  We prove the existence of $L^2$ solutions to the Helmholtz equation $(-\Delta
- 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to
$L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that
$\hat{f}$ vanishes on the unit sphere. This problem, and similar results for
the perturbed Helmholtz equation $(-\Delta -1)u = -Vu + f$, are connected to
the Limiting Absorption Principle for Schr\"odinger operators.
  The same techniques are then used to prove that a wide range of $L^p \mapsto
L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their
action on the closed subspace of functions whose Fourier transform vanishes on
the unit sphere.
