\(L^{p}\)-boundedness of wave operators for bi-Schrödinger operators on the line.(English)Zbl 1546.35048
Summary: This paper is devoted to establishing several types of \(L^{p}\)-boundedness of wave operators \(W_{\pm} = W_{\pm}(H, \mathrm{\Delta}^{2})\) associated with the bi-Schrödinger operators \(H = \mathrm{\Delta}^{2} + V(x)\) on the line \(\mathbb{R}\). Given suitable decay potentials \(V\), we firstly prove that the wave and dual wave operators are bounded on \(L^{p}(\mathbb{R})\) for all \(1 < p < \infty \):\[\| W_{\pm} f \|_{L^{p} (\mathbb{R})} + \| W_{\pm}^{\ast} f \|_{L^{p} (\mathbb{R})} \lesssim \| f \|_{L^{p} (\mathbb{R})},\]which are further extended to the \(L^{p}\)-boundedness on the weighted spaces \(L^{p}(\mathbb{R}, w)\) with general even \(A_{p}\)-weights \(w\) and to the boundedness on the Sobolev spaces \(W^{s, p} (\mathbb{R})\). For the limiting case, we prove that \(W_{\pm}\) are bounded from \(L^{1} (\mathbb{R})\) to \(L^{1, \infty} (\mathbb{R})\) as well as bounded from the Hardy space \(\mathcal{H}^{1} (\mathbb{R})\) to \(L^{1} (\mathbb{R})\). These results especially hold whatever the zero energy is a regular point or a resonance of \(H\). We also obtain that \(W_{\pm}\) are bounded from \(L^{\infty} (\mathbb{R})\) to \(\mathrm{BMO} (\mathbb{R})\) if zero is a regular point or a first kind resonance of \(H\). Next, we show that \(W_{\pm}\) are neither bounded on \(L^{1} (\mathbb{R})\) nor on \(L^{\infty} (\mathbb{R})\) even if zero is a regular point of \(H\). Moreover, if zero is a second kind resonance of \(H\), then \(W_{\pm}\) are shown to be even not bounded from \(L^{\infty} (\mathbb{R})\) to \(\mathrm{BMO} (\mathbb{R})\) in general. In particular, we remark that our results give a complete picture of the validity of \(L^{p}\)-boundedness of the wave operators for all \(1 \leq p \leq \infty\) in the regular case. Finally, as applications, we deduce the \(L^{p} - L^{q}\) decay estimates for the propagator \(e^{- itH} P_{\mathrm{ac}}(H)\) with pairs \((1/p, 1/q)\) belonging to a certain region of \(\mathbb{R}^{2}\), as well as establish the Hörmander-type \(L^{p}\)-boundedness theorem for the spectral multiplier \(f(H)\).
MSC:
| 35J30 | Higher-order elliptic equations |
| 31B30 | Biharmonic and polyharmonic equations and functions in higher dimensions |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Cite
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