There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space X equipped with a doubling measure ¯. A generalization of a Sobolev function and its gradient is a pair u 2 L 1 loc (X), 0 g 2 L p (X) such that for every ball B ae X the Poincar'e-type inequality Z B ju \Gamma uB j d¯ Cr `Z oeB g p d¯ ' 1=p holds, where r is the radius of B and oe 1, C ? 0 are fixed constants. Working in the above setting we show that basically...

We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. In this case our construction can be viewed as a far-reaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the non-standard wavelet representation of Calderón-Zygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.

Abstract. We establish local (L p, L q) mapping properties for averages on curves. The exponents are sharp except for endpoints. 1.

We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity solutions are indeed classical solutions. 2001 ditions scientifiques et mdicales Elsevier SAS Keywords: Nonlinear degenerate Kolmogorov equation, Interior regularity, Hrmander operators RSUM. -- Nous tudions la rgularit intrieure des solutions de viscosit d'une quation non linaire du second ordre dgnre que l'on rencontre en finance mathmatique. Nous tudions le problme par la thorie des oprateurs de Hrmander sans aucune hypothse sur la dgnerescence de l'algbre de Lie engendre. Nous montrons que la solution de viscosit est une solution classique. 2001 ditions scientifiques et mdicales Elsevier SAS 1.

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial di#erential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group. 1.

We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:...

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Abstract. Let p: C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define ¯ Zτp = ∂ ∂p + τ ∂¯z ∂¯z, a closed, densely-defined operator on L2 (C). If □τp = ¯ Zτp ¯ Z ∗ τp and τ> 0, we solve the heat equation ∂u + □τpu = 0, u(0, z) = f(z), on (0, ∞) × C. The solution comes via ∂s the heat semigroup e −s□τp, and we show that u(s, z) = e −s□τp [f](z) = ∫

Abstract. We develop classes of one-parameter families (OPF) of operators on C ∞ c (C) which characterize the behavior of operators associated to the ¯ ∂-problem in L 2 (C, e −2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from L q (C) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in C 2. 1.

Abstract. Let p: C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter.