Far Eastern Mathematical Journal

To content of the issue

The estimates of the approximation numbers of the Hardy operator acting in the Lorenz spaces in the case ${\it {\bf \max}(r,s)\leq q}$

Lomakina E.N., Nasyrova M.G., Nasyrov V.V.

2021, issue 1, Ñ. 71–88
DOI: https://doi.org/10.47910/FEMJ202107

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.

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

Download the article (PDF-file)


[1] C. Bennett, R. Sharpley, Interpolation of Operators. V. 129, Pure and Applied Mathematics, Academic Press, Boston, 1988.
[2] A. Pich, Operatornye idealy, Mir, M., 1982.
[3] 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.
[4] H. Konig, Eigenvalue distribution of compact operators. V. 16, Operator Theory: Advances and Applications, Birkh?auser Verlag, Basel, 1986.
[5] D.E. Edmunds, W.D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1987.
[6] B. Carl, I. Stephani, Entropy, compactness and the approximation of operators, Cambridge Univ. Press., Cambridge, 1990.
[7] D.E. Edmunds, W. D. Evans, D. J. Harris, “Approximation numbers of certain Volterra integral operators”, London Math. Soc. (2), 38 (1988), 471–489.
[8] D.E. Edmunds, V. Stepanov, “On singular numbers of certain Volterra integral operators”, J. Funct. Anal., 134 (1995), 222–246.
[9] D.E. Edmunds, W.D. Evans, D. J. Harris, “Two-sided estimates of the approximation numbers of certain Volterra integral operators”, Studia Math. (1), 124 (1997), 59–80.
[10] 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, 2000, 153–187.
[11] M.A. Lifshits, W. Linde, “Approximation and entropy numbers of Volterra operators with application to Brownian motion”, Mem. Am. Math. Soc., 745 (2002.), 1–87.
[12] E. Lomakina, V. Stepanov, “On the compactness and approximation numbers of Hardy type integral operators in Lorentz spases”, J. London Math. Soc. (2), 53 (1996), 369–382.
[13] E. Lomakina, V. Stepanov, “On the Hardy-type integral operators in Banach function spases”, Publicacions Matematiques, 42 (1998), 165–194.
[14] E.N. Lomakina, “ Ob otsenkakh norm operatora Khardi, deistvuiushchego v prostranstvakh Lorentsa”, Dal'nevostoch. matem. zhurn., 20:2 (2020), 191–211.
[15] E.T. Sawyer, “Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator”, Trans. Amer. Math. Soc., 281 (1984), 329–337.
[16] A. Pietsch, “s-Numbers of operators in Banach spaces”, Studia Math., 51 (1974), 201–223.

To content of the issue