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34
Elements of stochastic calculus via regularisation
, 2006
"... This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure Itô and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite quadratic variation processes, Dirichlet and weak Dirichlet p ..."
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Cited by 23 (10 self)
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This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure Itô and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite quadratic variation processes, Dirichlet and weak Dirichlet processes.
Riesz transform, Gaussian bounds and the method of wave equation
 Math. Z
"... Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We ..."
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Cited by 20 (1 self)
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Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the LaplaceBeltrami operator on Riemannian manifolds for p> 2. 1.
Pseudodifferential operators on locally compact abelian groups and Sjöstrand’s symbol class
 J. Reine Angew. Math
, 2006
"... We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the KohnNirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard ana ..."
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Cited by 11 (2 self)
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We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the KohnNirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis ” techniques are not available on locally compact abelian groups, a new timefrequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjöstrand’s original results are thus understood as a phenomenon of abstract harmonic analysis rather than “hard analysis ” and are proved in their natural context and generality. 1
Gaussian heat kernel upper bounds via the PhragmnLindelf theorem
 Proc. Lond. Math. Soc
"... Abstract. We prove that in presence of L 2 Gaussian estimates, socalled DaviesGaffney estimates, ondiagonal upper bounds imply precise offdiagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Contents ..."
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Cited by 7 (0 self)
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Abstract. We prove that in presence of L 2 Gaussian estimates, socalled DaviesGaffney estimates, ondiagonal upper bounds imply precise offdiagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Contents
WEIGHT FUNCTIONS IN TIMEFREQUENCY ANALYSIS
, 2006
"... We discuss the most common types of weight functions in harmonic analysis and how they occur in timefrequency analysis. As a general rule, submultiplicative weights characterize algebra properties, moderate weights characterize module properties, GelfandRaikovShilov weights determine spectral inv ..."
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Cited by 6 (2 self)
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We discuss the most common types of weight functions in harmonic analysis and how they occur in timefrequency analysis. As a general rule, submultiplicative weights characterize algebra properties, moderate weights characterize module properties, GelfandRaikovShilov weights determine spectral invariance, and BeurlingDomar weights guarantee the existence of compactly supported test functions.
Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces
, 2010
"... Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ · ‖) is a pconvex Banach space then for any Lipschitz function f: H → X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x) − f(y)‖ dW ..."
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Cited by 5 (3 self)
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Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ · ‖) is a pconvex Banach space then for any Lipschitz function f: H → X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x) − f(y)‖ dW (x, y) log log dW (x, y) log dW (x, y)
THE MIURA MAP ON THE LINE
, 2005
"... Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrödinger operators Lq = −d2 /dx2 + q where r and q are realvalued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R ..."
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Cited by 5 (1 self)
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Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrödinger operators Lq = −d2 /dx2 + q where r and q are realvalued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R) to the local Sobolev space H −1 loc (R) and the restriction of B to the Sobolev spaces Hβ (R) with β ≥ 0. For example, we prove that the image of B on L2 loc (R) consists exactly of those q ∈ H −1 loc (R) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application we prove an existence result for solutions of the Kortewegde Vries equation in H−1 (R) for initial data in the range B(L2 (R)) of the Miura
Second order rectifiability of integral varifolds of locally bounded first variation
 2008. OF PERTURBED ALLEN–CAHN EQUATIONS 23
"... Abstract. In this work it is shown that for every integral n varifold in an open subset U of R n+m, n, m ∈ N, of locally bounded first variation there exists a countable collection C of n dimensional submanifolds of U of class C 2 such that µ(U ∼ S C) = 0 and for each member M of C Hµ(x) = HM (x ..."
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Cited by 3 (0 self)
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Abstract. In this work it is shown that for every integral n varifold in an open subset U of R n+m, n, m ∈ N, of locally bounded first variation there exists a countable collection C of n dimensional submanifolds of U of class C 2 such that µ(U ∼ S C) = 0 and for each member M of C Hµ(x) = HM (x) for µ almost all x ∈ M. Contents