Oscillation inequalities, lower ball growth and Sobolev embeddings on metric measure spaces
We study the relation between lower ball growth, rearrangement oscillations, and Sobolev-type embeddings on metric measure spaces. We prove that a lower estimate for the ball function \[ h_μ(r):=\inf_{x\inΩ}μ(B(x,r)) \] implies pointwise symmetrization inequalities for averaged differences and Hajłasz gradients, without doubling, Poincaré-type assumptions, or extra topological hypotheses. Conversely, the validity of either family of inequalities for suitable cutoff functions recovers the corresponding lower growth estimate. Thus, up to multiplicative constants, the lower geometry of the measure is equivalent to the pointwise oscillation principle itself. Taking norms in rearrangement-invariant spaces yields normed oscillation inequalities and Sobolev embeddings for Hajłasz and averaged Besov spaces. This separates the metric-measure input from the choice of the final function-space target. In the power-growth model, the resulting targets are identified with Lorentz spaces. We also show how Sobolev inequalities between general rearrangement-invariant spaces force lower ball estimates through the fundamental functions of the source and target spaces. The results apply to general admissible growth functions and moduli of smoothness, beyond the classical power setting.