Norms of maximal functions between generalized and classical Lorentz spaces

Abstract

In this paper we calculate the norm of the generalized maximal operator $M_{\phi,\Lambda^{\alpha}(b)}$, defined with $0 < \alpha < \infty$ and functions $b,\,\phi: (0,\infty) \rightarrow (0,\infty)$ 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 ${\operatorname{G\Gamma}}(p,m,v)$ into $\Lambda^q(w)$. Here $\Lambda^{\alpha}(b)$ and ${\operatorname{G\Gamma}}(p,m,w)$ are the classical and generalized Lorentz spaces, defined as a set of all measurable functions $f$ defined on ${\mathbb R}^n$ for which $$ \|f\|_{\Lambda^{\alpha}(b)} = \bigg( \int_0^{\infty} [f^*(s)]^{\alpha} b(s)\,ds \bigg)^{\frac{1}{\alpha}} < \infty \quad \mbox{and} \quad \|f\|_{{\operatorname{G\Gamma}}(p,m,w)} = \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (\tau)]^p\,d\tau \bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} < \infty, $$ 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 $(0,\infty)$. Here $f^*$ 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 $(0,\infty)$ by $$ (T_{u,b} g)(t) : = \sup_{\tau \in [t,\infty)} \frac{u(\tau)}{B(\tau)} \int_0^{\tau} g(s)b(s)\,ds,\qquad t \in (0,\infty), $$ where $u$ and $b$ are appropriate weight functions on $(0,\infty)$ and the function $B(t) : = \int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every $t \in (0,\infty)$..