  We investigate strong pairing fluctuations and effects of a harmonic trap in
the superfluid phase of an ultracold Fermi gas. Including amplitude and phase
fluctuations of the inhomogeneous superfluid order parameter $\Delta(r)$ in a
trap within a combined $T$-matrix theory with the local density approximation,
we examine local properties of single-particle excitations and a thermodynamic
quantity in the BCS-BEC crossover region. Below the superfluid phase transition
temperature $T_{\rm c}$, we show that inhomogeneous pairing fluctuations lead
to a shell structure of the gas cloud in which the spatial region where the
ordinary BCS-type superfluid density of states appears is surrounded by the
region where the pseudogap associated with strong pairing fluctuations
dominates single-particle excitations. The former spatial region enlarges to
eventually cover the whole gas cloud far below $T_{\rm c}$. We also examine how
this shell structure affects the photoemission spectrum, as well as the local
pressure. Since a cold Fermi gas is always trapped in a harmonic potential, our
results would be useful for the study of strong-coupling superfluid physics,
including this realistic situation.
