Geodesic
Strong-Type Inequality for Convolution with Square Root of the Poisson Kernel
Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 580-591.

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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. 
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V. G. Krotov; I. N. Katkovskaya. Strong-Type Inequality for Convolution with Square Root of the Poisson Kernel. Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 580-591. http://geodesic.mathdoc.fr/item/MZM_2004_75_4_a6/

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