Generalized fractional maximal functions in Lorentz spaces

In this paper we give the complete characterization of the boundedness of the generalized fractional maximal operator $$M_{\varphi ,{\mathrm{\Lambda }}^{\alpha }(b)}f(x):=\underset{Q\ni x}{\sup }\frac{\mathrm{\parallel }f{\chi }_Q{\mathrm{\parallel }}_{{\mathrm{\Lambda }}^{\alpha }(b)}}{\varphi (\mathrm{\mid }Q\mathrm{\mid })}(x\in {\mathbb{R}}^n),$$ between the classical Lorentz spaces ${\mathrm{\Lambda }}^p(v)$ and ${\mathrm{\Lambda }}^q(w)$ for appropriate functions $\varphi$, where $0<p,q<\mathrm{\infty }$, $0 < \alpha \le r < \infty$,$v,w,,b$areweightfunctionson$(0,\infty)$suchthat$0 < B(x): = \int_0^x b < \infty$,$x > 0$,$B \in \Delta_2$and$B(t) / t^{\alpha / r}$ is quasi-increasing.