论文标题
Bi-Schrödinger运营商的$ l^p $ bundedness the Line
$L^p$-boundedness of wave operators for bi-Schrödinger operators on the line
论文作者
论文摘要
本文致力于建立波动运算符的几种类型的$ l^p $结合度$ w_ \ pm = w_ \ pm（h，δ^2）$与bi-schrödingeroberators $ h =Δ^Δ^{2}+v（x）$相关联。给定适当的衰变潜力$ v $，我们首先证明了$ 1 <p <\ iffty $：$ 1 \ | w_ \ | w_ \ | w_ \ pm f \ | _ {l^p <pm | _ {l^p（\ mathbb {r} {r}） f \ | _ {l^p（\ Mathbb {r}）}} \ Lessim \ | f \ | _ {l^p（\ Mathbb {r}）}，$$，这些$ $将进一步扩展到$ l^p $ bungedness $ l^p $ boungedness $ l^p（$ l^p（$ l^p（$ mathbb）$ $ $ $ $ p（$ pe）以及Sobolev Space上的有限度$ W^{S，P}（\ Mathbb {r}）$。对于限制情况，我们证明$ w_ \ pm $从$ l^1（\ r）$到$ l^{1，\ infty}（\ r）$，以及从Hardy Space $ \ H^1（\ r）$到$ l^1（\ l^1（\ r）$。这些结果尤其容纳零能量是$ h $的常规点或共振。我们还获得$ w_ \ pm $从$ l^\ infty（\ r）$到$ \ bmo（\ r）$的界限，如果零是常规点或$ h $的第一个共振。接下来，我们表明$ w_ \ pm $既不是$ l^1（\ mathbb {r}）$的界限，也不是$ l^\ infty（\ mathbb {r}）$上的限制，即使零是$ h $的常规点，也是如此。此外，如果零是$ h $的第二种共振，则$ w_ \ pm $甚至没有从$ l^\ infty（\ r）$到$ \ bmo（\ r）$的限制。特别是，我们指出的是，我们的结果给出了$ l^p $结合的完整图片，其中所有$ 1 \ le p \ le p \ le \ le \ infty $在常规情况下的有效性。最后，作为应用程序，我们将$ l^p $ - $ l^q $衰减估计值$ e^{ - ith} p _ {\ mathrm {ac}}}（h）$带有对$（1/p，1/q）$属于$ \ \ m i \ mathbb {r} $，以及建立的$（1/p，1/q）$光谱乘数$ f（h）$的$ l^p $ bundedness定理。
This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_\pm=W_\pm(H, Δ^2)$ associated with the bi-Schrödinger operators $H=Δ^{2}+V(x)$ on the line $\mathbb{R}$. Given suitable decay potentials $V$, we firstly prove that the wave and dual wave operators are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$: $$ \|W_\pm f\|_{L^p(\mathbb{R})}+\|W_\pm^* 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 $L^1(\R)$ to $L^{1,\infty}(\R)$ as well as bounded from the Hardy space $\H^1(\R)$ to $L^1(\R)$. 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 $L^\infty(\R)$ to $\BMO(\R)$ 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 $L^\infty(\R)$ to $\BMO(\R)$ 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 $1\le p\le \infty$ 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 $f(H)$.
