Norms of maximal functions between generalized and classical Lorentz spaces

Abstract

In this paper we calculate the norm of the generalized maximal operator $M_{\varphi ,{\Lambda }^{\alpha }(b)}$, defined with $0<\alpha <\infty$ and functions $b,\varphi :(0,\infty )\to (0,\infty )$ for all measurable functions $f$ on ${\mathbb{R}}^n$ by \begin{equation*} M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from $\mathrm{G\Gamma }(p,m,v)$ into ${\Lambda }^q(w)$. Here ${\Lambda }^{\alpha }(b)$ 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 $$ \|f\|_{Λ^α(b)} = \bigg( \int_0^{\infty} [f^*(s)]^α b(s)\,ds \bigg)^{\frac{1}α} < \infty \quad \mbox{and} \quad \|f\|_{\operatorname{GΓ}(p,m,w)} = \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} < \infty, $$ 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^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} \end{equation*} where $w$ and $v$ are weight functions on $(0,\infty )$. Here $f^{\ast }$ 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 $(0,\infty )$ by $$(T_{u,b}g)(t):=\underset{\tau \in [t,\infty )}{\sup }\frac{u(\tau )}{B(\tau )}{\int }_0^{\tau }g(s)b(s)ds,t\in (0,\infty ),$$ 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 $t\in (0,\infty )$..

30 pages. arXiv admin note: substantial text overlap with arXiv:2109.06745