$L^p$-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on $\mathbb{R}^3$
Haruya Mizutani, Zijun Wan, Xiaohua Yao
TL;DR
This work provides a comprehensive, sharp analysis of $L^p$-boundedness for the wave operators $W_\pm(H,\Delta^2)$ in $\mathbb{R}^3$ where $H=\Delta^2+V$ with decaying potential, focusing on all types of zero-energy threshold singularities. By developing precise stationary formulas, detailed low-energy resolvent expansions, and a delicate resonance-space calculus (via $S_j$ and $\mathcal{D}_j$), the authors derive exact $L^p$-boundedness ranges: the first-kind resonance yields $1<p<\infty$; the second-kind resonance yields $1<p<3$ (unbounded for $3\le p\le \infty$); and the third-kind resonance yields $1<p<3$ with boundedness for $p\ge3$ under specific orthogonality conditions and absence of $p$-wave resonances, otherwise unbounded for $p\ge3$. These results hinge on precise low-energy expansions of $M(\lambda)^{-1}$, the stationary representation, and a suite of kernel estimates for related integral operators. As applications, the paper also establishes $L^p$-$L^q$ decay estimates for the fourth-order Schrödinger and beam equations with zero-threshold resonances, clarifying the threshold-by-threshold behavior of higher-order dispersive dynamics in three dimensions.
Abstract
Let $H = Δ^2 + V$ be the fourth-order Schrödinger operator on $\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, then it was recently shown that the wave operators $W_\pm(H, Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ and unbounded at the endpoints $p=1$ and $p=\infty$. This paper is to further establish the $L^p$-boundedness of $W_\pm(H, Δ^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_\pm(H, Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < 3$, but not if $3 \le p \le \infty$. In the third kind resonance case, we also show that $W_\pm(H, Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<3$ and generically unbounded on $L^p(\R^3)$ for any $3\le p\le\infty$. Moreover, it is also shown that $W_\pm(H, Δ^2)$ are bounded on $L^p(\R^3)$ for all $3\le p<\infty$ if in addition $H$ has the zero eigenvalue, but no $p$-wave zero resonances and all zero eigenfunctions are orthogonal to $x_ix_jx_kV$ in $L^2(\R^3)$ for all $i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)\in \R^3$. These results describe precisely the validity of the $L^p$-boundedness of $W_\pm(H, Δ^2)$ in $\mathbb{R}^3$ for all types of singularities at the zero energy threshold with some exceptions for the endpoint cases $p=1,\infty$. As an application, $L^p$-$L^q$ decay estimates are also derived for the fourth-order Schrödinger equations and Beam equations with zero resonance singularities.