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

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.

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

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.