A strong type inequality for convolution with square root of the Poisson kernel

The boundary behaviour of convolutions with Poisson kernel and with square root from Poisson kernel is essentially differs. The first ones have only nontangential limit. For the last ones the convergence is over domains admittings a logarithmic order of the contact with the boundary . This result was generalized by authors on the spaces of homogeneous type. Here we prove the boundedness and some weighted estimates for the corresponding maximal operator. Earlier it was known only weak type inequality.