Annali di Matematica Pura ed Applicata最新文献

Let (Omega ) be a bounded non-smooth domain in (mathbb {R}^n) that satisfies the measure density condition. In this paper, the authors study the interrelations of three basic types of Besov spaces (B_{p,q}^s(Omega )), (mathring{B}_{p,q}^s(Omega )) and (widetilde{B}_{p,q}^s(Omega )) on (Omega ), which are defined, respectively, via the restriction, completion and supporting conditions with (p,qin [1,infty )) and (sin (0,1)). The authors prove that (B_{p,q}^s(Omega )=mathring{B}_{p,q}^s(Omega )=widetilde{B}_{p,q}^s(Omega )), if (Omega ) supports a fractional Besov–Hardy inequality, where the latter is proved under certain conditions on fractional Besov capacity or Aikawa’s dimension of the boundary of (Omega ).

We show that the set of “wild data”, meaning the initial data for which the barotropic Euler system admits infinitely many admissible entropy solutions, is dense in the (L^p)-topology of the phase space.

We improve the result by Feola and Iandoli (J Math Pures Appl 157:243–281, 2022), showing that quasilinear Hamiltonian Schrödinger type equations are well posed on (H^s({{mathbb {T}}}^d)) if (s>d/2+3). We exploit the sharp paradifferential calculus on ({{mathbb {T}}}^d) developed by Berti et al. (J Dyn Differ Equ 33(3):1475–1513, 2021).

In this paper, the orbital instability theorem is introduced for Hamiltonian partial differential equation (PDE) systems. Our specific focus lies on the Schrödinger system featuring quadratic nonlinearity, and we apply the theorem to analyze its behavior. Our theorem establishes the abstract instability theorem for a specific class of Hamiltonian PDE systems. We consider the energy functional to be of class (C^2) rather than (C^3), particularly when the second derivative of the energy exhibits multiple degenerate kernels. Using this theorem, we provide a comprehensive classification of the stability and instability of the semitrivial solution within the Hamiltonian PDE system featuring quadratic nonlinearity. This classification resolves an open problem previously posed by Colin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:2211–2226, 2009), specifically in cases of homogeneous nonlinearity. Additionally, we present proof of instability results for synchronous solutions of Hamiltonian PDE systems. We believe that this abstract theorem constitutes a novel contribution with potential applicability in various situations beyond those specifically discussed in this paper.

We study an extension problem for continuous linear maps in the setting of (LB)-spaces. More precisely, we characterize the pairs (E, Z), where E is a locally complete space with a fundamental sequence of bounded sets and Z is an (LB)-space, such that for every exact sequence of (LB)-spaces

the map

is surjective, meaning that each continuous linear map (X rightarrow E) can be extended to a continuous linear map (Y rightarrow E) via (iota ), under some mild conditions on E or Z (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24].

This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane (mathbb {C}^2) satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in (mathbb {C}^2), and every connected domain in (mathbb {C}^2) admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, M, closed discrete subset E of (mathring{M}=Msetminus bM), and compact subset (Ksubset mathring{M}setminus E) without holes in (mathring{M}), any (mathscr {C}^1) embedding (f:Mhookrightarrow mathbb {C}^2) which is holomorphic in (mathring{M}) can be approximated uniformly on K by holomorphic embeddings (F:Mhookrightarrow mathbb {C}^2) which map (Ecup bM) out of a given ball and satisfy some interpolation conditions.

Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit ball D of a Banach space ((V,Vert cdot Vert )) equipped with a particular Jordan structure, called a (textrm{JB}^*)-triple. We identify the horofunctions in the metric compactification of ((V,Vert cdot Vert )) and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space (V^*). Finally, we show that the exponential map (exp _0 :V longrightarrow D) at (0in D) extends to a homeomorphism between the metric compactifications of ((V,Vert cdot Vert )) and ((D,rho )), preserving the geometric structure, where (rho ) is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of ((V,Vert cdot Vert )).

We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs (m, n) the locus of polarised abelian surfaces of type (1, d) that contain two complementary elliptic curve of exponents m, n, denoted (mathcal {E}_d(m,n)) is non-empty. We show that if d is square-free, the locus (mathcal {E}_d(m,n)) is an irreducible surface (if non-empty). We also show that the loci (mathcal {E}_d(d,d)) can have many components if d is an odd square. As an application, we show that for a genus 3 curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings (f_i:Crightarrow E_i, i=1,2,3) satisfy ({{,textrm{lcm},}}(deg (f_1),deg (f_2))={{,textrm{lcm},}}(deg (f_1),deg (f_3))={{,textrm{lcm},}}(deg (f_2),deg (f_3))).

Let G be the semidirect product (N rtimes mathbb {R}), where N is a stratified Lie group and (mathbb {R}) acts on N via automorphic dilations. Homogeneous left-invariant sub-Laplacians on N and (mathbb {R}) can be lifted to G, and their sum (Delta ) is a left-invariant sub-Laplacian on G. In previous joint work of Ottazzi, Vallarino and the first-named author, a spectral multiplier theorem of Mihlin–Hörmander type was proved for (Delta ), showing that an operator of the form (F(Delta )) is of weak type (1, 1) and bounded on (L^p(G)) for all (p in (1,infty )) provided F satisfies a scale-invariant smoothness condition of order (s > (Q+1)/2), where Q is the homogeneous dimension of N. Here we show that, if N is a group of Heisenberg type, or more generally a direct product of Métivier and abelian groups, then the smoothness condition can be pushed down to the sharp threshold (s>(d+1)/2), where d is the topological dimension of N. The proof is based on lifting to G weighted Plancherel estimates on N and exploits a relation between the functional calculi for (Delta ) and analogous operators on semidirect extensions of Bessel–Kingman hypergroups.

We use the Leray spectral sequence for the sheaf cohomology groups with compact supports to obtain a vanishing result. The stalks of sheaves (R^{bullet }phi _{!}mathcal {O}) for the structure sheaf (mathcal {O}) on the total space of a holomorphic fiber bundle (phi ) has canonical topology structures. Using the standard Čech argument we prove a density lemma for QDFS-topology on this stalks. In particular, we obtain a vanishing result for holomorphic fiber bundles with Stein fibers. Using Künnet formulas, properties of an inductive topology (with respect to the pair of spaces) on the stalks of the sheaf (R^{1}phi _{!}mathcal {O}) and a cohomological criterion for the Hartogs phenomenon we obtain the main result on the Hartogs phenomenon for the total space of holomorphic fiber bundles with (1, 0)-compactifiable fibers.