The class of mappings with bounded specific oscillation,and integrability of mappings with bounded distortion on Carnot groups

This paper is the first of the author’s three articles on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\).In the present article we study integrability of mappings with bounded specific oscillation on spaces of homogeneous type. As an example, we consider mappings with bounded distortion on the Heisenberg group. We prove that a mapping with bounded distortion belongs to the Sobolev class W _p,loc ¹ , where p → ∞ as the distortion coefficient tends to 1.     

Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups

This article completes the authors’s series on stability in the Liouville theorem on the Heisenberg group. We show that every mapping with bounded distortion on a John domain of the Heisenberg group is approximated by a conformal mapping with order of closeness √K ? 1 in the uniform norm and with order of closeness K ? 1 in the Sobolev L _p ¹ -norm for all p < C/K?1. We construct two examples, demonstrating the asymptotic sharpness of our results.     

On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let \(\mathbb{S}\) be a cone in ? ⁿ . A bounded linear operator T: L _p (? ⁿ ) → L _p (? ⁿ ) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s ε W ? \(\mathbb{S}\)) ? (Tx) (s) = 0 (s ε W ? \(\mathbb{S}\)) is valid for any x ε L _p (? ⁿ ) and any open subset W\(\subseteq\) ? ⁿ . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator

$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$

in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the a _n are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).

     

The Calderón problem for Hilbert couples

We prove that intermediate Banach spaces\(\mathcal{A}\) and\(\mathcal{B}\) with respect to arbitrary Hilbert couples\(\bar {H}\) and\(\bar {K}\) are exact interpolation if and onlyif they are exactK-monotonic, i.e. the condition\(f^0 \in \mathcal{A}\) and the inequality\(K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0\), implyg⁰∈B and ‖g⁰‖B≤‖f⁰‖ _A (K is Peetre’sK-functional). It is well known that this property is implied by the following: for each ?>1 there exists an operator\(T:\bar {H} \to \bar {K}\) such thatTf⁰=g⁰, and\(K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0\). Verifying the latter property, it suffices to consider the “diagonal case” where\(\bar {H} = \bar {K}\) is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ?=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.     

Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

Let \(L=-\mathrm{div}(A\nabla )\) be a second order divergence form elliptic operator and A an accretive \(n\times n\) matrix with bounded measurable complex coefficients in \({\mathbb R}^n\). Let \(\nabla b\in L^n({\mathbb R}^n)\,(n>2)\). In this paper, we prove that the commutator generated by b and the square root of L, which is defined by \([b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)\), is bounded from the homogenous Sobolev space \({\dot{L}}_1^2({\mathbb R}^n)\) to \(L^2({\mathbb R}^n)\).     

Real-Linear Isometries on Spaces of Functions of Bounded Variation

Let X and Y be subsets of the real line with at least two points. We study the surjective real-linear isometries \({T:BV(X)\longrightarrow BV(Y)}\) between the spaces of functions of bounded variation on X and Y with respect to the supremum norm \({\|\cdot\|_\infty}\) and the complete norm \({\|\cdot\|:=\max(\|\cdot\|_\infty,\mathcal{V}(\cdot))}\), where \({\mathcal{V}(\cdot)}\) denotes the total variation of a function. Additively norm preserving maps between these spaces are also characterized as a corollary.     

On Filtered Polynomial Approximation on the Sphere

This paper considers filtered polynomial approximations on the unit sphere \(\mathbb {S}^d\subset \mathbb {R}^{d+1}\), obtained by truncating smoothly the Fourier series of an integrable function f with the help of a “filter” h, which is a real-valued continuous function on \([0,\infty )\) such that \(h(t)=1\) for \(t\in [0,1]\) and \(h(t)=0\) for \(t\ge 2\). The resulting “filtered polynomial approximation” (a spherical polynomial of degree \(2L-1\)) is then made fully discrete by approximating the inner product integrals by an N-point cubature rule of suitably high polynomial degree of precision, giving an approximation called “filtered hyperinterpolation”. In this paper we require that the filter h and all its derivatives up to \(\lfloor \tfrac{d-1}{2}\rfloor \) are absolutely continuous, while its right and left derivatives of order \(\lfloor \tfrac{d+1}{2}\rfloor \) exist everywhere and are of bounded variation. Under this assumption we show that for a function f in the Sobolev space \(W^s_p(\mathbb {S}^d),\ 1\le p\le \infty \), both approximations are of the optimal order \( L^{-s}\), in the first case for \(s>0\) and in the second fully discrete case for \(s>d/p\), conditions which in both cases cannot be weakened.     

