Strong-Type Inequality for Convolution with Square Root of the Poisson Kernel

The boundary behavior of convolutions with Poisson kernel and with square root of the Poisson kernel is essentially different. The former has only a nontangential limit. The latter involves convergence over domains admitting the logarithmic order of tangency with the boundary (P. Sjögren, J.-O. Rönning). This result was generalized by the authors to spaces of homogeneous type. Here we prove the boundedness in $L^p$, $p > 1$, of the corresponding maximal operator. Only a weak-type inequality was known before.