Sampling Theorems Associated with Singular q-Sturm Liouville Problems

In this paper we introduce three sampling theorems for transformations defined in terms of Jackson q-integration when the kernel of the transformation is a solution or the Green’s function of singular q-Sturm–Liouville problems. We consider the problem when the q-Sturm–Liouville problem is singular either at infinity or at zero with detailed investigations when the singular point is infinity. This approach allows the derivation of sampling representations for transforms whose kernels are linear combinations of q-Bessel functions, not just a single one as previously established.     

Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems

We show that entire positive solutions exist for the semilinear elliptic system Δu = p(x)v^α, Δv = q(x)u^β on R^N, N ≥ 3, for positive α and β, provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity. We also show that entire solutions fail to exist if the functions p and q are of slow decay.     

Basis Properties of the System of Root Functions of the Sturm–Liouville Operator with Degenerate Boundary Conditions: I

The spectral problem for the Sturm–Liouville operator with an arbitrary complex-valued potential q(x) of the class L¹(0, π) and degenerate boundary conditions is considered. We prove that the system of root functions of this operator is not a basis in the space L²(0, π).     

On the fractional parts of the natural powers of a fixed number

Let ξ ≠ = 0 and α > 1 be reals. We prove that the fractional parts {ξ αⁿ}, n = 1, 2, 3, ..., take every value only finitely many times except for the case when α is the root of an integer: α = q ^1/d, where q ≥ 2 and d ≥ 1 are integers and ξ is a rational factor of a nonnegative integer power of α.     

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u ^q, q > 1, on spherically symmetric noncompact Riemannian manifolds.     

Monotone solutions of the parametric linear complementarity problem

The parametric linear complementarity problem under study here is given by the conditionsq + αp + Mz ≥ 0,α ≥ 0,z ≥ 0,z ^T (q + αp + Mz) = 0 whereq is nonnegative. This paper answers three questions including the following one raised by G. Maier: What are the necessary and sufficient conditions onM guaranteeing that for every nonnegative starting pointq and every directionp the components of the solution to the parametric linear complementarity problem are nondecreasing functions of the parameterα?     

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

We define nonnegative quasi-nearly subharmonic functions on so called locally uniformly homogeneous spaces. We point out that this function class is rather general. It includes quasi-nearly subharmonic (thus also subharmonic, quasisubharmonic and nearly subharmonic) functions on domains of Euclidean spaces \mathbbRⁿ{{\mathbb{R}}^n}, n ≥ 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \mathbbRⁿ{{\mathbb{R}}^n}, n ≥ 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball B ⁿ of \mathbbRⁿ{{\mathbb{R}}^n}, the M{{\mathcal{M}}}-invariant measure on the unit ball B ²ⁿ of \mathbbCⁿ{{\mathbb{C}}^n}, n ≥ 1, and the quasihyperbolic measure on any domain D ì \mathbbRⁿ{D\subset {\mathbb{R}}^n}, D 1 \mathbbRⁿ{D\ne {\mathbb{R}}^n}. Moreover, we show that if u is a quasi-nearly subharmonic function on a locally uniformly homogeneous space and the space satisfies a mild additional condition, then also u ^p is quasi-nearly subharmonic for all p > 0.     

Criteria of solvability for multidimensional Riccati equations

We study the solvability problem for the multidimensional Riccati equation ??u=|?u|^q+ω, whereq>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation ?Δu?ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions onR ⁿ in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type?Lu=f(x, u, ?u)+ω where , andL is a uniformly elliptic operator.     

Large solutions of mixed sublinear/superlinear elliptic equations

We consider the equation Δu=p(x)uα+q(x)uβ on RN (N?3) where p, q are nonnegative continuous functions and 0<α?β. We establish conditions sufficient to ensure the existence and nonexistence of nonnegative entire large solutions of the equation.     

Existence of positive solutions for some nonlinear parabolic systems in exteriors domains

We prove some existence results of positive continuous solutions to the semilinear parabolic system , in an unbounded domain D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f, g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the parabolic Kato class J^∞(D).     

Plurisubharmonic functions in calibrated geometry and q-convexity

Let (M, ω) be a Kähler manifold. An integrable function ${\varphi}Let (M, ω) be a K?hler manifold. An integrable function j{\varphi} on M is called ω ^q -plurisubharmonic if the current dd^cjùw^q-1{dd^c\varphi\wedge \omega^{q-1}} is positive. We prove that j{\varphi} is ω ^q -plurisubharmonic if and only if j{\varphi} is subharmonic on all q-dimensional complex subvarieties. We prove that a ω ^q -plurisubharmonic function is q-convex, and admits a local approximation by smooth, ω ^q -plurisubharmonic functions. For any closed subvariety Z ì M{Z\subset M} , dim_\mathbbC Z ￡ q-1{\dim_\mathbb{C} Z\leq q-1} , there exists a strictly ω ^q -plurisubharmonic function in a neighbourhood of Z (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony’s lemma on integrability of positive closed (p, p)-forms which are integrable outside of a complex subvariety of codimension ≥  p + 1.     

