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Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
On the spectrum of Schrödinger operators with quasiperiodic algebrogeometric KdV potentials
 J. Analyse Math. 95
, 2005
"... Dedicated with great pleasure to Vladimir A. Marchenko on the occasion of his 80th birthday. Abstract. We characterize the spectrum of onedimensional Schrödinger operators H = −d 2 /dx 2 + V in L 2 (R; dx) with quasiperiodic complexvalued algebrogeometric potentials V (i.e., potentials V which s ..."
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Cited by 4 (3 self)
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Dedicated with great pleasure to Vladimir A. Marchenko on the occasion of his 80th birthday. Abstract. We characterize the spectrum of onedimensional Schrödinger operators H = −d 2 /dx 2 + V in L 2 (R; dx) with quasiperiodic complexvalued algebrogeometric potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg–de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The corresponding problem appears to have been open since the midseventies. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green’s function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semiinfinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These
Heat Kernels and maximal L^p  L^q Estimates for Parabolic Evolution Equations
"... Let A be the generator of an analytic semigroup T on L 2(\Omega\Gamma1 where \Omega is a homogeneous space with doubling property. We prove maximal L p \Gamma L q apriori estimates for the solution of the parabolic evolution equation u 0 (t) = Au(t) + f(t); u(0) = 0 provided T may be repre ..."
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Cited by 3 (1 self)
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Let A be the generator of an analytic semigroup T on L 2(\Omega\Gamma1 where \Omega is a homogeneous space with doubling property. We prove maximal L p \Gamma L q apriori estimates for the solution of the parabolic evolution equation u 0 (t) = Au(t) + f(t); u(0) = 0 provided T may be represented by a heatkernel satisfying certain bounds (and in particular a Gaussian bound). 1991 Mathematics Subject Classification: 35K22, 58D25, 47D06 1 Introduction Consider an inhomogeneous initial value problem of the form (1:1) u 0 (t) = Au(t) + f(t); t 2 [0; 1) u(0) = u 0 ; where A is the generator of an analytic semigroup on some Banach space X. It is a well known fact that, in general, the derivative u 0 of a solution u of the above Cauchy problem is less regular than the right hand side f , even though the problem is of parabolic type. This phenomena, sometimes called 'loss of regularity', causes problems in the treatment of quasilinear parabolic problems. It is therefore d...
MAXIMAL PARABOLIC REGULARITY FOR DIVERGENCE OPERATORS INCLUDING MIXED BOUNDARY CONDITIONS
, 903
"... Abstract. We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to q ..."
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Cited by 2 (1 self)
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Abstract. We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with nonsmooth data are presented. 1.
Wiener Regularity and Heat Semigroups on Spaces of Continuous Functions
, 1998
"... Let IR N be open. It is shown that the Dirichlet Laplacian generates a (holomorphic) C 0 semigroup on C 0 ( if and only if is regular in the sense of Wiener. The same result remains true for elliptic operators in divergence form. ..."
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Let IR N be open. It is shown that the Dirichlet Laplacian generates a (holomorphic) C 0 semigroup on C 0 ( if and only if is regular in the sense of Wiener. The same result remains true for elliptic operators in divergence form.
unknown title
, 2004
"... On necessary and sufficient conditions for L pestimates of Riesz transforms associated to elliptic operators on R n and related estimates ..."
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On necessary and sufficient conditions for L pestimates of Riesz transforms associated to elliptic operators on R n and related estimates
THE SQUARE ROOT PROBLEM FOR SECOND ORDER, DIVERGENCE FORM OPERATORS WITH MIXED BOUNDARY CONDITIONS ON L p
"... Abstract. We show that, under very general conditions on the domain Ω and the Dirichlet partD of the boundary, the operator ( −∇·µ∇+1) 1/2 with mixed boundary conditions provides a topological isomorphism between W 1,p D (Ω) and Lp (Ω), if p ∈]1,2]. hal00737614, version 2 23 Jan 2013 ..."
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Abstract. We show that, under very general conditions on the domain Ω and the Dirichlet partD of the boundary, the operator ( −∇·µ∇+1) 1/2 with mixed boundary conditions provides a topological isomorphism between W 1,p D (Ω) and Lp (Ω), if p ∈]1,2]. hal00737614, version 2 23 Jan 2013