Far Eastern Mathematical Journal

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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, P. 71–88 DOI: https://doi.org/10.47910/FEMJ202107

Abstract

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.

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

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References

[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.

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