On weighted strong type inequalities for the generalized weighted mean operator

Abstract

The generalized weighted mean operator ${\mathbf{M}}_w^g$ is given by $$[{\mathbf{M}}_w^{g}f](x)=g^{-1}\left(\frac{1}{W(x)}{\int }_0^{x}w(t)g(f(t))\mathrm{d}t\right),$$ with $$W(x)={\int }_0^{x}w(s)\mathrm{d}s,\text{for}x\in (0,+\infty ),$$ where $w$ is a positive measurable function on $(0,+\infty )$ and $g$ is a real continuous strictly monotone function with its inverse $g^{-1}$. We give some sufficient conditions on weights $u,v$ on $(0,+\infty )$ for which there exists a positive constant $C$ such that the weighted strong type $(p,q)$ inequality $${\left({\int }_0^{\infty }u(x)([{\mathbf{M}}_w^{g}f](x){)}^q\mathrm{d}x\right)}^{\frac{1}{q}}\le C{\left({\int }_0^{\infty }v(x)f(x{)}^p\mathrm{d}x\right)}^{\frac{1}{p}}$$ holds for every measurable non-negative function $f$, where the positive reals $p,q$ satisfy certain restrictions.