Band 36, Heft 3

The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial.

The group of matrices P 1 {P_{1}} of Pauli is a finite 2-group of order 16 and plays a fundamental role in quantum information theory, since it is related to the quantum information on the 1-qubit. Here we show that both P 1 {P_{1}} and the Pauli 2-group P 2 {P_{2}} of order 64 on 2-qubits, other than in quantum computing, can also appear in dynamical systems which are described by non-self-adjoint Hamiltonians. This will allow us to represent P 1 {P_{1}} and P 2 {P_{2}} in terms of pseudofermionic operators.

We prove that three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat, provided that the electric field and the gradient of the lapse function are linearly dependent. Consequently, a three-dimensional electrostatic manifold admits a local warped product structure with a one-dimensional base and a constant curvature surface fiber.

In this paper, we will study the behavior of the solutions of the linear parabolic equation with Dirichlet conditions when the domain is perturbed in the C 1 {C^{1}} topology. More precisely, it is shown that the solutions of this equation are stable under such perturbations.

The aim of this paper is to describe the topological K -ring in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K -ring, and hence the Grothendieck ring of algebraic vector bundles over flag Bott–Samelson varieties.

We study Seshadri constants of certain ample vector bundles on projective varieties. Our main motivation is the following question: Under what conditions are the Seshadri constants of ample vector bundles at least 1 at all points of the variety? We exhibit some conditions under which this question has an affirmative answer. We primarily consider ample bundles on projective spaces and Hirzebruch surfaces. We also show that Seshadri constants of ample vector bundles can be arbitrarily small.

In this article, we derive the restriction theorem for the special Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions associated with the special Hermite operator. Further, we discuss the optimal behavior of the constant as a limit of a large number of functions.

We are interested in the one-dimensional nonlinear wave equations with multiple wave speeds by the energy method. By choosing different multipliers corresponding to the different wave speeds, we show that the one-dimensional nonlinear wave equations also have globally smooth solutions provided that the nonlinearities satisfy certain structural conditions when the initial data are small. Furthermore, we can prove that the global solutions will converge to the solutions of the linearized system based on the decay properties of the nonlinearities.

We establish existence results for a class of mixed anisotropic and nonlocal p -Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, the Harnack inequality for solutions and the weak Harnack inequality for supersolutions.

The aim of this paper is to obtain the boundedness of commutators of Hardy operators with rough kernels on grand variable Herz spaces K ˙ q ⁢ ( ⋅ ) a ⁢ ( ⋅ ) , u , θ ⁢ ( ℝ n ) {\dot{K}^{a(\,\cdot\,),u,\theta}_{q(\,\cdot\,)}(\mathbb{R}^{n})} by applying some properties of variable exponent. Moreover, by using the idea of grand variable Herz–Morrey spaces, we will prove the boundedness of Hardy operators on these spaces.

Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the s -Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where s is the Hausdorff dimension of its support and obtain their exact upper bound of the densities.

The aim of this article is to give an alternative proof of Tataru’s dispersive estimates for wave equations posed on the hyperbolic space. Based on the formula for the wave kernel on ℍ n {\mathbb{H}^{n}} , we give the proof from the perspective of Bessel potentials, by exploiting various facts about Gamma functions, modified Bessel functions, and Bessel potentials. This leads to our proof being more self-contained than that in [D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 2001, 2, 795–807].

We study a “ p -powered” version K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} of the well-known Bohr radius problem for the family F ⁢ ( R ) {F(R)} of holomorphic functions f : R → X {f:R\to X} satisfying ∥ f ∥ < ∞ {\lVert f\rVert<\infty} , where ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} is a norm in the function space F ⁢ ( R ) {F(R)} , R ⊂ ℂ n {R\subset{\mathbb{C}}^{n}} is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0 {p>0} , we describe in full detail the asymptotic behavior of K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} , where F ⁢ ( R ) {F(R)} is: (a) the Hardy space of X -valued holomorphic functions defined in the open unit polydisk 𝔻 n {{\mathbb{D}}^{n}} ; and (b) the space of bounded X -valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B ⁢ ( l t n ) {B(l_{t}^{n})} of the Minkowski space l t n {l_{t}^{n}} . We give an alternative definition of the optimal cotype for a complex Banach space X in the light of these results. In addition, the best possible versions of two theorems from [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results.

