Recent zbMATH articles in MSC 47G30https://zbmath.org/atom/cc/47G302021-06-15T18:09:00+00:00WerkzeugGrowth of Sobolev norms for abstract linear Schrödinger equations.https://zbmath.org/1460.353072021-06-15T18:09:00+00:00"Bambusi, Dario"https://zbmath.org/authors/?q=ai:bambusi.dario"Grébert, Benoît"https://zbmath.org/authors/?q=ai:grebert.benoit"Maspero, Alberto"https://zbmath.org/authors/?q=ai:maspero.alberto"Robert, Didier"https://zbmath.org/authors/?q=ai:robert.didierSummary: We prove an abstract theorem giving a \(\langle t\rangle^\epsilon\) bound (for all \(\epsilon > 0)\) on the growth of the Sobolev norms in linear Schrödinger equations of the form \(\mathrm i \dot{\psi} = H_0 \psi + V(t) \psi\) as \(t \to \infty\). The abstract theorem is applied to several cases, including the cases where (i) \(H_0\) is the Laplace operator on a Zoll manifold and \(V(t)\) a pseudodifferential operator of order smaller than 2; (ii) \(H_0\) is the (resonant or nonresonant) Harmonic oscillator in \(\mathbb R^d\) and \(V(t)\) a pseudodifferential operator of order smaller than that of \(H_0\) depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of \textit{A. Maspero} and the last author [J. Funct. Anal. 273, No. 2, 721--781 (2017; Zbl 1366.35153)].Complete systems of eigenfunctions of the Vladimirov operator in \(L^2(B_r)\) and \(L^2(\mathbb{Q}_p)\).https://zbmath.org/1460.470482021-06-15T18:09:00+00:00"Bikulov, A. Kh."https://zbmath.org/authors/?q=ai:bikulov.albert-khakimovich"Zubarev, A. P."https://zbmath.org/authors/?q=ai:zubarev.aleksandr-petrovichSummary: We construct new bases of real functions from \(L^2(B_r)\) and from \(L^2(\mathbb{Q}_p)\). These functions are eigenfunctions of the \(p\)-adic pseudo-differential Vladimirov operator, which is defined on a compact set \(B_r \subset \mathbb{Q}_p\) of the field of \(p\)-adic numbers \(\mathbb{Q}_p\) or, respectively, on the entire field \(\mathbb{Q}_p\). A~relation between the basis of functions from \(L^2(\mathbb{Q}_p)\) and the basis of \(p\)-adic wavelets from \(L^2(\mathbb{Q}_p)\) is found. As an application, we consider the solution of the Cauchy problem with the initial condition on a compact set for a pseudo-differential equation with a general pseudo-differential operator that is diagonal in the basis constructed.Commutators of bilinear pseudo-differential operators on local Hardy spaces with variable exponents.https://zbmath.org/1460.420212021-06-15T18:09:00+00:00"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghuiSummary: The aim of this paper is to establish the boundedness of the commutator \([b_1, b_2,T_{\sigma}]\) generated by the bilinear pseudo-differential operator \(T_{\sigma}\) with smooth symbols and \(b_1,b_2\in \mathrm{BMO}(\mathbb{R}^n)\) on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator \(T_{\sigma}\) is bounded from the Lebesgue space \(L^p(\mathbb{R}^n)\) into \(h^{p_1(\cdot)}(\mathbb{R}^n)\times h^{p_2(\cdot)}(\mathbb{R}^n)\). Moreover, the boundedness of the commutator \([b_1, b_2, T_{\sigma}]\) on product of local Hardy spaces with variable exponents is also obtained.