arXiv subjects
Publications and source records attributed to Zhirayr Avetisyan.
We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very weak solutions to establish the existence and uniqueness of solutions to the heat equation with singular, anisotropic, time-dependent diffusivity.
On a compact Lie group $G$, we consider the reproducing kernel Hilbert space $\mathcal{H}_K$ associated with the integral kernel $K$ of a left-invariant, positive, symmetric, trace class integral operator on $L^2(G)$. We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of $\mathcal{H}_K$ into the space $C(G)$ of continuous functions on $G$.
On a complete, connected, non-compact Riemannian manifold, with Ricci curvature bounded from below, we establish exponential decay estimates at infinity for the spherical sums of the resolvent kernel, i.e., the integral kernel of the resolvent $(-Δ+λ)^{-1}$ of the Laplace-Beltrami operator for $λ>0$. The exponential decay rate in these estimates is optimal, in the sense that it cannot be improved based solely on a uniform bound on the Ricci curvature. In addition to technical results, the paper offers two conceptual takeaways. Firstly, in pursuit of optimal decay estimates, it may be worth shifting the emphasis from the resolvent kernel to its density - a geometrically more natural object. Secondly, contrary to expectations, the exponential decay of the resolvent kernel is driven mainly by the area/volume growth, and the bottom of the spectrum of $Δ$ is not a decisive factor. This is in contrast with the heat kernel decay, and shows that optimal resolvent kernel estimates cannot be derived from heat kernel estimates.
We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very weak solutions to establish the existence and uniqueness of solutions to the heat equation with singular, anisotropic, time-dependent diffusivity.
In this paper, we consider a two-dimensional operator with an antisymmetric integral kernel, recently introduced by Z. Avetisyan and A. Karapetyants in connection to the study of general homogeneous operators. This is the unique two-dimensional operator that has an antisymmetric kernel homogeneous with respect to all orientation-preserving linear transformations of the plane. It is shown that the operator under consideration interacts naturally, both in Cartesian and polar coordinates, with projective tensor products of some classical functional spaces, such as Lebesgue, Hardy, and Hölder spaces; conditions for their boundedness as operators acting from these spaces to Banach lattices of measurable functions and estimates of their norms are given.
In this paper we prove the existence and uniqueness of positive mild solutions for the semilinear parabolic equations of the form $u_t+\mathcal{L}u=f+h\cdot G(u)$, where $h$ is a positive function and $G$ a positive concave function (for example, $G(u)=u^α$ for $0<α<1$). In contrast with the case of convex $G$, where the Fujita exponent appears, and only existence of a positive solution for special data is achieved, in this concave case we obtain the existence and uniqueness for any positive data. The method relies on proving the existence and uniqueness of solutions for a certain class of Hammerstein-Volterra-type integral equations with a concave non-linear term, in the very general setting of arbitrary measure spaces. As a consequence, we obtain results for the existence and uniqueness of mild solutions to semilinear parabolic equations with concave nonlinearities under rather general assumptions on $f, h$ and $G$, in a variety of settings: e.g. for $\mathcal{L}$ being the Laplacian on complete Riemannian manifolds (e.g., symmetric spaces of any rank) or Hörmander sum of squares on general Lie groups. We also study the corresponding equations in the presence of a damping term, in which case assumptions can be relaxed even further.
We describe a general framework of functional and Fourier analysis on domains with a free action of an Abelian Lie group $G$. Namely, on a domain of the form $G\times Y$ we introduce the appropriate spaces of distributions and measurable functions, establishing their most basic properties. Then we consider the half-Fourier transform $f(x,y)\mapsto\hat f(ξ,y)$ in the first variable, and discuss the behaviour of function spaces on $G\times Y$ and $\hat G\times Y$ under this transform. We introduce general mixed-Fourier-norm spaces on $G\times Y$, and the subspaces of holomorphic functions among them, and give an explicit descriptions of the Fourier images of these spaces.
We introduce and study weighted spaces of functions with mixed norm on the upper half-plane, defined in terms of Fourier transform. We give a characterization of analytic functions within these spaces, and in particular, we provide an analog of the Paley-Wiener theorem in this setting. As an application, we consider Toeplitz operators with vertical symbols in these new spaces.
The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions.
An almost Abelian group is a non-Abelian Lie group with a codimension 1 Abelian subgroup. This paper investigates invariant Hermitian and Kähler structures on connected complex almost Abelian groups. We find explicit formulas for the left and right Haar measures, the modular function, and left and right generator vector fields on simply connected complex almost Abelian groups. From the generator fields, we obtain invariant vector and tensor field frames, allowing us to find an explicit form for all invariant tensor fields. Namely, all such invariant tensor fields have constant coefficients in the invariant frame. From this, we classify all invariant Hermitian forms on complex simply connected almost Abelian groups, and we prove the nonexistence of invariant Kähler forms on all such groups. Via constructions involving the pullback of the quotient map, we extend the explicit description of invariant Hermitian structures and the nonexistence of Kähler structures to connected almost Abelian groups.
