The spectrum of differential operators in \(H^p\) spaces (Q2567488)

This paper is concerned with linear partial differential operators with constant coefficients in \(H^p(\mathbb R^n)\). In the case \(0<p \leq 1\), the authors establish the spectral mapping property and determine completely the essential spectrum, point spectrum, approximate point spectrum, continuous spectrum and residual spectrum of such differential operators. In the case \(p>2\), they show that the point spectrum of such differential operators in \(L^p(\mathbb R^n)\) is the empty set where \(2< p < 2n/(n-1)\). Let \(P: {\mathbb R}^n \rightarrow {\mathbb C}\) be a polynomial of degree \(m>0\). The corresponding PDO in \(H^p\) is defined by \[ P_p f = {\mathcal F}^{-1}(P \widehat{f})\quad {\text{with}}\quad D(P_p) = \{ f \in H^p ; {\mathcal F}^{-1}(P \widehat{f}) \in H^p \}. \] We give some notations: \(H^p_s = \{ f \in H^p\); \({\mathcal F}^{-1}( (1 + | \cdot |^2)^{s/2} \widehat{f}) \in H^p \}\). \({\mathcal M}_p = \{ u \in L^{\infty}\); \(\| u \|_{{\mathcal M}_p} < \infty \}\) where \(\| u \|_{{\mathcal M}_p} = \sup\{ \| {\mathcal F}^{-1}(u \widehat{f}) \|_{H^p}\); \(f \in {\mathcal S} \cap H^p\), \(\| f \|_{H^p} \leq 1 \}\). Denote by \(\rho(P_p)\) the resolvent set of \(P_p\), that is, \[ \begin{aligned} \rho(P_p)= \{ \lambda \in {C}; \;&{\text{the range}}\;R(\lambda - P_p)\;{\text{is dense in}}\;H^p \;{\text{and there exists}} \\ &M>0 \;{\text{such that}}\;\| f \|_{H^p} \leq M \| (\lambda - P_p)f \|_{H^p}\;{\text{for}}\;f \in D(P_p) \}. \end{aligned} \] The spectrum of \(P_p\) is \(\sigma(P_p) = {\mathbb C} \setminus \rho(P_p)\). The authors consider the following cases: (1) \(\overline{P({\mathbb R}^n)} \subset \sigma(P_p)\), where \(P({\mathbb R}^n) =\{ P(\xi)\); \(\xi \in {\mathbb R}^n \}\); (2) \(\lambda \in \rho(P_p)\) if and only if \((\lambda - P )^{-1} \in {\mathcal M}_p\); (3) \(P_p\) has no eigenvalues; (4) \(P_p\) has no compact resolvents; (5) \(P_p\) has no eigenvalues if \(2 < p < \frac{2n}{n-1}\).