Abstract:
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 Lp, p>1, of the corresponding maximal operator. Only a weak-type inequality was known before.
Citation:
V. G. Krotov, I. N. Katkovskaya, “Strong-Type Inequality for Convolution with Square Root of the Poisson Kernel”, Mat. Zametki, 75:4 (2004), 580–591; Math. Notes, 75:4 (2004), 542–552