  We investigate $L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$ dispersive
estimates for the Schr\"odinger operator $H=-\Delta+V$ when there are
obstructions, a resonance or an eigenvalue, at zero energy. In particular, we
show that if there is a resonance or an eigenvalue at zero energy then there is
a time dependent, finite rank operator $F_t$ satisfying $\|F_t\|_{L^1\to
L^\infty} \lesssim 1/\log t$ for $t>2$ such that
$$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim t^{-1},\,\,\,\,\,\text{for}
t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no
resonance at zero energy. We then develop analogous dispersive estimates for
the solution operator to the four dimensional wave equation with potential.
