Sobolev spaces and symbolic calculus

We use elementary theory of distributions and geometry of Euclidean spaces to obtain a theorem on the symbolic calculus of several variables in spaces of Fourier series with weights.     

On the finite Mellin transform in quantum calculus and application

The aim of the present paper is to introduce and study a new type of q-Mellin transform [11], that will be called q-finite Mellin transform. In particular, we prove for this new transform an inversion formula and q-convolution product. The application of this transform is also earlier proposed in solving procedure for a new equation with a new fractional differential operator of a variational type.     

Applications of the Mellin transform in quantum calculus

In this paper, we study in quantum calculus the correspondence between poles of the q-Mellin transform (see [A. Fitouhi, N. Bettaibi, K. Brahim, The Mellin transform in Quantum Calculus, Constr. Approx. 23 (3) (2006) 305-323]) and the asymptotic behaviour of the original function at 0 and ∞. As applications, we give a new technique (in q-analysis) to derive the asymptotic expansion of some functions defined by q-integrals or by q-harmonic sums. Finally, a q-analogue of the Mellin-Perron formula is given.     

The homology of path spaces and Floer homology with conormal boundary conditions

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions. Dedicated to Felix E. Browder     

Differential equations with conical degeneration in spaces with asymptotics

We define a space with asymptotics isomorphic to the direct sum of the corresponding Sobolev spaces for any values of their indices. In spaces with asymptotics, we consider problems for differential operators with conical degeneration and state conditions providing their Fredholm property.     

Elastic potentials at corners in Sobolev spaces with asymptotics

This paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

On the symbolic calculus in homogeneous Banach algebras

We construct a strongly homogeneous Banach algebraB so that for an appropriate positive integern, (1)A(T) ⊊B ⊊C(T); (2)n-times continuously differentiable functions operate onB. This work was supported in part by NSF Grant MCS75-08552-A02.     

Differential equations on manifolds with edge singularities in spaces with asymptotics

The present paper deals with differential equations with edge degeneration in spaces with asymptotics. We give the definition of an edge space with asymptotics, prove the continuity of operators with edge degeneration in the scale of these spaces, present statements of problems with edge operators, and state conditions providing their Fredholm property.     

Polarizations and differential calculus in affine spaces

Within the framework of mappings between affine spaces, the notion of nth polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as linearity, iterability, Leibniz and chain rules are shared - at the finite level - by the polarization operators. We give these results by means of explicit general formulae, which are valid at any order n, and are based on combinatorial identities. The infinitesimal limits of the nth polarizations of a function will yield its nth derivatives (without resorting to the usual recursive definition), and the afore-mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula.     

Mellin and Green symbols for boundary value problems on manifolds with edges

We introduce the algebra of smoothing Mellin and Green symbols in a pseudodifferential calculus for manifolds with edges. In addition, we define scales of weighted Sobolev spaces with asymptotics based on the Mellin transform and analyze the mapping properties of the operators on these spaces. This will allow us to obtain complete information on the regularity and asymptotics of solutions to elliptic equations on these spaces.     

Sobolev type spaces in quantum calculus

In this paper q-Sobolev type spaces are defined on Rq by using the q-cosine Fourier transform and its inverse. In particular, embedding results for these spaces are established. Next we define the q-cosine potential and study some of its properties.     

Subdifferential calculus in Asplund generated spaces

We extend the definition of the limiting Fréchet subdifferential and the limiting Fréchet normal cone from Asplund spaces to Asplund generated spaces. Then we prove a sum rule, a mean value theorem, and other statements for this concept.     

Short‐time asymptotics in Dirichlet spaces

This work is involved with the short‐time asymptotics of the heat semigroup in a general setting. A generalization of Fang's version of Varadhan's formula is proved. A spectral gap or the possibility of obtaining one by an appropriate change of measure is required. © 2001 John Wiley & Sons, Inc.     

Malliavin calculus and decoupling inequalities in Banach spaces

We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided Lp-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.     

Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators

Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.     

Metric regularity and subdifferential calculus in Banach spaces

In this paper we give verifiable conditions in terms of limiting Fréchet subdifferentials ensuring the metric regularity of a multivalued functionF(x)=–g(x)+D. We apply our results to the study of the limiting Fréchet subdifferential of a composite function defined on a Banach space.     

Complete spaces and zero-one measures

It is the purpose of this paper to characterize the complete spaces in the sense of [6] by measure-theoretic properties. Let (X,) be a measurable space and let be a subpaving of satisfying certain closure properties, then X is-complete iff every 0,1-valued-regular measure on is a Dirac measure. In particular, we obtain Hewitt's well-known theorem that a completely regular space X is realcompact iff every 0,1-valued Baire measure on X is a Dirac measure. The main tool for our investigations is an extension theorem for measures due to Topsoe [10].