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: 1

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.