  The Assouad and lower dimensions and dimension spectra quantify the
regularity of a measure by considering the relative measure of concentric
balls. On the other hand, one can quantify the smoothness of an absolutely
continuous measure by considering the $L^p$ norms of its density. We establish
sharp relationships between these two notions. Roughly speaking, we show that
smooth measures must be regular, but that regular measures need not be smooth.