This paper is concerned with the following fractional (N/s)-Laplacian Choquard equation: ε N ⁢ ( - Δ ) N / s s ⁢ u + V ⁢ ( x ) ⁢ | u | N s - 2 ⁢ u = ε μ ⁢ ( 1 | x | N - μ ∗ F ⁢ ( u ) ) ⁢ f ⁢ ( u ) , x ∈ ℝ N , \varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)\lvert u\rvert^{\frac{N}{s}-2}u=% \varepsilon^{\mu}\Bigl{(}\frac{1}{\lvert x\rvert^{N-\mu}}\ast F(u)\Bigr{)}f(u)% ,\quad x\in{\mathbb{R}}^{N}, where ( - Δ ) N / s s {(-\Delta)_{N/s}^{s}} denotes the (N/s)-Laplacian operator, 0 < μ < N {0<\mu<N} , and V and f are continuous real functions satisfying some mild assumptions. Applying the weak growth conditions on the exponential critical nonlinearity f and without using the strictly monotone condition, we use some refined analysis and develop the arguments in the existing results to establish the existence of the ground state solution of the fractional (N/s)-Laplacian Choquard equation. Moreover, we also study the concentration phenomenon of the ground state solutions. As far as we know, our results seem to be new concerning the fractional (N/s)-Laplacian equation with the Choquard reaction.

We consider the solutions of the Klein–Gordon equation in the 2 + 1 2{+}1 -dimensional space-time which gravity is analyzed, i.e., the manifold ℝ t × ℝ + × ℝ / 2 ⁢ π ⁢ α ⁢ ℤ {\mathbb{R}_{t}\times\mathbb{R}_{+}\times\mathbb{R}/2\pi\alpha\mathbb{Z}} created by a massive point particle. Using the Schwartz kernel of resolvent and spectral measure for Schrödinger operator on the spinless cone, we prove the dispersive estimates and Strichartz estimates for the Klein–Gordon equation. In a future paper, we will consider the problem on the spinning cone.

Due to its applications in signal analysis and image processing, the quaternionic Fourier analysis has received increasing attention. In particular, quaternionic Gabor frames (QGFs) attracted some mathematicians’ interest. From the literatures, some results on QGFs are based on quaternionic Fourier orthonormal bases. But those used so-called quaternionic Fourier orthonormal bases have a gap that they are all incomplete. In this paper, we present a class of quaternionic Fourier orthonormal bases, and using them derive the corresponding Gabor orthonormal bases.

In this paper, we combine the arguments of [L. Fanelli, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operators in critical magnetic fields, Int. Math. Res. Not. IMRN 2023), 10.1093/imrn/rnac362] and [Y. Sire, C. D. Sogge, C. Wang and J. Zhang, Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications, Comm. Partial Differential Equations 47 2022, 6, 1124–1132] to prove the maximal estimates for fractional Schrödinger equations ( i ⁢ ∂ t + ℒ 𝐀 α 2 ) ⁢ u = 0 {(i\partial_{t}+\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})u=0} in the purely magnetic fields which includes the Aharonov–Bohm fields. The proof is based on the cluster spectral measure estimates. In particular, for α = 1 {\alpha=1} , the maximal estimate for wave equation is sharp up to the endpoint.

This paper concerns the scalar field equation - Δ ⁢ u = λ ⁢ u + | u | q - 2 ⁢ u + a ⁢ | u | 4 ⁢ u + b ⁢ ( I 2 ∗ | u | 5 ) ⁢ | u | 3 ⁢ u   in  ⁢ ℝ 3 -\Delta u=\lambda u+|u|^{q-2}u+a|u|^{4}u+b(I_{2}\ast|u|^{5})|u|^{3}u\quad\text% {in }\mathbb{R}^{3} under the normalized constraint ∫ ℝ 3 u 2 ⁢ 𝑑 x = c 2 {\int_{\mathbb{R}^{3}}u^{2}\,dx=c^{2}} , where a , b , c > 0 {a,\,b,\,c>0} , 2 < q < 10 3 {2<q<\frac{10}{3}} and I 2 {I_{2}} is the Riesz potential. We prove that for small prescribed mass c the above equation has a positive ground state solution and an infinite sequence of normalized solutions with negative energies tending to zero. Asymptotic properties of ground state solutions as a → 0 + {a\to 0^{+}} and as b → 0 + {b\to 0^{+}} are also studied.