On the absolute Cesàro summability factors of infinite series

The author has established that if {λ _n | is a convex sequence such that the series\(\sum {\frac{{\lambda _n }}{n}} \) is convergent and if Σa _n is bounded [R, logn, 1] with indexk, then\(\sum {a_n \lambda _n } \) is summable |C, 1|k fork>1. The casek=1 of the theorem is due to Pati [3].     

The limit case of a domination property

The domination number γ(G) of a connected graph G of order n is bounded below by(n+2-e(G))/ 3 , where (G) denotes the maximum number of leaves in any spanning tree of G. We show that (n+2-e(G))/ 3 = γ(G) if and only if there exists a tree T ∈ T ( G) ∩ R such that n1(T ) = e(G), where n1(T ) denotes the number of leaves of T1, R denotes the family of all trees in which the distance between any two distinct leaves is congruent to 2 modulo 3, and T (G) denotes the set composed by the spanning trees of G. As a consequence of the study, we show that if (n+2-e(G))/ 3 = γ(G), then there exists a minimum dominating set in G whose induced subgraph is an independent set. Finally, we characterize all unicyclic graphs G for which equality (n+2-e(G))/ 3= γ(G) holds and we show that the length of the unique cycle of any unicyclic graph G with (n+2-e(G))/ 3= γ(G) belongs to {4} ∪ {3 , 6, 9, . . . }.     

Linear extensions of some Baire-one functions

Let X be a Hausdorff topological space, and let \({\mathscr {B}}_1(X)\) denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let \({\mathscr {F}}(A)\) be the set of functions \(A\rightarrow {\mathbb R}\) with a property \({\mathscr {F}}\) making \({\mathscr {F}}(A)\) a linear subspace of \({\mathscr {B}}_1(A)\). We give a sufficient condition for the existence of a linear extension operator \(T_A:{\mathscr {F}}(A)\rightarrow {\mathscr {F}}(X)\), where \({\mathscr {F}}\) means to be piecewise continuous on a sequence of closed and \(G_\delta \) subsets of X and is denoted by \({\mathscr {P}_0}\). We show that \(T_A\) restricted to bounded elements of \({\mathscr {F}}(A)\) endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.     

On rational isomorphisms of Lie algebras

Let \(\mathfrak{n}\) be a finite-dimensional noncommutative nilpotent Lie algebra for which the ring of polynomial invariants of the coadjoint representation is generated by linear functions. Let \(\mathfrak{g}\) be an arbitrary Lie algebra. We consider semidirect sums \(\mathfrak{n} \dashv _\rho \mathfrak{g}\) with respect to an arbitrary representation ρ: \(\mathfrak{g}\) → der \(\mathfrak{n}\) such that the center z \(\mathfrak{n}\) of \(\mathfrak{n}\) has a ρ-invariant complement.We establish that some localization \(\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})\) of the Poisson algebra of polynomials in elements of the Lie algebra \(\mathfrak{n} \dashv _\rho \mathfrak{g}\) is isomorphic to the tensor product of the standard Poisson algebra of a nonzero symplectic space by a localization of the Poisson algebra of the Lie subalgebra \((z\mathfrak{n}) \dashv \mathfrak{g}\). If \([\mathfrak{n},\mathfrak{n}] \subseteq z\mathfrak{n}\), then a similar tensor product decomposition is established for the localized universal enveloping algebra of the Lie algebra \(\mathfrak{n} \dashv _\rho \mathfrak{g}\). For the case in which \(\mathfrak{n}\) is a Heisenberg algebra, we obtain explicit formulas for the embeddings of \(\mathfrak{g}_P \) in \(\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})\). These formulas have applications, some related to integrability in mechanics and others to the Gelfand-Kirillov conjecture.     

Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces \({\mathcal{M}^{p}}\) and Stepanoff spaces \({\mathcal{S}^p}\), 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (\({M^{p}_{0}}\)). We construct a function f that belongs to these spaces and has the property that all approximate identities \({\phi_\varepsilon * f}\) converge to f pointwise but they never converge in norm.     

Single and Double Layer Potentials on Domains with Conical Points I: Straight Cones

Let \({\Omega = \mathbb{R}^+ \omega}\) be an open straight cone in \({\mathbb{R}^n, n\geq3}\) , where \({\omega \subset S^{n-1}}\) is a smooth subdomain of the unit sphere. Denote by K and S the double and single layer potential operators associated to Ω and the Laplace operator Δ. Let r be the distance to the origin. We consider a natural class of dilation invariant operators on ?Ω, called Mellin convolution operators and show that \({K_a :=r^{a}Kr^{-a}}\) and \({S_b := r^{b-\frac{1}{2}}Sr^{-b-\frac{1}{2}}}\) are Mellin convolution operators for \({a \in (-1, n-1)}\) and \({b \in (\frac{1}{2}, n-\frac{3}{2})}\) . It is known that a Mellin convolution operator T is invertible if, and only if, its Mellin transform \({\hat T( \lambda)}\) is invertible for any real λ. We establish a reduction procedure that relates the Mellin transforms of K _a and S _b to the single and, respectively, double layer potential operators associated to some other elliptic operators on ω, which can be shown to be invertible using the classical theory of layer potential operators on smooth domains. This reduction procedure thus allows us to prove that \({\frac{1}{2}\pm K}\) and S are invertible between suitable weighted Sobolev spaces. A classical consequence of the invertibility of these operators is a solvability result in weighted Sobolev spaces for the Dirichlet problem on Ω.     

Measures under the flat norm as ordered normed vector space

The space of real Borel measures \(\mathcal {M}(S)\) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone \(\mathcal {M}_+(S)\) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of \(\mathcal {M}_+(S)\) are compact and semiflows on \(\mathcal {M}_+(S)\) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because \(\mathcal {M}(S)\) is rarely complete and \(\mathcal {M}_+(S)\) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on \(\mathcal {M}_+(S)\) and continuous semiflows. Both topics prepare for a dynamical systems theory on \(\mathcal {M}_+(S)\).     

Canonical Representations and Overgroups for Hyperboloids

For the hyperboloid \(X = G/H\), where G = SO₀(p, q) and H = SO₀(p, q ? 1), we define canonical representations R_λ,ν λ ∈ ?, ν = 0, 1, as the restrictions to G of representations \(\tilde R\lambda ,\nu\), associated with a cone, of the group \(\tilde G = \operatorname{SO} _0 (p + 1,q)\). They act on functions on the direct product Ω of two spheres of dimensions p ? 1 and q ? 1. The manifold Ω contains two copies of \(X\) as open G-orbits. We explicitly describe the interaction of the Lie operators of the group \({\tilde G}\) in \(\tilde R\lambda ,\nu\) with the Poisson and Fourier transforms associated with the canonical representations. These transforms are operators intertwining the representations R_λ,ν with representations of G associated with a cone.     

