The integrability of conjugate functions

For any functionf of L(0, 2), we prove that there is a function L(0, 2) such that ¦(x)¦ = ¦f(x)¦ almost everywhere and L(0, 2), where is the conjugate of.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 461–465, October, 1968.     

Darboux-Jouanolou-Ghys integrability for one-dimensional foliations on toric varieties

We use the existence of homogeneous coordinates for simplicial toric varieties to prove a result analogous to the Darboux-Jouanolou-Ghys integrability theorem for the existence of rational first integrals for one-dimensional foliations.     

On the integrability of radial basis functions

In this paper we investigate the integrability of certain radial basis functions. From the following forms of function σ, $$\varphi \left( r \right) = \left\{ \begin{gathered} \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} + g(r) } r > A, \hfill \\ \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} \ln r + g(r), } r > A. \hfill \\ \end{gathered} \right.$$ where A≧0 and $g \circ || \circ || \in L^1 \left( {R^d } \right)$ , we construct the function $$\psi (t) = \sum\limits_{j \in J} {a_j \varphi \left( {||t - t_j ||} \right),} $$ where J is a finite index set, $\left\{ {a_j } \right\}_{j \in J} \subseteq R$ and $\left\{ {t_j } \right\}_{j \in J} \subseteq R^d $ . We show that if $\hat \psi $ is continuous at the origin, the ψ is integrable in R^d.     

Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

We first prove a quantitative estimate of the volume of the sublevel sets of a plurisubharmonic function in a hyperconvex domain with boundary values 0 (in a quite general sense) in terms of its Monge–Ampère mass in the domain. Then we deduce a sharp sufficient condition on the Monge–Ampère mass of such a plurisubharmonic function φ for exp(−2φ) to be globally integrable as well as locally integrable.     

A higher integrability result for nondivergence elliptic equations

The aim of this paper is to establish a higher integrability result of the second derivatives of solutions to nondivergence elliptic equations of the type . We assume that the coefficients a _ij are bounded and have small BMO-norm.      

Global integrability of p-superharmonic functions on metric spaces

We consider a metric space equipped with a doubling measure and a length metric. We prove that p-superharmonic functions are integrable with a small exponent on Hölder domains of the space.     

On Birkhoff integrability for scalar functions and vector measures

The Bartle–Dunford–Schwartz integral for scalar functions with respect to vector measures is characterized by means of Riemann-type sums based on partitions of the domain into countably many measurable sets. In this setting, two natural notions of integrability (Birkhoff integrability and Kolmogoroff integrability) turn out to be equivalent to Bartle–Dunford–Schwartz integrability. A. Fernández, F. Mayoral and F. Naranjo were supported by MEC and FEDER (project MTM2006–11690–C02–02) and La Junta de Andalucía. J. Rodríguez was supported by MEC and FEDER (project MTM2005-08379), Fundación Séneca (project 00690/PI/04) and the Juan de la Cierva Programme (MEC and FSE).     

On Birkhoff integrability for scalar functions and vector measures

The Bartle–Dunford–Schwartz integral for scalar functions with respect to vector measures is characterized by means of Riemann-type sums based on partitions of the domain into countably many measurable sets. In this setting, two natural notions of integrability (Birkhoff integrability and Kolmogoroff integrability) turn out to be equivalent to Bartle–Dunford–Schwartz integrability.     

A lemma on the higher integrability of functions with applications to the regularity theory of two-dimensional generalized Newtonian fluids

In this paper we prove a lemma on the higher integrability of functions and discuss its applications to the regularity theory of two-dimensional generalized Newtonian fluids.Mathematics Subject Classification (2000): 76M30, 49N60, 35J50, 35Q30Acknowledgement Part of this work was done when X.Z. visited the Department of Mathematics, Saarland University, Germany, in June of 2003. He would like to thank the institute for the hospitality. X.Z. was partially supported by the Academy of Finland, project 207288.     

Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

We study integrable geodesic flows on Stiefel varieties V _n,r ?=?SO(n)/SO(n?r) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on V _n,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T ^* V _n,r )/SO(r). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian G _n,r and on a sphere S ^n?1 in presence of Yang–Mills fields or a magnetic monopole field. Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety W _n,r ?=?U(n)/U(n?r), the matrix analogs of the double and coupled Neumann systems.     

A regularity result for polyharmonic maps with higher integrability

We prove a regularity result for critical points of the polyharmonic energy in with and p > 1. Our proof is based on a Gagliardo–Nirenberg-type estimate and avoids the moving frame technique. In view of the monotonicity formulae for stationary harmonic and biharmonic maps, we infer partial regularity in theses cases.     

Local higher integrability for parabolic quasiminimizers in metric spaces

Using purely variational methods, we prove in metric measure spaces local higher integrability for minimal p-weak upper gradients of parabolic quasiminimizers related to the heat equation. We assume the measure to be doubling and the underlying space to be such that a weak Poincaré inequality is supported. We define parabolic quasiminimizers in the general metric measure space context, and prove an energy type estimate. Using the energy estimate and properties of the underlying metric measure space, we prove a reverse Hölder inequality type estimate for minimal $p$ -weak upper gradients of parabolic quasiminimizers. Local higher integrability is then established based on the reverse Hölder inequality, by using a modification of Gehring’s lemma.     

Global higher integrability for parabolic quasiminimizers in nonsmooth domains

We study the global higher integrability of the gradient of a parabolic quasiminimizer with quadratic growth conditions. We show that if the lateral boundary satisfies a capacity density condition and if boundary and initial values are smooth enough, then quasiminimizers globally belong to a higher Sobolev space than assumed a priori. We derive estimates near the lateral and the initial boundaries.     

On global integrability of BMO functions on general domains

Extending results of Staples and Smith-Stegenga, we characterize measurable subsets of a given domainD ⊃R ⁿ on which BMO(D) functions areL ^p integrable or exponentially integrable. In particular, we characterize uniform domains by the integrability of BMO functions. We also remark on the boundedness of domains satisfying a certain integrability condition for the quasihyperbolic metric.     

Stability and higher integrability of derivatives of solutions in double obstacle problems

Higher integrability of the derivatives of solutions to double obstacle problems associated with the second-order quasilinear elliptic differential equation ∇·A(x,∇u)=0 is obtained under natural assumptions on obstacles. This result is used to prove a stability result for solutions to double obstacle problems for varying equations.     

Global higher integrability of solutions to subelliptic double obstacle problems

In this paper we consider the double obstacle problems associated with nonlinear subelliptic equation \[X^*A(x,u,Xu)+ B(x,u,Xu)=0, \ \ x\in\Omega,\] where $X=(X_1,\ldots,X_m)$ is a system of smooth vector fields defined in $\mathbb{R}^n$ satisfying H\"{o}rmander"s condition. The global higher integrability for the gradients of the solutions is obtained under a capacitary assumption on the complement of the domain $\Omega$.     

Global higher integrability for the parabolic equations in Reifenberg domains

In this paper we generalize global L^p‐type gradient estimates to Orlicz spaces for weak solutions of the parabolic equations with small BMO coefficients in Reifenberg flat domains (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)