Far Eastern Mathematical Journal

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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


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

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