Abstract: Abstract For maps of one complex variable, f, given as the sum of a degree n power map and a degree d polynomial, we provide necessary and sufficient conditions that the geometric limit as n approaches infinity of the set of points that remain bounded under iteration by f is the closed unit disk or the unit circle. We also provide a general description, for many cases, of the limiting set. PubDate: 2020-08-07

Abstract: In this paper, we shall obtain some inequalities for the polar derivative of polynomial having all zeros in \( z \le k, k \ge 1.\) Our results sharpen some well-known results of Turán, Dubinin and others. PubDate: 2020-06-29

Abstract: Although the Bergman projection operator \({\mathbf{B}}_{\Omega }\) is defined on \(L^2(\Omega )\), its behavior on other \(L^p(\Omega )\) spaces for \(p\not =2\) is an active research area. We survey some of the recent results on \(L^p\) estimates on the Bergman projection. PubDate: 2020-06-06

Abstract: We present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions to sums of squares of vector fields satisfying the Hörmander condition. PubDate: 2020-06-03

Abstract: We investigate regularity of CR-mappings between real-analytic infinite type hypersurfaces in \(\mathbb {C}^{2}\). We show that, under the Fuchsian type condition, all (respectively formal or smooth) CR-diffeomorphisms between them are automatically analytic. The Fuchsian condition appears to be in a certain sense optimal for the regularity problem. PubDate: 2020-05-28

Abstract: Here we shall discuss analyticity results for several important partial differential equations. This includes the analytic regularity of sub-Laplacians under the finite type condition; the analyticity of the solution in both variables to the Cauchy problem for the Camassa–Holm equation with analytic initial data by using the Ovsyannikov theorem, which is a Cauchy–Kowalevski type theorem for nonlocal equations; the Cauchy problem for BBM with analytic initial data; the Cauchy problem for KdV with analytic initial data examining the evolution of uniform radius of spatial analyticity; and finally the time regularity of KdV solutions, which is Gevrey 3. PubDate: 2020-05-25

Abstract: We prove that the tangential Cauchy–Riemann operator \({\bar{\partial }}_b\) has closed range on Levi-pseudoconvex CR manifolds that are embedded in a q-convex complex manifold X. Our result generalizes the known case when X is a Stein manifold (in particular, when \(X={\mathbb C}^n\)). PubDate: 2020-05-25

Abstract: This note reviews briefly the classes of involutive systems \({\varvec{L}} =\left( L_{1},\ldots ,L_{\nu }\right)\) of analytic vector fields for which necessary and sufficient conditions for the local solvability, or local exactness in a given degree of the associated differential complex, are known. We point out that these conditions, at the microlocal level, can be reformulated in terms of a locally exact \(\left( 1,0\right)\)-form on the \(complex\) characteristic bundle of \({\varvec{L}}\) and of the primitive of its imaginary part. The results in the known cases raise natural questions about their generalization. PubDate: 2020-05-25

Abstract: This paper is the second of a multi-part series in which we explore geometric and analytic properties of the Kohn–Laplacian and its inverse on general quadric submanifolds of \({\mathbb{C}}^n\times {\mathbb{C}}^m\). We have two goals in this paper. The first is to give useable sufficient conditions for a map T between quadrics to be a Lie group isomorphism that preserves \(\Box _b\), and the second is to establish a framework for which appropriate derivatives of the complex Green operator are continuous in \(L^p\) and \(L^p\)-Sobolev spaces (and hence are hypoelliptic). We apply the general results to codimension two quadrics in \({\mathbb{C}}^4\). PubDate: 2020-05-14

Abstract: The three-dimensional case of the structure equations of Webster are summarized and then used to determine the sub-Riemannian geodesics of the left-invariant CR structures on the group \(SU(2,{\mathbf{R}}) =S^3\). These geodesics are studied in detail for the standard left-invariant CR structure. PubDate: 2020-04-27

