An abstract transmission problem in a thin layer, I: Sharp estimates

We study an elliptic transmission problem in Banach spaces. The problem is considered on the juxtaposition of two intervals, one of which of small length δ, and models physical phenomena in media constituted by two parts with different physical characteristics. We obtain results of existence, uniqueness, maximal regularity and optimal dependence on the parameter δ for Lp solutions of the problem. The main tools of our approach are impedance and admittance operators (i.e. Dirichlet-to-Neumann and Neumann-to-Dirichlet operators) and H^∞ functional calculus for sectorial operators in Banach spaces.     

Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω⊂Cn. We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n−2, then compactness of the Hankel operator Hβ implies that β is holomorphic “along” analytic discs in the boundary. Furthermore, when Ω is convex in C² we show that the condition on β is necessary and sufficient for compactness of Hβ.     

Groupoids and pseudodifferential calculus on manifolds with corners

We associate to any manifold with corners (even with non-embedded hyperfaces) a (non-Hausdorff) longitudinally smooth Lie groupoid, on which we define a pseudodifferential calculus. This calculus generalizes the b-calculus of Melrose, defined for manifolds with embedded corners. The groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension. Using tools from the theory of C∗-algebras of groupoids, we also obtain new proofs for the study of b-calculus.     

Convergence of a fixed point iteration method for the OSV model

In image processing, image denoising and texture extraction are important problems in which many new methods recently have been developed. One of the most important models is the OSV model [S. Osher, A. Solé, L. Vese, Image decomposition and restoration using total variation minimization and the H^-1 norm, Multiscale Model. Simul. A SIAM Interdisciplinary J. 1(3) (2003) 349-370] which is constructed by the total variation and H^-1 norm. This paper proves the existence of the minimizer of the functional from the OSV model and analyzes the convergence of an iterative method for solving the problems. Our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated Euler-Lagrange equation, and the limit of our iterations satisfies the minimizer of the functional from the OSV model. In numerical experiments, we compare the numerical results of our works with those of the ROF model [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259-268].     

Solutions of nonlinear evolution inclusions

In this paper, the solution of the nonlinear evolution inclusion problem of the form u^′(t)+B(t,u(t))∋f(t) is studied. In this problem, the operators are of type (M) or type (S₊), which are different from those of pseudo-monotone operators that had been studied by many authors. At the same time, we study the perturbation problem. In fact, many kinds of evolution equations can be generalized by this problem. The former results are improved and generalized by our conclusions, and we will give more applications.     

Maximal ergodic theorems for some group actions

We prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative Lp-space. We also obtain the corresponding individual ergodic theorems. Applying these results to actions of a free group on a von Neumann algebra, we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of Nevo-Stein.     

Separable states and positive maps

Using the natural duality between linear functionals on tensor products of C^∗-algebras with the trace class operators on a Hilbert space H and linear maps of the C^∗-algebra into B(H), we study the relationship between separability, entanglement and the Peres condition of states and positivity properties of the linear maps.     

Conservation laws for some compacton equations using the multiplier approach

This paper is an application of the variational derivative method to the derivation of the conservation laws for partial differential equations. The conservation laws for (1+1) dimensional compacton k(2,2) and compacton k(3,3) equations are studied via multiplier approach. Also the conservation laws for (2+1) dimensional compacton Zk(2,2) equation are established by first computing the multipliers.     

Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via an SPDE approach

The main objective of this paper is to prove the essential self-adjointness of Dirichlet operators in L²(μ) where μ is a Gibbs measure on an infinite volume path space C(R,Rd). This operator can be regarded as a perturbation of the Ornstein-Uhlenbeck operator by a nonlinearity and corresponds to a parabolic stochastic partial differential equation (= SPDE, in abbreviation) on R. In view of quantum field theory, the solution of this SPDE is called a P₁(?)-time evolution.     

Continuous block implicit hybrid one-step methods for ordinary and delay differential equations

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and Aω-stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.     

