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We consider spherical averages of the Fourier transform of fractal measuresand improve both the upper and lower bounds on the rate of decay. Maximalestimates with respect to fractal measures are deduced for the Schr\"odingerand wave equations. This refines the almost everywhere convergence of thesolution to its initial datum as time tends to zero. A consequence is that thesolution to the wave equation cannot diverge on a $(d-1)$-dimensional manifoldif the data belongs to the energy space $\dot{H}^1(\mathbb{R}^d)\timesL^2(\mathbb{R}^d)$.

Average decay of the Fourier transform of measures with applications