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16
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 61 (15 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
The Laplacian with Wentzell{Robin boundary conditions on spaces of continuous functions, Semigroup Forum 67
, 2003
"... We investigate the Laplacian ∆ on a smooth bounded open set Ω ⊂ Rn with WentzellRobin boundary condition βu+ ∂u ∂ν +∆u = 0 on the boundary Γ. Under the assumption β ∈ C(Γ) with β ≥ 0, we prove that ∆ generates a differentiable positive contraction semigroup on C(Ω̄) and study some monotonicity pr ..."
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Cited by 24 (5 self)
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We investigate the Laplacian ∆ on a smooth bounded open set Ω ⊂ Rn with WentzellRobin boundary condition βu+ ∂u ∂ν +∆u = 0 on the boundary Γ. Under the assumption β ∈ C(Γ) with β ≥ 0, we prove that ∆ generates a differentiable positive contraction semigroup on C(Ω̄) and study some monotonicity properties and the asymptotic behaviour.
Heat Kernels and maximal L^p  L^q Estimates for Parabolic Evolution Equations
"... Let A be the generator of an analytic semigroup T on L²(Ω), where Ω 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 heatk ..."
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Cited by 22 (2 self)
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Let A be the generator of an analytic semigroup T on L²(Ω), where Ω 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).
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 15 (5 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.
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 8 (4 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
Gaussian estimates for a heat equation on a network
 Netw. Heter. Media
, 2007
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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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Cited by 6 (2 self)
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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
Heat kernel bounds for elliptic partial differential operators in divergence form with Robintype boundary conditions
 J. Analyse Math
"... Dedicated with great pleasure to E. Brian Davies on the occasion of his 70th birthday. Abstract. The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for selfadjoint L2 (Ω; dnx)realizations, n ∈ N, n � 2, of divergence form elliptic partial differential ..."
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Cited by 5 (1 self)
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Dedicated with great pleasure to E. Brian Davies on the occasion of his 70th birthday. Abstract. The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for selfadjoint L2 (Ω; dnx)realizations, n ∈ N, n � 2, of divergence form elliptic partial differential expressions L with (nonlocal) Robintype boundary conditions in bounded Lipschitz domains Ω ⊂ Rn, where n∑ Lu = − ∂jaj,k∂ku. j,k=1 The (nonlocal) Robintype boundary conditions are then of the form ν · A∇u + Θ [ u ∣ ] = 0 on ∂Ω, ∂Ω where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ∂Ω of Ω, and ν denotes the outward pointing normal unit vector on ∂Ω. hal00830563, version 1 5 Jun 2013
POSITIVE SEMIGROUPS OF KERNEL OPERATORS
"... Dedicated to the memory of H.H. Schaefer. Abstract. Extending results of Davies and of Keicher on `p we show that the peripheral point spectrum of the generator of a positive bounded C0semigroup of kernel operators on Lp is reduced to 0. It is shown that this implies convergence to an equilibrium i ..."
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Cited by 3 (0 self)
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Dedicated to the memory of H.H. Schaefer. Abstract. Extending results of Davies and of Keicher on `p we show that the peripheral point spectrum of the generator of a positive bounded C0semigroup of kernel operators on Lp is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space nontrivial. The results are applied to elliptic operators. Irreducibility is a fundamental notion in PerronFrobenius Theory. It had been introduced in a direct way by Perron and Frobenius for matrices, but it was H. H. Schaefer who gave the definition via closed ideals. This turned out to be most fruitful and led to a wealth of deep and important results. For applications Ouhabaz’