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- W3210772013 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 ${operatorname{GGamma}}(p,m,v)$ into $Lambda^q(w)$. Here $Lambda^{alpha}(b)$ and ${operatorname{GGamma}}(p,m,w)$ are the classical and generalized Lorentz spaces, defined as a set of all measurable functions $f$ defined on ${mathbb R}^n$ for which $$ |f|_{Lambda^{alpha}(b)} = bigg( int_0^{infty} [f^*(s)]^{alpha} b(s),ds bigg)^{frac{1}{alpha}} < infty quad mbox{and} quad |f|_{{operatorname{GGamma}}(p,m,w)} = bigg( int_0^{infty} bigg( int_0^x [f^* (tau)]^p,dtau bigg)^{frac{m}{p}} v(x),dx bigg)^{frac{1}{m}} < infty, $$ 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),dxbigg)^{frac{1}{q}} le C , bigg( int_0^{infty} bigg( int_0^x [f^* (tau)]^p,dtau 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 $t in (0,infty)$.." @default.
- W3210772013 created "2021-11-08" @default.
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- W3210772013 date "2023-01-01" @default.
- W3210772013 modified "2023-10-16" @default.
- W3210772013 title "Norms of Maximal Functions between Generalized and Classical Lorentz Spaces" @default.
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- W3210772013 doi "https://doi.org/10.59849/2218-6816.2023.2.51" @default.
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