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We prove $L^p$-analogues of the classical tauberian theorem of Ingham andKaramata, and its variations giving rates of decay. These results are appliedto derive $L^p$-decay of operator families arising in the study of the decay ofenergy for damped wave equations and local energy for wave equations inexterior domains. By constructing some examples of critical behaviour we showthat the $L^p$-rates of decay obtained in this way are best possible under ourassumptions.

L^p-tauberian theorems and L^p-rates for energy decay