On the $L^p$ boundedness of the wave operators for fourth order Schrödinger operators

We consider the fourth order Schrödinger operator H = ? 2 + V ( x stretchy="false">) H=\Delta ^2+V(x) in three dimensions with real-valued potential V"> encoding="application/x-tex">V . Let 0 squared"> 0 encoding="application/x-tex">H_0=\Delta ^2 , if decays sufficiently and there are no eigenvalues or resonances absolutely continuous spectrum of H"> encoding="application/x-tex">H then wave operators W Subscript plus-or-minus Baseline s reverse-solidus comma en-dash limit Underscript t right-arrow infinity Endscripts e Superscript i minus 0"> W ± s \,– movablelimits="true" form="prefix">lim t stretchy="false">?<!-- ? mathvariant="normal">?<!-- ? </mml:munder> e i ?<!-- ? encoding="application/x-tex">W_{\pm }= s\text {\,–}\lim _{t\to \pm \infty } e^{itH}e^{-itH_0} extend to bounded on L p double-struck R cubed L p mathvariant="double-struck">R 3 encoding="application/x-tex">L^p(\mathbb R^3) for all alttext="1 greater-than infinity"> 1 &gt; encoding="application/x-tex">1&gt;p&gt;\infty