The Helmholtz equation with $L^p$ data and Bochner-Riesz multipliers

We prove the existence of $L^{2}$ solutions to the Helmholtz equation $(-\Delta

1)u = f $i n$ {\mathbb R}^n $a s s u m i n g t h e g i v e n d a t a$ f $b e l o n g s t o$ L^{(2n+2)/(n+5)}({\mathbb R}^n) $a n d s a t i s f i e s t h e " F r e d h o l m c o n d i t i o n " t h a t$ \hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(- Δ - 1) u = - V u + f$ , 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.

We prove the existence of $L^{2}$ solutions to the Helmholtz equation $(-\Delta

1)u = f $i n$ {\mathbb R}^n $a s s u m i n g t h e g i v e n d a t a$ f $b e l o n g s t o$ L^{(2n+2)/(n+5)}({\mathbb R}^n) $a n d s a t i s f i e s t h e " F r e d h o l m c o n d i t i o n " t h a t$ \hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(- Δ - 1) u = - V u + f$ , 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.

The Helmholtz equation with $L^{p}$ data and Bochner-Riesz multipliers