Kähler geometry of hyperbolic type on the manifold of nondegenerate <Emphasis Type="Italic">m</Emphasis>-pairs

A nondegenerate m-pair (A, Ξ) in an n-dimensional projective space ?P _n consists of an m-plane A and an (n ? m ? 1)-plane Ξ in ?P _n , which do not intersect. The set \(\mathfrak{N}_m^n \) of all nondegenerate m-pairs ?P _n is a 2(n ? m)(n ? m ? 1)-dimensional, real-complex manifold. The manifold \(\mathfrak{N}_m^n \) is the homogeneous space \(\mathfrak{N}_m^n = {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(m + 1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(m + 1,\mathbb{R})}} \times GL(n - m,\mathbb{R})\) equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold \(\mathfrak{N}_m^n \) is a hyperbolic analogue of the complex Grassmanian ?G _m,n = U(n+1)/U(m+1) × U(n?m). In particular, the manifold of 0-pairs \(\mathfrak{N}_m^n {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(1,\mathbb{R})}} \times GL(n,\mathbb{R})\) is a hyperbolic analogue of the complex projective space ?P _n = U(n+1)/U(1) × U(n). Similarly to ?P _n , the manifold \(\mathfrak{N}_m^n \) is a Kähler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense, \(\mathfrak{N}_0^n \) is a hyperbolic spatial form. It was proved in [6] that the manifold of 0-pairs \(\mathfrak{N}_0^n \) is globally symplectomorphic to the total space T*?P _n of the cotangent bundle over the projective space ?P _n . A generalization of this result (see [7]) is as follows: the manifold of nondegenerate m-pairs \(\mathfrak{N}_m^n \) is globally symplectomorphic to the total space T*?G _m,n of the cotangent bundle over the Grassman manifold ?G _m,n of m-dimensional subspaces of the space ?P _n .In this paper, we study the canonical Kähler structure on \(\mathfrak{N}_m^n \). We describe two types of submanifolds in \(\mathfrak{N}_m^n \), which are natural hyperbolic spatial forms holomorphically isometric to manifolds of 0-pairs in ?P _m +1 and in ?P _n?m , respectively. We prove that for any point of the manifold \(\mathfrak{N}_m^n \), there exist a 2(n ? m)-parameter family of 2(m + 1)-dimensional hyperbolic spatial forms of first type and a 2(m + 1)-parameter family of 2(n ? m)-dimensional hyperbolic spatial forms of second type passing through this point. We also prove that natural hyperbolic spatial forms of first type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifold \(\mathfrak{N}_{m + 1}^n \) and natural hyperbolic spatial forms of second type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifolds \(\mathfrak{N}_{m + 1}^n \).     

Expansion over the signum system on a sigma-finite space

A generalization of the notion of expansion over the signum system to the case of a sigma-finite space is considered in the paper. Let {E _k } _k=1 ^∞ be an exhaustion of X, then under the condition \(\sum\limits_{k = 1}^\infty {\tfrac{1}{{\mu E_k }}} = \infty \) the expansion converges to the expanded function in the metric of L ₂, and under the conditions \(\sum\limits_{k = 1}^\infty {\tfrac{1}{{\mu E_k }}} = \infty \) and \(\mathop {\lim }\limits_{k \to \infty } \tfrac{{\mu E_{k - 1} }}{{\mu E_k }} = 1\) the convergence holds almost everywhere for functions from L _∞(X).     

The regularity of conformal target harmonic maps

We establish that any weakly conformal \(W^{1,2}\) map from a Riemann surface S into a closed oriented sub-manifold \(N^n\) of an euclidian space \({\mathbb {R}}^m\) realizes, for almost every sub-domain, a stationary varifold if and only if it is a smooth conformal harmonic map form S into \(N^n\).     

On subordination of some analytic functions

We define V (α, β) (α < 1 and β > 1), the new subclass of analytic functions with bounded positive real part, \(V\left( {\alpha ,\beta } \right): = \left\{ {f \in A:\alpha < \operatorname{Re} \left\{ {{{\left( {\frac{z}{{f\left( z \right)}}} \right)}^2}f'\left( z \right)} \right\} < \beta } \right\}\), and study some properties of V (α, β). We also study the class U (γ) (γ > 0): \(u\left( \gamma \right): = \left\{ {f \in A:\left| {{{\left( {\frac{z}{{f\left( z \right)}}} \right)}^2}f'\left( z \right)} \right| - 1 < \gamma } \right\}\), where A is the class of normalized functions.