Estimates of L p -oscillations of functions for p > 0

Abstract

We prove a number of inequalities for the mean oscillations $$\mathcal{O}_\theta (f,B,I) = \left( {\frac{1} {{\mu (B)}}\int_B {\left| {f(y) - I} \right|^\theta d\mu (y)} } \right)^{1/\theta } ,$$ , where θ > 0, B is a ball in a metric space with measure µ satisfying the doubling condition, and the number I is chosen in one of the following ways: I = f(x) (x ∈ B), I is the mean value of the function f over the ball B, and I is the best approximation of f by constants in the metric of L θ (B). These inequalities are used to obtain L p -estimates (p > 0) of the maximal operators measuring local smoothness, to describe Sobolev-type spaces, and to study the self-improvement property of Poincare-Sobolev-type inequalities.