The Liouville Field Theory Zero-Mode Problem

We quantize the canonical free-field zero modes p, q on the half-plane p > 0 for both Liouville field theory and its reduced Liouville particle dynamics. We describe the particle dynamics in detail, calculate one-point functions of particle vertex operators, deduce their zero-mode realization on the half-plane, and prove that the particle vertex operators act self-adjointly on the Hilbert space L ²(₊) because of symmetries generated by the S-matrix. Similarly, we obtain the self-adjointness of the corresponding Liouville field theory vertex operator in the zero-mode sector by applying the Liouville reflection amplitude, which is derived by the operator method.     

Asymptotic formulae for eigenvalues and eigenfunctions of q‐Sturm‐Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q‐Sturm‐Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q‐trigonometric functions as well as fundamental sets of solutions of the associated second order q‐difference equation. As in classical Sturm‐Liouville theory, the eigenvalues behave like zeros of q‐trigonometric functions and the eigenfunctions behave like q‐trigonometric functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

Classes of Quasi-nearly Subharmonic Functions

It is well known and important that if u ≥ 0 is subharmonic on a domain Ω in ℝ ⁿ and p > 0, then there is a constant C(n,p) ≥ 1 such that for each open ball B(x,r) ⊂ Ω. The definition of a relatively new function class, quasi-nearly subharmonic functions, is based on such a generalized mean value inequality. It is pointed out that the obtained function class is natural. It has important and interesting properties and, at the same time, it is large: In addition to nonnegative subharmonic functions, it includes, among others, Hervé’s nearly subharmonic functions, functions satisfying certain natural growth conditions, especially certain eigenfunctions, polyharmonic functions and generalizations of convex functions. Further, some of the basic properties of quasi-nearly subharmonic functions are stated in a unified form. Moreover, a characterization of quasi-nearly subharmonic functions with the aid of the quasihyperbolic metric and two weighted boundary limit results are given.      

Positivity of the generalized translation associated with the q-Hankel transform and applications

A new formulation of the Graf-type addition formula related to the third Jackson q-Bessel function gives a solution for the problem of the positivity of the generalized q-translation operator associated with the q-Hankel transform. Next some applications in q-theory are treated, for instance the relationship between the q-Bessel- positive and negative-definite functions. We also show how the positivity of the q-Bessel translation operator plays a central role in q-Fourier analysis, namely in the study of Markov operators in the q-context. The paper concludes with the nonnegative product linearization of the q^?2-Lommel polynomials.     

Existence of positive bounded solutions for some nonlinear elliptic systems

We prove some existence results of positive bounded continuous solutions to the semilinear elliptic system Δu=λp(x)g(v), Δv=μq(x)f(u) in domains D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f,g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the Kato class K(D).     

A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems

We prove that the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) where 0<α?1, 0<β?1, and p and q are nonnegative continuous functions has a nonnegative entire radial solution satisfying lim|x|→∞u(x)=lim|x|→∞v(x)=∞ if and only if the functions p and q satisfy

     

q-Taylor and interpolation series for Jackson q-difference operators

We prove q-Taylor series for Jackson q-difference operators. Absolute and uniform convergence to the original function are proved for analytic functions. We derive interpolation results for entire functions of q-exponential growth which is less than lnq⁻¹, 0<q<1, from its values at the nodes , a is a non-zero complex number with absolute and uniform convergence criteria.     

Gradient Estimates of q-Harmonic Functions of Fractional Schrödinger Operator

We study gradient estimates of q-harmonic functions u of the fractional Schrödinger operator Δ ^α/2?+?q, α?∈?(0, 1] in bounded domains D???? ^d . For nonnegative u we show that if q is Hölder continuous of order η?>?1???α then $\nabla u(x)$ exists for any x?∈?D and $|\nabla u(x)| \le c u(x)/ ({\rm dist}(x,\partial D) \wedge 1)$ . The exponent 1???α is critical i.e. when q is only 1???α Hölder continuous $\nabla u(x)$ may not exist. The above gradient estimates are well known for α?∈?(1, 2] under the assumption that q belongs to the Kato class $\mathcal{J}^{\alpha - 1}$ . The case α?∈?(0, 1] is different. To obtain results for α?∈?(0, 1] we use probabilistic methods. As a corollary, we obtain for α?∈?(0, 1) that a weak solution of Δ ^α/2 u?+?q u?=?0 is in fact a strong solution.