On boundedness of the Hardy operator in Morrey-type spaces

In this paper we study the boundedness of the Hardy operator $H_\alpha$ in local and global Morrey-type spaces $LM_{p\theta,w(\cdot)}$, $GM_{p\theta,w(\cdot)}$ respectively, characterized by numerical parameters $p,\theta$ and a functional parameter $w$. We reduce this problem to the problem of a continuous embedding of one local Morrey-type space to another one. This allows obtaining, for all admissible values of the numerical parameters $\alpha,p_1,p_2,\theta_1,\theta_2$, sufficient conditions on the functional parameters $w_1$ and $w_2$ ensuring the boundedness of $H_\alpha$ from $LM_{p_1\theta_1,w_1(\cdot)}$ to $LM_{p_2\theta_2,w_2(\cdot)}$ and from $GM_{p_1\theta_1,w_1(\cdot)}$ to $GM_{p_2\theta_2,w_2(\cdot)}$. Moreover, for a certain range of the numerical parameters and under certain a priori assumptions on $w_1$ and $w_2$ these sufficient conditions coincide with the necessary ones.