arXiv subjects
Publications and source records attributed to Lizhong Peng.
We prove that the KP-I initial-value problem \begin{eqnarray*} \begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 {on}{\R}^2_{x,y}\times {\R}_t; u(x,y,0)=ϕ(x,y), \end{cases} \end{eqnarray*} is locally well-posed in the space \begin{eqnarray*} H^{1,0}(\R^2)=\{ϕ\in L^2(\R^2): \ \normϕ_{H^{1,0}(\R^2)}\approx\normϕ_{L^2}+\norm{\partial_xϕ}_{L^2}<\infty\}. \end{eqnarray*}
In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{itϕ(\sqrt{-Δ})}$, where $ϕ: \mathbb{R}^+\to \mathbb{R}$ is smooth away from the origin. Especially, the decay estimates for the solutions of the Klein-Gordon equation and the beam equation are simplified and slightly improved.
In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schrödinger operator $P=-Δ+V(x)$ on $\mathbb{R}^n, n\geq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q \geq n/2$. Let $T_1=(-Δ+V)^{-1}V,\ T_2=(-Δ+V)^{-1/2}V^{1/2}$ and $T_3=(-Δ+V)^{-1/2}\nabla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(\mathbb{R}^n)$ when $p$ ranges in a interval, where $b \in \mathbf{BMO}(\mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness.
Consider a root system of type $BC_1$ on the real line $\mathbb R$ with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an $L^2$-space on $\mathbb R$ to a $L^2$-space of $\mathbb C^2$-valued functions on $\mathbb R^+$ with the Harish-Chandra measure $|c(\lam)|^{-2}d\lam$. By introducing a weight function of the form $\cosh^{-\sig}(t)\tanh^{2k} t$ on $\mathbb R$ we find an orthogonal basis for the $L^2$-space on $\mathbb R$ consisting of even and odd functions expressed in terms of the Jacobi polynomials (for each fixed $\sig$ and $k$). We find a Rodrigues type formula for the functions in terms of the Cherednik operator. We compute explicitly their Cherednik-Opdam transforms. We discover thus a new family of $\mathbb C^2$-valued orthogonal polynomials. In the special case when $k=0$ the even polynomials become Wilson polynomials, and the corresponding result was proved earlier by Koornwinder.