Embeddings between weighted Tandori and Cesàro function spaces

Year 2021, Volume: 70 Issue: 2, 837 - 848, 31.12.2021

Tuğçe Unver Yıldız

https://doi.org/10.31801/cfsuasmas.869893

Cited By: 2

Abstract

We characterize the weights for which the two-operator inequality
$$\Vert ({\int }_0^{x}f\left(t{)}^{p}v\right(t{)}^{p}dt{)}^{\frac{1}{p}}{\Vert }_{q,u,(0,\infty )}\le c\Vert ess\underset{t\in (x,\infty )}{sup}f\left(t\right){\Vert }_{r,w,(0,\infty )}$$
holds for all non-negative measurable functions on $(0,\infty )$, where $0<p<q\le \infty$ and $0<r<\infty$, namely, we find the least constants in the embeddings between weighted Tandori and Ces\`{a}ro function spaces. We use the combination of duality arguments for weighted Lebesgue spaces and weighted Tandori spaces with weighted estimates for the iterated integral operators.

Keywords

Cesàro function spaces , Copson function spaces , Tandori function spaces , embeddings , weighted inequalities , Hardy operator , Copson operator , iterated operators

References

• Astashkin, S. V., Maligranda, L., Structure of Cesàro function spaces, Indag. Math. (N.S.), 20(3) (2009), 329-379. https://dx.doi.org/10.1016/S0019-3577(10)00002-9
• Astashkin, S. V., Maligranda, L., Structure of Cesàro function spaces: a survey, Banach Center Publications, 102 (2014), 13-40. https://dx.doi.org/10.4064/bc102-0-1
• Barza, S., Marcoci, A. N., Marcoci, L. G., Factorizations of weighted Hardy inequalities, Bull. Braz. Math. Soc. (N.S.), 49(4) (2018), 915-932. https://dx.doi.org/10.1007/s00574-018-0087-7
• Bennett, G., Factorizing the classical inequalities, Mem. Amer. Math. Soc., 120 (576) (1996). https://dx.doi.org/10.1090/memo/0576
• Boas, R. P., Jr., Some integral inequalities related to Hardy's inequality, J. Analyse Math., 23 (1970), 53-63. https://dx.doi.org/10.1007/BF02795488
• Carro, M., Gogatishvili, A., Martín, J., Pick, L., Weighted inequalities involving two Hardy operators with applications to embeddings of function spaces, J. Operator Theory, 59(2) (2008), 309-332.
• Evans, W. D., Gogatishvili, A., Opic, B., The reverse Hardy inequality with measures, Math. Inequal. Appl., 11(1) (2008), 43-74. https://dx.doi.org/10.7153/mia-11-03
• Gogatishvili, A., Mustafayev, R. Ch., Persson, L. E., Some new iterated Hardy-type inequalities, J. Funct. Spaces Appl., Art. ID 734194, (2012). https://dx.doi.org/10.1155/2012/734194
• Gogatishvili, A., Mustafayev, R. Ch., Ünver, T., Embeddings between weighted Copson and Cesàro function spaces, Czechoslovak Math. J., 67(4) (2017), 1105-1132. https://dx.doi.org/10.21136/CMJ.2017.0424-16
• Gogatishvili, A., Mustafayev, R. Ch., Ünver, T., Pointwise multipliers between weighted Copson and Cesàro function spaces, Math. Slovaca, 69(6) (2019), 1303-1328. https://dx.doi.org/10.1515/ms-2017-0310
• Grosse-Erdmann, K.-G., The blocking technique, weighted mean operators and Hardy's inequality, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1998. https://dx.doi.org/10.1007/BFb0093486
• Hassard, B. D., Hussein, D. A., On Cesàro function spaces, Tamkang J. Math., 4 (1973), 19-25.
• Kaminska, A., Kubiak, D., On the dual of Cesàro function space, Nonlinear Anal., 75(5) (2012), 2760-2773. https://dx.doi.org/10.1016/j.na.2011.11.019
• Kolwicz, P., Le´snik, K., Maligranda, L., Symmetrization, factorization and arithmetic of quasi-Banach function spaces, J. Math. Anal. Appl., 470(2) (2019), 1136-1166. https://dx.doi.org/10.1016/j.jmaa.2018.10.054
• Kufner, A., Persson, L. -E., Samko, N., Weighted Inequalities of Hardy Type, Second Ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. https://dx.doi.org/10.1142/10052
• Le´snik, K., Maligranda, L., Abstract Cesàro spaces. Duality, J. Math. Anal. Appl., 424(2) (2015), 932-951. https://dx.doi.org/10.1016/j.jmaa.2014.11.023
• Le´snik, K., Maligranda, L., Abstract Cesàro spaces. Optimal range, Integral Equations Operator Theory, 81(2) (2015), 227-235. https://dx.doi.org/10.1007/s00020-014-2203-4
• Le´snik, K., Maligranda, L., Interpolation of abstract Cesàro, Copson and Tandori spaces, Indag. Math. (N.S.), 27(3) (2016), 764-785. https://dx.doi.org/10.1016/j.indag.2016.01.009
• Mustafayev, R. Ch., Ünver, T., Reverse Hardy-type inequalities for supremal operators with measures, Math. Inequal. Appl., 18(4) (2015), 1295-1311. https://dx.doi.org/10.7153/mia-18-101
• Shiue, J. -S., A note on Cesàro function space, Tamkang J. Math., 1 (1970), 91-95.
• Sy, P. W., Zhang, W. Y., Lee, P. Y., The dual of Cesàro function spaces, Glas. Mat. Ser. III, 22(42) (1) (1987), 103-112.
• Tandori, K., Über einen speziellen Banachschen Raum, Publ. Math. Debrecen, 3 (1954), 263-268.