$L^p$ estimate for (powers of) a Laplacian with inverse square potential

Question

I need an estimate of the form $$ \|v\|_{L^p} \le C \|(K-\Delta- c|x|^{-2})^s v\|_{L^p} $$ where $K>0$ can be large if necessary, $c$ is positive but below the Hardy constant $(n-2)^2/4$, where $n$ is the space dimension. The power $s>0$ can be equal to 1 or another integer if this simplifies the proof.

This is trivial when $p=2$ and $s=1/2$, but I need $p$ near 4. Note that the heat kernel of the Laplacian plus inverse square potential does not satisfy the usual gaussian bounds, and this makes it more difficult to apply the standard techniques (even complex interpolation is not completely obvious). Any ideas?

Answer 1

See Lemma 5.3 in the paper V.F. Kovalenko, M.A. Perelmuter, Yu.A. Semenov, Schrödinger Operators with $L^{l/2}_w (R^l)$-Potentials, J. Math. Phys., Vol. 22, No. 5, 1981, pp. 1033-1044

You can use the similar approach: Neumann series and norm of the operator $|x|^{-2}(-\Delta)^{-1}$ in $L^p$.