On weighted strong type inequalities for the generalized weighted mean operator

Abstract

The generalized weighted mean operator $\mathbf{M}^{g}_{w}$ is given by $$[\mathbf{M}^{g}_{w}f](x)= g^{-1}\left(\frac{1}{W(x)}\int_{0}^{x}w(t)g(f(t))\,\mathrm{d}t\right),$$ with $$W(x)=\int_{0}^{x} w(s)\,\mathrm{d}s, \quad \textrm{for} x \in (0, +\infty),$$ 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 $$\left(\int_{0}^{\infty} u(x)\Bigl([\mathbf{M}^{g}_{w}f](x)\Bigr)^{q}\,\mathrm{d}x \right)^{1 \over q} \leq C \left(\int_{0}^{\infty}v(x)f(x)^{p}\,\mathrm{d}x \right)^{1 \over p} $$ holds for every measurable non-negative function $f$, where the positive reals $p,q$ satisfy certain restrictions.