On applications of the Taylor formula in some quasispaces

We consider some metric spaces with quasimetric (quasispaces) comprising uniformly regular (equiregular) Carnot — Carathéodory quasispaces whose quasimetric is induced by C ^ϒ−1-smooth vector fields of formal degree not higher than ϒ. For these spaces, some analogues of the Campbell — Hausdorff formula are derived, which allows us to prove a theorem on a nilpotent tangent cone, a theorem on isomorphism of various nilpotent tangent cones defined at a common point, and a local approximation theorem.     

Existence of T-solution for degenerated problem via Minty’s lemma

We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div（a（x, u,△↓u）） = F in Ω, where Ω is a bounded domain of R^N, N≥2, a ：Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′（Ω, w^＊）. The existence result is proved by using the L^1-version of Minty＇s lemma.     

Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Carathéodory metrics, the singular Carathéodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂⁿ are studied. Translated fromMathematicheskie Zametski, Vol. 67, No. 2, pp. 230–240, February, 2000.     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

A New Approach to Investigation of Carnot�CCarath��odory Geometry

We develop a new approach to studying the geometry of Carnot–Carathéodory spaces under minimal assumptions on the smoothness of basis vector fields. We obtain quantitative comparison estimates for the local geometries of two different local Carnot groups, as well as of a local Carnot group and the original space. As corollaries, we deduce some results that are well-known and basic for the “smooth” case: the generalized triangle inequality for d _∞, the local approximation theorem for the quasimetric d _∞, the Rashevskiǐ–Chow theorem, the ball-box theorem, and so on.     

A multiplicity theorem for p-superlinear p-Laplacian equations using critical groups and morse theory

We study a nonlinear elliptic equation driven by the Dirichlet p-Laplacian and with a Carathéodory nonlinearity. We assume that the nonlinearity exhibits a p-superlinear growth near infinity but need not satisfy the Ambrosetti–Rabinowitz condition. Using truncation techniques, minimax methods and Morse theory, we show that the problem admits at least three nontrivial solutions, two of which have constant sign (one positive, the other negative).     

Carathéodory - Fejér Problem for Pairs of Meromorphic Functions

We continue our study of families of pairs of matrix-valued meromorphic functions P(ρ,P) depending on two parameters p and P introduced in [2]. These include as special cases the projective Schur, Nevanlinna and Carathéodory classes. A two sided Carathéodory Fejér interpolation problem is defined and solved in P(ρ,P), using the fundamental matrix inequality method. A corresponding Schur algorithm is studied. Finally we also consider the case of functions (as opposed to pairs).     

Nonlinear degenerate parabolic equations for Baouendi–Grushin operators

In this paper, we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equations: and Here, Ω is a Carnot–Carathéodory metric ball in R ^N and V ∈ L ¹_loc(Ω). The critical exponents m * and p * are found, and the nonexistence results are proved for m * ≤ m < 1 and p * ≤ p < 2. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The game for the speed of convergence in repeated games of incomplete information

We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v _∞ (p) for all 0≤p≤1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v _∞ (p) (and will converge from above to v _∞ (p) if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payoffs-to the value v _∞ (p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG _∞ (p), and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v _n (p)− v _∞ (p) is of the order of magnitude of . In that case the value of SG _∞ (p) is of the order of magnitude of . We then show a class of games for which the value does not exist. Given any infinite martingale 𝔛^∞={X _k }^∞ _k=1, one defines for each n : V _n (𝔛^∞) ≔E∑ⁿ _k=1 |X _k+1 − X _k|. For our first result we prove that for a uniformly bounded, infinite martingale 𝔛^∞, V _n (𝔛^∞) can be of the order of magnitude of n ^1/2−ε, for arbitrarily small ε>0. Received January 1999/Final version April 2002     

Carathéodory balls and norm balls of the domainH={(z1,z2)∈C2∶|z1|+|z2|<1}

LetH be the domain inC ² defined byH={Z=(z ₁,z ₂):║Z║₁=│z║₁│+│z║₂│<1}. LetC _H(z,w) be the Carathéodory distance ofH,z,w∈H. The Carathéodory ballB _C(z_C,α;H) with centerz _C,z_C∈H, and radius α, 0<α<1, is defined byB _c(z_C,α;H)={z∶C_H(z,z_C)<arc tanh α}. The norm ballB _N(z_N,r) with centerz _N,z_N∈H, and radiusr, 0<r<1-‖z _N‖₁, is defined byB _N(z_N,r)={z∶ ‖z−z_N‖₁<r}. Theorem:The only Carathéodory balls of H which are also norm balls are those with their center at the origin.     