An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian subgroup. We show that all discrete subgroups of complex simply connected almost Abelian groups are finitely generated. The topology of connected almost Abelian Lie groups is studied by expressing each connected almost Abelian Lie group as a quotient of its universal covering group by a discrete normal subgroup. We then prove that no complex connected almost Abelian group is compact, and give conditions for the compactness of connected subgroups of such groups. Towards studying the homotopy type of complex connected almost Abelian groups, we investigate the maximal compact subgroups of such groups.
We call an element $U$ conditionally universal for a sequential convergence space $\mathbfΩ$ with respect to a minimal system $\{φ_n\}_{n=1}^\infty$ in a continuously and densely embedded Banach space $\mathcal{X}\hookrightarrow\mathbfΩ$ if the partial sums of its phase-modified Fourier series is dense in $\mathbfΩ$. We will call the element $U$ almost universal if the change of phases (signs) needs to be performed only on a thin subset of Fourier coefficients. In this paper we prove the existence of an almost universal element under certain assumptions on the system $\{φ_n\}_{n=1}^\infty$. We will call a function $U$ asymptotically conditionally universal in a space $L^1(\mathcal{M})$ if the partial sums of its phase-modified Fourier series is dense in $L^1(F_m)$ for an ever-growing sequence of subsets $F_m\subset\mathcal{M}$ with asymptotically null complement. Here we prove the existence of such functions $U$ under certain assumptions on the system $\{φ_n\}_{n=1}^\infty$. Moreover, we show that every integrable function can be slightly modified to yield such a function $U$. In particular, we establish the existence of almost universal functions for $L^p([0,1])$, $p\in(0,1)$, and asymptotically conditionally universal functions for $L^1([0,1])$, with respect to the trigonometric system.
An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media - cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified and studied in details. However, a systematic investigation of almost Abelian Lie groups has not been carried out yet, and the present paper is devoted to an explicit description of properties of this wide and diverse class of groups. The subject of investigation are real almost Abelian Lie groups with their Lie group theoretical aspects, such as the exponential map, faithful matrix representations, discrete and connected subgroups, quotients and automorphisms. The emphasis is put on explicit description of all technical details.
An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media - cosmology, crystallography etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics perspective has not been carried out yet, and the present paper is the first step in addressing this wide and diverse class of groups and algebras. The present paper studies the structure and important algebraic properties of almost Abelian Lie algebras of arbitrary dimension over any field of scalars. A classification of almost Abelian Lie algebras is given. All Lie subalgebras and ideals, automorphisms and derivations, Lie orthogonal operators and quadratic Casimir elements are described exactly.
In our previous article Phys. Rev. Lett. 127 (2021) 271601, we announced a novel 'democratic' Lagrangian formulation of general nonlinear electrodynamics in four dimensions that features electric and magnetic potentials on equal footing. Here, we give an expanded and more detailed account of this new formalism, and then proceed to push it significantly further by building the corresponding Lagrangian theories of higher form field interactions in arbitrary dimensions. Special attention is given to interactions of chiral 2k-forms in 4k+2 dimensions, with further details for 2-forms in 6 dimensions and 4-forms in 10 dimensions. We comment more broadly on the structure of covariant equations of motion for chiral fields, and on the place of our Lagrangian theories in this context. The Lagrangian theories we develop are simple and explicit, and cover a much broader class of interactions than all past attempts in the literature.
We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples - classes of homogeneous integral operators on various metric spaces, and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article - various specific cases - illustrate general statements and results given in the paper and at the same time are of interest in their own way.
We construct a Lagrangian for general nonlinear electrodynamics that features electric and magnetic potentials on equal footing. In the language of this Lagrangian, discrete and continuous electric-magnetic duality symmetries can be straightforwardly imposed, leading to a simple formulation for theories with the $SO(2)$ duality invariance. When specialized to the conformally invariant case, our construction provides a manifestly duality-symmetric formulation of the recently discovered ModMax theory. We briefly comment on a natural generalization of this approach to $p$-forms in $2p+2$ dimensions.
The subject of the paper is the geometry and topology of cosmological spacetimes and vector bundles thereon, which are used to model physical fields propagating in the universe. Global hyperbolicity and factorization properties of the spacetime and the vector bundle that are usually independently assumed to hold, are now derived from a minimal set of assumptions based on the recent progress in differential geometry and topology.