Far Eastern Mathematical Journal

To content of the issue

On estimates for the norms of the Hardy operator acting in the Lorenz spaces

Lomakina E.N.

2020, issue 2, Ñ. 191–211
DOI: https://doi.org/10.47910/FEMJ202019

In the paper conditions are found under which the compact operator $Tf(x)=\varphi(x)\int_0^xf(\tau)v(\tau)\,d\tau,$ $x>0,$ acting in weighted Lorentz spaces $T:L^{r,s}_{v}(\mathbb{R^+})\to L^{p,q}_{\omega}(\mathbb{R^+})$ in the domain $1<\max (r,s)\le \min(p,q)<\infty,$ belongs to operator ideals $\mathfrak{S}^{(a)}_\alpha$ and $\mathfrak{E}_\alpha$, $0<\alpha<\infty$. And estimates are also obtained for the quasinorms of operator ideals in terms of integral expressions which depend on operator weight functions.

operator ideal, Hardy operator, compact operator, Lorentz spaces, approximation numbers, entropy numbers

Download the article (PDF-file)


[1] A. Pich, Operatornye idealy, Mir, M., 1982.
[2] H. K?onig, Eigenvalue distribution of compact operators, Birkh?auser, Boston, 1986.
[3] C. Bennett, R. Sharpley, Interpolation of Operators, v. 129, Pure. Appl. Math., 1988.
[4] S. Barza, V. Kolyada V., J. Soria, “Sharp constants related to the triangle inequality in Lorentz spaces”, Trans. Amer. Math. Soc., 361:10, (2009), 5555–5574.
[5] D. E. Edmunds, W. D. Evans, D. J. Harris, “Two-sided estimates of the approximation numbers of certain Volterra integral operators”, Studia Math, 24:1, (1997), 59–80.
[6] E. Lomakina, V. Stepanov, “On asymptotic behaviour of the approximation numbers and estimates of Schatten – von Neumann norms of the Hardy-type integral operators”, Function spaces and application, Narosa Publishing Hause, New Delhi, 2000.
[7] V. D. Stepanov, “On the singular numbers of certain Volterra integral operators”, J. London Math. Soc., 61:2, (2000), 905–922.
[8] E. P. Ushakova, “ Otsenki singuliarnykh chisel preobrazovanii tipa Stil't'esa”, Sib. matem. zhurn., 52:1, (2011), 201–209.
[9] E. Lomakina, V. Stepanov, “On the compactness and approximation numbers of Hardy type integral operators in Lorentz spases”, J. London Math. Soc., 53:2, (1996), 369–382.
[10] H. M. Chung, R. A. Hunt, D. S. Kurtz, “The Hardy-Littlewood maximal function on L(p, q) spaces with weights”, Indiana Univ. Math. J., 31, (1982), 109–120.
[11] D. E. Edmunds, P. Gurka, L. Pick, “Compactness of Hardy-type integral operators in weighted Banach function spaces”, Studia Math., 109, (1994), 73–90.
[12] E. T. Sawyer, “Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator”, Trans. Amer. Math. Soc., 281, (1984), 329–337.
[13] L. Grafakos, Classical fourier analysis, Springer-Verlag, New York, 2008.
[14] D. E. Edmunds, V. D. Stepanov, “On the singular numbers of certain Volterra integral Funct. Anal., 134:1, (1995), 222–246.

To content of the issue