Norms of maximal functions between generalized and classical Lorentz spaces

In this paper we calculate the norm of the generalized maximal operator $M_{\varphi ,{\mathrm{\Lambda }}^{\alpha }(b)}$, defined with $0<\alpha <\mathrm{\infty }$ and functions $b,\varphi :(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ for all measurable functions $f$ on ${\mathbb{R}}^n$ by \begin{equation*} M_{\phi,\Lambda^{\alpha}(b)}f(x) : = \sup_{Q \ni x} \frac{|f \chi_Q|{\Lambda^{\alpha}(b)}}{\phi (|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from $\mathrm{G\Gamma }(p,m,v)$ into ${\mathrm{\Lambda }}^q(w)$. Here ${\mathrm{\Lambda }}^{\alpha }(b)$ and $\mathrm{G\Gamma }(p,m,w)$ are the classical and generalized Lorentz spaces, defined as a set of all measurable functions $f$ defined on ${\mathbb{R}}^n$ for which $$\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathrm{\Lambda }}^{\alpha }(b)}=({\int }_0^{\mathrm{\infty }}[f^{\ast }(s){]}^{\alpha }b(s)ds{)}^{\frac{1}{\alpha }}<\mathrm{\infty }and\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{G\Gamma }(p,m,w)}=\left({\int }_0^{\mathrm{\infty }}\right({\int }_0^x[f^{\ast }(\tau ){]}^{p}d\tau {)}^{\frac{m}{p}}v(x)dx{)}^{\frac{1}{m}}<\mathrm{\infty },$$ respectively. We reduce the problem to the solution of the inequality \begin{equation*} \bigg( \int_0^{\infty} \big[ T{u,b}f^* (x)\big]^q , w(x),dx\bigg)^{\frac{1}{q}} \le C , \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (\tau)]^p,d\tau \bigg)^{\frac{m}{p}} v(x),dx \bigg)^{\frac{1}{m}} \end{equation*} where $w$ and $v$ are weight functions on $(0,\mathrm{\infty })$. Here $f^{\ast }$ is the non-increasing rearrangement of $f$ defined on ${\mathbb{R}}^n$ and $T_{u,b}$ is the iterated Hardy-type operator involving suprema, which is defined for a measurable non-negative function $f$ on $(0,\mathrm{\infty })$ by $$(T_{u,b}g)(t):=\underset{\tau \in [t,\mathrm{\infty })}{\sup }\frac{u(\tau )}{B(\tau )}{\int }_0^{\tau }g(s)b(s)ds,t\in (0,\mathrm{\infty }),$$ where $u$ and $b$ are appropriate weight functions on $(0,\mathrm{\infty })$ and the function $B(t) : = \int_0^t b(s),ds$satisfies$0 < B(t) < \infty$forevery$t \in (0,\infty)$…