Valuations and multiplier ideals

We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollár. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on the realization of ideals as multiplier ideals.

     

Sobolev-type functions with variable integrability exponent on metric measure spaces

We consider various embedding theorems for Sobolev-type function classes with variable integrability exponent on a metric space.     

Function spaces of variable smoothness and integrability

In this article we introduce Triebel-Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Using it we derive molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition we prove optimal trace theorem in the variable indices case.     

Symmetric stable processes in parabola-shaped regions

We identify the critical exponent of integrability of the first exit time of the rotation invariant stable Lévy process from a parabola-shaped region.

     

Scaling exponents of random walks in random sceneries

We compute the exact asymptotic normalizations of random walks in random sceneries, for various null recurrent random walks to the nearest neighbours, and for i.i.d., centered and square integrable random sceneries. In each case, the standard deviation grows like n with . Here, the value of the exponent is determined by the sole geometry of the underlying graph, as opposed to previous examples, where this value reflected mainly the integrability properties of the steps of the walk, or of the scenery. For discrete Bessel processes of dimension d[0;2[, the exponent is . For the simple walk on some specific graphs, whose volume grows like n^d for d[1;2[, the exponent is =1−d/4. We build a null recurrent walk, for which without logarithmic correction. Last, for the simple walk on a critical Galton–Watson tree, conditioned by its nonextinction, the annealed exponent is . In that setting and when the scenery is i.i.d. by levels, the same result holds with .     

Composition operators that improve integrability on weighted Bergman spaces

Composition operators between weighted Bergman spaces with a smaller exponent in the target space are studied. An integrability condition on a generalized Nevanlinna counting function of the inducing map is shown to characterize both compactness and boundedness of such an operator. Composition operators mapping into the Hardy spaces are included by making particular choices for the weights.

     

Higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients

In this paper, based on the theory of variable exponent spaces, we study the higher integrability for a class of nonlinear elliptic equations with variable growth and discontinuous coefficients. Under suitable assumptions, we obtain a local gradient estimate in Orlicz space for weak solution.     

The porous medium equation with measure data

We study the existence of solutions to the porous medium equation with a nonnegative, finite Radon measure on the right-hand side. We show that such problems have solutions in a wide class of supersolutions. These supersolutions are defined as lower semicontinuous functions obeying a parabolic comparison principle with respect to continuous solutions. We also consider the question of how the integrability of the gradient of solutions is affected if the measure is given by a function in L ^s , for a small exponent s > 1.     

Gradient integrability and rigidity results for two-phase conductivities in two dimensions

This paper deals with higher gradient integrability for σ-harmonic functions u with discontinuous coefficients σ  , i.e. weak solutions of div(σ∇u)=0$\mathrm{div}(\sigma \mathrm{\nabla }u)=0$ in dimension two. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and σ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries.     

Conditional mean convergence theorems of conditionally dependent random variables under conditions of integrability

We give the conditionally residual h-integrability with exponent r for an array of random variables and establish the conditional mean convergence of conditionally negatively quadrant dependent and conditionally negative associated random variables under this integrability. These results generalize and improve the known ones.     

An improvement to Darboux integrability theorem for systems having a center

We consider planar polynomial differential systems of degree m with a center at the origin and with an arbitrary linear part. We show that if the system has m(m + 1)/2 − [(m + 1)/2] algebraic solutions or exponential factors then it has a Darboux integrating factor. This result is an improvement of the classical Darboux integrability theorem and other recent results about integrability.     

Morrey global bounds and quasilinear Riccati type equations below the natural exponent

We obtain global bounds in Lorentz–Morrey spaces for gradients of solutions to a class of quasilinear elliptic equations with low integrability data. The results are then applied to obtain sharp existence results in the framework of Morrey spaces for Riccati type equations with a gradient source term having growths below the natural exponent of the operator involved. A special feature of our results is that they hold under a very general assumption on the nonlinear structure, and under a mild natural restriction on the boundary of the ground domain.     

Интерполяционные те оремы для классов фун кций с доминирующей смеша нной производной

If a function belongs to two functional spaces with a dominating mixed derivative, then it also belongs to the intermediate spaces (in the sense of the order of differentiation and the integrability exponent). An interpolation theorem is proved for the operators on such spaces. A linear operator is considered which is bounded on each of the two periodic functional spaces with a dominating mixed derivative. Boundedness of the operator on the intermediate functional spces is proved and the corresponding estimates of the norms of the operator are deduced.     

A note on the de La Vallée Poussin criterion for uniform integrability

In this paper, we present new versions of the classical de La Vallée Poussin criterion for uniform integrability. Our results concern the uniform integrability of a continuous function relative to a sequence of distribution functions. We apply our results to obtain a result on the convergence of a sequence of integrals which we illustrate with an example.     

Local variability of non-smooth functions

With the help of Müntz powers a formula of the Taylor type for non-smooth functions is presented. The approximation provides a local study for the variability of some curves which do not have a derivative. The approach includes the classical case but, at the same time, other non-analytical and non-differentiable mappings. In the first place, a Müntz curve representing the local variability of a function is defined. The coefficient and exponent of the model allow a numerical characterization of the relative extremes and differentiability of the map. The introduction of exponents of higher order provides a generalization of the Taylor’s formula including some cases of non-differentiability. In the last part, a series expansion of non necessarily integer powers representing the function is presented. Several properties of convergence, continuity, integrability and density are studied.     

一类具非标准增长条件和零阶项的抛物方程的存在性结果

在变指数背景下,我们考虑了一类具零阶项的抛物方程解的存在性结果. 存在性的证明在本质上依赖于选取合适的检验函数,这些检验函数要同时兼顾方程右端源项的可积性和零阶项. 利用先验估计和极限分析,借助于Young测度方法确立了非线性项的弱收敛元.     

A class of Nevanlinna functions related to singular Sturm-Liouville problems

The class of Nevanlinna functions consists of functions which are holomorphic off the real axis, which are symmetric with respect to the real axis, and whose imaginary part is nonnegative in the upper halfplane. The Kac subclass of Nevanlinna functions is defined by an integrability condition on the imaginary part. In this note a further subclass of these Kac functions is introduced. It involves an integrability condition on the modulus of the Nevanlinna functions (instead of the imaginary part). The characteristic properties of this class are investigated. The definition of the new class is motivated by the fact that the Titchmarsh-Weyl coefficients of various classes of Sturm-Liouville problems (under mild conditions on the coefficients) actually belong to this class.

     

Whitney’s jets for Sobolev functions

We present two fundamental facts from the jet theory for Sobolev spaces W ^{m, p} . One of these facts is that the formal differentiation of the k-jets theory is compatible with the pointwise definition of Sobolev (m − 1)-jet spaces on regular subsets of the Euclidean spaces ℝⁿ. The second result describes the Sobolev imbedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 345–358, March, 2007.     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.     

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Abstract. – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.?Our stability result generalizes those by Lochak-Neishtadt and P?schel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).?On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system. Manuscrit reĉu le 30 décembre 2001. In memoriam Michael R. Herman The present article is the result of a collaboration with Michael Herman, which started in October 1999. He had had the idea of studying the Nekhoroshev theory in the Gevrey category and, using a lemma of his, of producing new examples of unstable orbits for which the instability time could be compared with the distance of the system to integrability. Together we improved both the stability and instability results which he had already obtained, in view of making them match. Michael Herman’s sudden death in November 2000 prevented him from participating to the last developments and to the final writing of a work the main contributor of which he was.