Pseudoconvexity and Gromov hyperbolicity

We give an estimate for the distance functions related to the Bergman, Carathéodory, and Kobayashi metrics on a bounded strictly pseudoconvex domain with C²-smooth boundary. Our formula relates the distance function on the domain with the Carnot- Carathéodory metric on the boundary. As a corollary we conclude that the domain equipped with the any of the standard invariant distances is hyperbolic in the sense of Gromov. When the boundary of the domain is C³-smooth, our estimate is exact up to a fixed additive term.     

On p-superlinear equations with a nonhomogeneous differential operator

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator and with a (p?1)-superlinear Carathéodory reaction. Our formulation incorporates as a special case equations monitored by the p-Laplacian. Using variational methods coupled with truncation techniques and comparison principles, we show that the problem has at least five nontrivial smooth solutions.     

Positive solutions for nonlinear periodic problems with concave terms

We consider a nonlinear periodic problem, driven by the scalar p-Laplacian, with a parametric concave term and a Carathéodory perturbation whose potential (primitive) exhibits a p-superlinear growth near +∞, without satisfying the usual in such cases Ambrosetti-Rabinowitz condition. Using critical point theory and truncation techniques, we prove a bifurcation-type theorem describing the nonexistence, existence and multiplicity of positive solutions as the parameter varies.     

On the Schröder-Bernstein problem for carathéodory vector lattices

In this note we prove that there exists a Carathéodory vector lattice V such that V ≌ V ³ and V ?V ². This yields that V is a solution of the Schröder-Bernstein problem for Carathéodory vector lattices. We also show that no Carathéodory Banach lattice is a solution of the Schröder-Bernstein problem.     

Simultaneous Packing and Covering in Sequence Spaces

We adapt a construction of Klee (1981) to find a packing of unit balls in ℓ _p (1≤p<∞) which is efficient in the sense that enlarging the radius of each ball to any R>2^1−1/p covers the whole space. We show that the value 2^1−1/p is optimal. Dedicated to the memory of Victor Klee.     

Singular integrals associated to the Laplacian on the affine groupax+b

We consider singular integral operators of the form (a)Z ₁L⁻¹Z₂, (b)Z ₁Z₂L⁻¹, and (c)L ⁻¹Z₁Z₂, whereZ ₁ andZ ₂ are nonzero right-invariant vector fields, andL is theL ²-closure of a canonical Laplacian. The operators (a) are shown to be bounded onL ^p for allp∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for anyp∈[1, ∞). Research supported by the Australian Research Council. Research carried out as a National Research Fellow.     

Strong Solvability of a Unilateral Boundary Value Problem for Nonlinear Parabolic Operators

Strong solvability in Sobolev spaces is proved for a unilateral boundary value problem for nonlinear parabolic operators. The operator is assumed to be of Carathéodory type and to satisfy a suitable ellipticity condition; only measurability with respect to the independent variable X is required. The main tools of the proof are an estimate for the second derivatives of functions which satisfy the unilateral boundary conditions and the monotonicity of the operator − u _t with respect to Δu for the same functions.     

Spaces of analytic functions of Hardy-Bloch type

For 0<p≤∞ and 0<q≤∞, the space of Hardy-Bloch type ℬ(p,q) consists of those functionsf which are analytic in the unit diskD such that (1−r)M _p (r,f′)⊂L ^q (dr/(1−r)). We note that ℬ(∞,∞) coincides with the Bloch space ℬ and that ℬ⊂ℬ(p,∞) for allp. Also, the space ℬ(p,p) is the Dirichlet spaceD _p−1 ^p . We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about the mean growth of the ℬ(p,q). In particular, we prove that iff is an analytic function inD and 2≤p<∞, then the conditionM _p (r,f′)=O((1−r)⁻¹), asr→1, implies that

On the Constants in the Meyer Inequality

 We study the growth of the constants in the Meyer inequality as p → 1 and p → ∞. Both constants grow, within constant factors, like (p − 1)⁻¹ and like p respectively. Received August 10, 2001; in revised form February 5, 2002