Fourth-order Schr\"odinger type operator with singular potentials

TL;DR

This paper investigates the spectral and functional calculus properties of a biharmonic operator with a singular inverse fourth-order potential, establishing boundedness of associated semigroups and Riesz transforms in certain L^p spaces.

Contribution

It extends the analysis of fourth-order Schrödinger operators with singular potentials, providing new results on semigroup boundedness and Riesz transform boundedness in L^p spaces.

Findings

Semigroup generated by -A is bounded and holomorphic on L^p for p in [p'_0, p_0]

Boundedness of the Riesz transform ΔA^{-1/2} on L^p for p in (p'_0, 2]

Identification of the range of p for which these properties hold

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 $L^2(\mathbb{R}^N)$, $N\geq5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p(\mathbb{R}^N)$ for $p\in [p^{'}_0,p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{'}$ 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 $p\in(p_0^{'},2]$.