Growth conditions, compact perturbations and operator subdecomposability, with applications to generalized Cesàro operators

We adapt recent results of Albrecht and Ricker to obtain conditions under which growth constraints on the left resolvent of a Banach space operator are preserved under suitable perturbations. As an application, we establish Bishop's property (β) for certain generalized Cesàro operators on the classical Hardy spaces Hp, 1<p<∞. Our methods also apply to unilateral weighted shifts whose weight sequence converges sufficiently rapidly as well as to perturbations of restrictions of a class of generalized scalar operators.     

Rough oscillatory singular integral operators of nonconvolution type

In this paper, we study a classes of oscillatory singular integral operators of nonconvolution type with phases more general than polynomials. We prove that such operators are bounded on Lp provided their kernels satisfy a very weak condition. In addition, we also study the related truncated oscillatory singular integral operators. Moreover, we present a class of unbounded oscillatory singular integral operators.     

Spectral properties and their applications of frequency response operators in linear continuous-time periodic systems

Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the H_∞ norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors.     

The algebra of harmonic functions for a matrix-valued transfer operator

We analyze matrix-valued transfer operators. We prove that the fixed points of transfer operators form a finite-dimensional C^∗-algebra. For matrix weights satisfying a low-pass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algorithms.     

Multilinear extrapolation and applications to the bilinear Hilbert transform

We present two extrapolation methods for multi-sublinear operators that allow us to derive estimates for general functions from the corresponding estimates on characteristic functions. Of these methods, the first is applicable to general multi-sublinear operators while the second requires working with the so-called (ε,δ)-atomic operators. Among the applications, we discuss some new endpoint estimates for the bilinear Hilbert transform.     

Operator identities involving the bivariate Rogers-Szegö polynomials and their applications to the multiple q-series identities

In this paper, we first give several operator identities involving the bivariate Rogers-Szegö polynomials. By applying the technique of parameter augmentation to the multiple q-binomial theorems given by Milne [S.C. Milne, Balanced summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997) 93-187], we obtain several new multiple q-series identities involving the bivariate Rogers-Szegö polynomials. These include multiple extensions of Mehler's formula and Rogers's formula. Our U(n+1) generalizations are quite natural as they are also a direct and immediate consequence of their (often classical) known one-variable cases and Milne's fundamental theorem for An or U(n+1) basic hypergeometric series in Theorem 1.49 of [S.C. Milne, An elementary proof of the Macdonald identities for , Adv. Math. 57 (1985) 34-70], as rewritten in Lemma 7.3 on p. 163 of [S.C. Milne, Balanced summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997) 93-187] or Corollary 4.4 on pp. 768-769 of [S.C. Milne, M. Schlosser, A new An extension of Ramanujan's summation with applications to multilateral An series, Rocky Mountain J. Math. 32 (2002) 759-792].     

Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces Lp(Rn;X) of X -valued functions on Rn. We characterize Kato's square root estimates and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative Lp space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the Lp theory of square roots of elliptic operators, as well as an Lp version of Carleson's inequality.     

The quantitative difference between countable compactness and compactness

We establish here some inequalities between distances of pointwise bounded subsets H of RX to the space of real-valued continuous functions C(X) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C(X). We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure of H to C(X) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C(X): here distance refers to the metric of uniform convergence in RX. We study the quantitative behavior of sequences in H approximating points in . As a particular case we obtain the results known about angelicity for these Cp(X) spaces obtained by Orihuela. We indeed prove our results for spaces C(X,Z) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.     

Convergence of a stochastic particle approximation for fractional scalar conservation laws

We are interested in a probabilistic approximation of the solution to scalar conservation laws with fractional diffusion and nonlinear drift. The probabilistic interpretation of this equation is based on a stochastic differential equation driven by an α-stable Lévy process and involving a nonlinear drift. The approximation is constructed using a system of particles following a time-discretized version of this stochastic differential equation, with nonlinearity replaced by interaction. We prove convergence of the particle approximation to the solution of the conservation law as the number of particles tends to infinity whereas the discretization step tends to 0 in some precise asymptotics.