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29
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 53 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
The Green function estimates for strongly elliptic systems of second order
, 2007
"... We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain Ω ⊆ R n, n ≥ 3, under the assumption that solutions of the system satisfy De GiorgiNash type local Hölder continuity estimates. In par ..."
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Cited by 37 (13 self)
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We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain Ω ⊆ R n, n ≥ 3, under the assumption that solutions of the system satisfy De GiorgiNash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.
Operator Theory and Harmonic Analysis
, 1996
"... Contents 1. Spectral Theory of Bounded Operators (A) Spectra and resolvents of bounded operators on Banach spaces (B) Holomorphic functional calculi of bounded operators 2. Spectral Theory of Unbounded Operators (C) Spectra and resolvents of closed operators in Banach spaces (D) Holomorphic function ..."
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Cited by 32 (8 self)
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Contents 1. Spectral Theory of Bounded Operators (A) Spectra and resolvents of bounded operators on Banach spaces (B) Holomorphic functional calculi of bounded operators 2. Spectral Theory of Unbounded Operators (C) Spectra and resolvents of closed operators in Banach spaces (D) Holomorphic functional calculi of operators of type S!+ 3. Quadratic Estimates (E) Quadratic norms of operators of type S!+ in Hilbert spaces (F) Boundedness of holomorphic functional calculi 4. Operators with Bounded Holomorphic Functional Calculi (G) Accretive operators (H) Operators of type S! and spectral projections 5. Singular Integrals (I) Convolutions and the functional calculus of \Gammai d dx (J) The Hilbert transform and Hardy spaces 6. Calder'onZygmund Theory (K) Maximal functions and the Calder'onZygmund decomposition (L) Singular integral operators 7. Functional Ca
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 21 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Pointwise estimates for transition probabilities of random walks in infinite graphs
 in: Trends in mathematics: Fractals in Graz 2001
, 2002
"... walks on infinite graphs ..."
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A review of Hardy inequalities
 Eds.), The Maz'ya Anniversary Collection
, 1999
"... We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation. ..."
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Cited by 12 (0 self)
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We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation.
Gaussian estimates for fundamental solutions of second order parabolic systems with timeindependent coefficients
, 2007
"... Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on R n. In particular, in the case when n = 2 they obtained Gaussian upper bound estimates for the heat kernel without imposing further assum ..."
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Cited by 9 (7 self)
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Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on R n. In particular, in the case when n = 2 they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, timeindependent coefficients, and extend their results to the systems of parabolic equations.
ANALYTICITY OF LAYER POTENTIALS AND L 2 SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR DIVERGENCE FORM ELLIPTIC EQUATIONS WITH COMPLEX L ∞ COEFFICIENTS
, 705
"... Abstract. We consider divergence form elliptic operators of the form L= − div A(x)∇, defined inR n+1 ={(x, t)∈R n ×R}, n≥2, where the L ∞ coefficient matrix A is (n+ 1)×(n+1), uniformly elliptic, complex and tindependent. We show that for such operators, boundedness and invertibility of the corresp ..."
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Cited by 8 (6 self)
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Abstract. We consider divergence form elliptic operators of the form L= − div A(x)∇, defined inR n+1 ={(x, t)∈R n ×R}, n≥2, where the L ∞ coefficient matrix A is (n+ 1)×(n+1), uniformly elliptic, complex and tindependent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L2 (Rn)=L 2 (∂Rn+1 +), is stable under complex, L ∞ perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L2 (Rn) whenever A(x) is real and symmetric (and thus, by our stability result, also when A is complex,‖A − A0‖ ∞ is small enough and A0 is real, symmetric, L ∞ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L2 (resp. ˙L 2 1) data, for small complex perturbations of a real symmetric matrix. Previously, L2 solvability results for complex (or even real but nonsymmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A j,n+1 = 0=An+1, j, 1 ≤ j≤n, which corresponds to the Kato square root problem.
On the Green’s matrices of strongly parabolic systems of second order
"... Abstract. We establish existence and various estimates of fundamental matrices and Green’s matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior Hölder continuity estimate. We pres ..."
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Cited by 8 (5 self)
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Abstract. We establish existence and various estimates of fundamental matrices and Green’s matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior Hölder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases. 1.