The Modulus of Smoothness in Metric Spaces and Related Problems

We define a general variant of the modulus of smoothness in metric spaces and show that under mild condition it is equivalent to the K-functional of a couple of Besov type spaces which in special cases coincide with spaces defined by Korevaar and Schoen. We prove various symmetrization inequalities which involve the modulus, the K-functional and the isoperimetric estimators. We also characterize the Hajłasz-type Sobolev spaces defined not necessarily on doubling measure spaces by means of generalized Poincaré inequalities. This require to study of some variants of the Fefferman–Stein sharp functions as well as the Hardy–Littlewood maximal operators.