Fourth-order Schrödinger type operator with singular potentials

Abstract

In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A={\Delta }^2-V={\Delta }^2-c|x{|}^{-4}$ where $c$ is any constant such that $c<{\left(\frac{N(N-4)}{4}\right)}^2$. The semigroup generated by $-A$ in $L^2({\mathbb{R}}^N)$, $N\ge 5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p({\mathbb{R}}^N)$ for $p\in [p_0^{{}^{\prime }},p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{{}^{\prime }}$ is its dual exponent. Furthermore, we study the boundedness of the Riesz transform $\Delta A^{-1/2}$ on $L^p({\mathbb{R}}^N)$ for all $p\in (p_0^{{}^{\prime }},2]$.

Cited by in corpus (2)

• Bi-Laplacians on graphs and networks
• Fourth-order Schrödinger type operator with unbounded coefficients in $L^2({\mathbb{R}}^N)$