Recent zbMATH articles in MSC 47Ghttps://zbmath.org/atom/cc/47G2021-06-15T18:09:00+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.Eigenvalues and eigenfunctions of double layer potentials.https://zbmath.org/1460.470222021-06-15T18:09:00+00:00"Miyanishi, Yoshihisa"https://zbmath.org/authors/?q=ai:miyanishi.yoshihisa"Suzuki, Takashi"https://zbmath.org/authors/?q=ai:suzuki.takashiSummary: Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let \( \Omega \) be a \( C^2\) bounded region in \(\mathbb{R}^n\) (\(n=2, 3\)). The double layer potential \(K: L^2(\partial \Omega) \rightarrow L^2 (\partial \Omega ) \) is defined by
\[
(K \psi )(x) \equiv \int _{\partial \Omega } \psi (y)\cdot \nu _{y} E(x, y) \, ds_y,
\]
where
\[
E(x, y) =
\begin{cases}
\frac {1}{\pi } \log \frac {1}{| x-y|} \quad &\text{if } n=2,\\
\frac{1}{2\pi}\frac{1}{| x-y|} &\text{if } n=3,
\end{cases}
\]
where \(ds_y\) is the line or surface element and \( \nu _y\) is the outer normal derivative on \( \partial \Omega\). It is known that \( K\) is a compact operator on \( L^2 (\partial \Omega)\) and consists of at most a countable number of eigenvalues, with 0 as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of \( \partial \Omega\).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.Weighted estimates for bilinear Fourier multiplier operators with multiple weights.https://zbmath.org/1460.420112021-06-15T18:09:00+00:00"Hu, Guoen"https://zbmath.org/authors/?q=ai:hu.guoen"Wang, Zhidan"https://zbmath.org/authors/?q=ai:wang.zhidan"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingying"Yabuta, Kôzô"https://zbmath.org/authors/?q=ai:yabuta.kozoSummary: In the weighted theory of multilinear operators, the weights class which usually has been considered is the product of \(A_p\) weights. However, it is known that \(\prod_{k=1}^2A_{p_k}(\mathbb{R}^n)\varsubsetneq A_{\vec{p}}(\mathbb{R}^{2n})\), and \(\vec{w}=(w_1,\,w_2)\in A_{\vec{p}}(\mathbb{R}^{2n})\) does not imply that \(w_k\in L^1_{\text{loc}}(\mathbb{R}^n)\) for \(k=1,\,2\). Therefore, it is very interesting to study the weighted theory of multilinear operators with the weights in \(A_{\vec{p}}(\mathbb{R}^{2n})\). In this paper, we consider the weights class \(A_{\vec{p}/\vec{r}}(\mathbb{R}^{2n})\), which is more general than \(A_{\vec{p}}(\mathbb{R}^{2n})\). If \(\vec{w}=(w_1,\,w_2)\in A_{\vec{p}/\vec{r}}(\mathbb{R}^{2n})\), we show that the bilinear Fourier multiplier operator \(T_{\sigma}\) is bounded from \(L^{p_1}(w_1)\times L^{p_2}(w_2)\) to \(L^p(\nu_{\vec{w}})\) when the symbol \(\sigma\) satisfies the Sobolev regularity that \(\sup_{\kappa\in\mathbb{Z}}\Vert \sigma_k\Vert_{W^{s_1,s_2}(\mathbb{R}^{2n})}<\infty\) with \(s_1,s_2\in (\frac{n}{2},\,n]\).Norm dynamic inequalities and theorems of factorization of weighted Cesàro and Copson spaces.https://zbmath.org/1460.260312021-06-15T18:09:00+00:00"Saker, S. H."https://zbmath.org/authors/?q=ai:saker.samir-h"Abuelwafa, M. M."https://zbmath.org/authors/?q=ai:abuelwafa.m-m"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donal"Agarwal, R. P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: In this paper, we establish some factorization theorems for weighted Cesàro and Copson spaces, obtain two sided norm dynamic inequalities, and give conditions for the boundedness of the Hardy and Copson dynamic operators on the weighted space \(L_\lambda^p(\mathbb{T})\). We obtain, as special cases, the classical integral inequalities on \(\mathbb{R}\) and the discrete inequalities on \(\mathbb{N}\).