Abstract: We compare various definitions of nondegeneracy of the Levi map for real submanifolds of higher codimension in \({\mathbb {C}}^N\) and discuss the generalization to higher codimension of the 2-jet determination for biholomorphisms in the hypersurface case proved by Chern and Moser (Acta Math 133:219–271, 1974). PubDate: 2020-04-24

Abstract: Given a second-order partial differential operator \(P = P^\mathrm{o} + X + a\) on an open set \(\Omega \) in \({\mathbb {R}}^n\), where \(P^\mathrm{o}\) is the principal part and X is a real vector field, with non-negative real characteristic form, we study the \(s'\)-Gevrey regularity on an open subset \(\omega \) of \(\Omega \), of s-Gevrey vectors of P on \(\omega \). For that we associate to any subset \(A \subset \Omega \) an integer (finite or \(+\infty \)) named the type of A with respect to \(P^\mathrm{o}\) and denoted \(\tau (A; P^\mathrm{o}\)) (see in the next sections, precise definitions, facts and remarks about it). Denoting the space of s-Gevrey vectors of P in \(\omega \) by \(G^s(\omega ,P)\), we prove that \(G^{s}(\omega ; P) \subset G^{s'} (\omega )\), with \(s' = \tau (\omega ; P)\cdot s\) under the assumption that the coefficients of P are in \(G^{s} (\omega )\). Moreover, \(s'\) is optimal. PubDate: 2020-04-16

Abstract: In the present paper, we shall provide new examples of Stein manifolds enjoying the (algebraic) volume density property and compute their homology groups. PubDate: 2020-04-11

Abstract: We explore some links between the holomorphic extendability of CR functions on a hypersurface and the validity of the strong maximum principle for continuous CR functions. PubDate: 2020-04-06

Abstract: In this paper, we establish the Qi-Keng Lu uniformization theorem for Stein spaces with possibly isolated normal singularities. PubDate: 2020-04-06

Abstract: In this paper, by using the \(G_{2(2)}\)-structure on \(\hbox {Im}(\mathbf{Ca'})\cong {\mathbb {R}}^{3,4}\) of the purely imaginary Cayley’s split-octaves \(\mathbf{Ca'}\), the \(G_{2(2)}\)-bi-normal motion of curves \(\gamma _t(s)\) in the pseudo-Euclidean space \({\mathbb {R}}^{3,4}\) is studied. The motion is closely related to Schrödinger flows into homogeneous pseudo-Riemannian manifolds \({\mathbb {S}}^{2,4}(1)=G_{2(2)}/SU(1,2)\) of signature (2, 4) and \({\mathbb {S}}^{3,3}(\sqrt{-1})=G_{2(2)}/SL(3,{\mathbb {R}})\) of signature (3, 3), in which \({\mathbb {S}}^{2,4}(1)\) (resp. \({\mathbb {S}}^{3,3}(\sqrt{-1}))\) admits the almost complex (resp. para-complex) structure. Furthermore, the motion of spacelike (resp. timelike) curves is also shown to be equivalent to the nonlinear Schrödinger-like system (resp. the nonlinear coupled heat equations) in three unknown functions, which generalizes the correspondence between the bi-normal motion of timelike (resp. spacelike) curves in \({\mathbb {R}}^{2,1}\) and the defocusing nonlinear Schrödinger equation (resp. the nonlinear heat equation). To show this correspondence, \(G_{2(2)}\)-frame field on the curve is used. PubDate: 2020-03-12

Abstract: We develop a link between degree estimates for rational sphere maps and compressed sensing. We provide several new ideas and many examples, both old and new, that amplify connections with linear programming. We close with a list of ten open problems. PubDate: 2020-02-11

Abstract: We solve the Levi-flat Plateau problem in the following case. Let \(M \subset {\mathbb {C}}^{n+1}\), \(n \ge 2\), be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose M is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in \({\mathbb {C}}^n \times {\mathbb {R}}\) with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary M. We also study boundary regularity of CR automorphisms of domains in \({\mathbb {C}}^n \times {\mathbb {R}}\). PubDate: 2020-01-13