Analysis and Mathematical Physics最新文献

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

We establish the monotonicity of positive solutions to the problem

where (p>2), (qge p-1) and a, f are locally Lipschitz continuous functions such that f is positive on ((0,+infty )) and it is either sublinear or superlinear near 0. The main tool we use is the refined method of moving planes for quasilinear elliptic problems in half-spaces.

In this paper we initiate the study of the forward and backward shifts on the discrete generalized Hardy space of a tree and the discrete generalized little Hardy space of a tree. In particular, we investigate when these shifts are bounded, find the norm of the shifts if they are bounded, characterize the trees in which they are an isometry, compute the spectrum in some concrete examples, and completely determine when they are hypercyclic.

Let (mathbf {E_{mathbb {X}}}) be a unit ball on complex Banach space (mathbb {X}) and (Phi ) be a convex function such that (Phi (0)=1) and (Re Phi (xi )>0) on (mathbb {D}={zin mathbb {C}:|z|<1}). In this paper, we continue the work related to the class (Q_textbf{B}^{Phi }(mathbf {E_{mathbb {X}}})) of quasi-convex mappings of type (textbf{B}) which have a (Phi )-parametric representation on (mathbf {E_{mathbb {X}}}), where the mappings (fin Q_textbf{B}^{Phi }(mathbf {E_{mathbb {X}}})) are k-fold symmetric, (kin mathbb {N}.) We give the improved Fekete-Szegö inequalities for the class (Q_textbf{B}^{Phi }(mathbf {E_{mathbb {X}}})) and establish the sharp bounds of all terms of homogeneous polynomial expansions for some subclasses of (Q_textbf{B}^{Phi }(mathbf {E_{mathbb {X}}})). Our main results are closely related to the Bieberbach conjecture in higher dimensions.

We describe a general method for constructing Heisenberg uniqueness pairs ((Gamma ,Lambda )) in the euclidean space (mathbb {R}^{n}) based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary (Gamma ) of a bounded convex set (Omega ) and a sphere (Lambda ) is an Heisenberg uniqueness pair if and only if the square of the radius of (Lambda ) is not an eigenvalue of the Laplacian on (Omega ). The main ingredients for the proofs are the Paley–Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in (mathbb {C}^{n}). Denjoy’s theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.

In the present paper we introduce a mechanism for generation a new class of linear transformations that preserve real roots of polynomials by using the theory of variation diminishing kernel.

In the present article, we extend the notion of vacuum static equations on almost coKähler manifolds and rename them as nearly vacuum static equations. It is shown that if an (eta )-Einstein almost coKähler manifold admits a non-trivial solution of a nearly vacuum static equation, then the solution must be a constant. In ((kappa ,mu ))-almost coKähler manifolds, the non-trivial solutions of nearly vacuum static equations do not exist. We also apply nearly vacuum static equations on perfect fluid spacetimes as well as generalized Robertson–Walker spacetimes. Among others, it is shown that a perfect fluid spacetime admitting nearly vacuum static equations is of constant scalar curvature and a generalized Robertson–Walker spacetime obeying nearly vacuum static equations represents a dark matter era.

We obtain estimates of Neumann eigenvalues of the divergence form elliptic operators in Sobolev extension domains. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. The connection between Neumann eigenvalues of elliptic operators and the smallest-circle problem (initially suggested by J. J. Sylvester in 1857) is given.

Semi-Riemannian manifolds that satisfy (homogeneous) linear differential conditions of arbitrary order r on the curvature are analyzed. They include, in particular, the spaces with (r th-order) recurrent curvature, (r th-order) symmetric spaces, as well as entire new families of semi-Riemannian manifolds rarely, or never, considered before in the literature—such as the spaces whose derivative of the Riemann tensor field is recurrent, among many others. Definite proof that all types of such spaces do exist is provided by exhibiting explicit examples of all possibilities in all signatures, except in the Riemannian case with a positive definite metric. Several techniques of independent interest are collected and presented. Of special relevance is the case of Lorentzian manifolds, due to its connection to the physics of the gravitational field. This connection is discussed with particular emphasis on Gauss–Bonnet gravity and in relation with Penrose limits. Many new lines of research open up and a handful of conjectures, based on the results found hitherto, is put forward.