type in Λp,q spaces and weighted mixed inequalities for singular integral operators, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 1, 157–170. 10.1017/S0308210500023556Search in Google Scholar

[31] M. a. Mastył o, Interpolation of linear operators in Calderón–Lozanovskiĭ spaces, Comment. Math. Prace Mat. 26 (1986), no. 2, 247–256. Search in Google Scholar

[32] V. G. Maz’ja, Sobolev Spaces. Translated from the Russian by T. O. Shaposhnikova, Springer, Berlin, 1985. Search in Google Scholar

[33] S. J. Montgomery-Smith, Comparison of Orlicz–Lorentz spaces, Studia Math. 103 (1992), no. 2, 161–189. 10.4064/sm-103-2-161-189Search in Google Scholar

[34] R. Oĭnarov, Weighted inequalities for a class of integral operators, Dokl. Akad. Nauk SSSR 319 (1991), no. 5, 1076–1078. Search in Google Scholar

[35] Y. Rakotondratsimba, On the boundedness of classical operators on weighted Lorentz spaces, Georgian Math. J. 5 (1998), no. 2, 177–200. 10.1007/BF02767995Search in Google Scholar

[36] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monogr. Textb. Pure Appl. Math. 146, Marcel Dekker, New York, 1991. Search in Google Scholar

[37] E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), no. 1, 329–337. 10.1090/S0002-9947-1984-0719673-4Search in Google Scholar

[38] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), no. 2, 145–158. 10.4064/sm-96-2-145-158Search in Google Scholar

[39] J. Soria, Lorentz spaces of weak-type, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 193, 93–103. 10.1093/qmathj/49.1.93Search in Google Scholar

[40] V. D. Stepanov, Weighted norm inequalities for integral operators and related topics, Nonlinear Analysis, Function Spaces and Applications. Vol. 5 (Prague 1994), Prometheus, Prague (1994), 139–175. Search in Google Scholar

[41] V. D. Stepanov, Weighted norm inequalities of Hardy type for a class of integral operators, J. Lond. Math. Soc. (2) 50 (1994), no. 1, 105–120. 10.1112/jlms/50.1.105Search in Google Scholar

Received: 2017-07-03
Published Online: 2018-01-10
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 5.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2017-0142/html
Scroll to top button