English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we calculate the norm of the generalized maximal operator$M_{\phi,\Lambda^{\alpha}(b)}$, defined with $0 < \alpha < \infty$ andfunctions $b,\,\phi: (0,\infty) \rightarrow (0,\infty)$ for all measurablefunctions $f$ 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 theclassical and generalized Lorentz spaces, defined as a set of all measurablefunctions $f$ 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 isdefined 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 $b$ areappropriate weight functions on $(0,\infty)$ and the function $B(t) : =\int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every $t \in (0,\infty)$..

Norms of maximal functions between generalized and classical Lorentz spaces