Boundary estimates for the fractional spherical maximal function

In this article, we study the fractional spherical maximal function and its lacunary counterpart. We study the necessary and sufficient conditions for $L^p-L^q$ boundedness of both maximal functions. In particular, we prove the restricted weak type estimate for both full and lacunary fractional spherical maximal functions at the boundary of the maximal $L^p-L^q$ bounded regions.