On the Continuity of Best Approximations by Constants on Balls in Metric Measure Spaces

Mathematical Notes - Conditions for the constants of best approximation in the metric of the spaces Lp(B) to be continuous or semicontinuous as functions of the center of a ball B of fixed radius...     

The Fatou Property for General Approximate Identities on Metric Measure Spaces

Mathematical Notes - Abstract approximate identities on metric measure spaces are considered in this paper. We find exact conditions on the geometry of domains for which the convergence of...     

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.