Results 1  10
of
16
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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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Plancherel type estimates and sharp spectral multipliers
 J. FUNCT. ANAL
, 2002
"... We study general spectral multiplier theorems for selfadjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmandertype spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of ..."
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We study general spectral multiplier theorems for selfadjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmandertype spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmandertype spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R³ and new spectral multiplier theorems for the Laguerre and Hermite expansions.
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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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
Sharp Bounds on Heat Kernels of Higher Order Uniformly Elliptic Operators
 J. Operator Theory
, 1996
"... this paper is to obtain precise quantitative bounds on the constants c 2 ..."
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Cited by 4 (1 self)
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this paper is to obtain precise quantitative bounds on the constants c 2
On the Lptheory of C0semigroups associated with second order . . .
"... We study positive C0semigroups on Lp associated with second order uniformly elliptic divergence type operators with singular lower order terms, subject to a wide class of boundary conditions. We obtain an interval (pmin; pmax) in the Lpscale where these semigroups can be defined, including the ca ..."
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We study positive C0semigroups on Lp associated with second order uniformly elliptic divergence type operators with singular lower order terms, subject to a wide class of boundary conditions. We obtain an interval (pmin; pmax) in the Lpscale where these semigroups can be defined, including the case 2 62 (pmin; pmax). We present an example showing that the result is optimal. We also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p, for the whole range of p where the semigroups are defined.
ON A QUADRATIC ESTIMATE RELATED TO THE KATO CONJECTURE AND BOUNDARY VALUE PROBLEMS
, 810
"... Abstract. We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L 2 boundary data. We develop the application to the Kato conjecture and to a ..."
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Abstract. We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L 2 boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms. MSC classes: 35J25, 35J55, 47N20, 47F05, 42B25 Keywords: LittlewoodPaley estimate, functional calculus, boundary value problems, second order elliptic equations and systems, square root problem 1.
PROPERTIES OF CENTERED RANDOM WALKS ON LOCALLY COMPACT GROUPS AND LIE GROUPS
, 2007
"... ABSTRACT. The basic aim of this paper is to study asymptotic properties of the convolution powers K (n) = K ∗K ∗ · · ·∗K of a possibly nonsymmetric probability density K on a locally compact, compactly generated group G. If K is centered, we show that the Markov operator T associated with K is an ..."
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ABSTRACT. The basic aim of this paper is to study asymptotic properties of the convolution powers K (n) = K ∗K ∗ · · ·∗K of a possibly nonsymmetric probability density K on a locally compact, compactly generated group G. If K is centered, we show that the Markov operator T associated with K is analytic in L p (G) for 1 < p < ∞, and establish DaviesGaffney estimates in L 2 for the iterated operators T n. These results enable us to obtain various Gaussian bounds on K (n). In particular, when G is a Lie group we recover and extend some estimates of Alexopoulos and of Varopoulos for convolution powers of centered densities and for the heat kernels of centered sublaplacians. Finally, in case G is amenable, we discover that the properties of analyticity or DaviesGaffney estimates hold only if K is centered.
Lpanalyticity of Schrödinger semigroups on Riemannian manifolds
"... We address the problems of extrapolation, analyticity, and Lpspectral independence for C0semigroups in the abstract context of metric spaces with exponentially bounded volume. The main application of the abstract result is Lpanalyticity of angle π of Schrödinger semigroups on Riemannian manifolds ..."
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We address the problems of extrapolation, analyticity, and Lpspectral independence for C0semigroups in the abstract context of metric spaces with exponentially bounded volume. The main application of the abstract result is Lpanalyticity of angle π of Schrödinger semigroups on Riemannian manifolds with Ricci curvature 2 bounded below, under the condition of form smallness of the negative part of the potential. MSC 2000: 47D06, 47N20, 35J10 The final aim of the present paper is the following result. Let M be a Riemannian manifold with Ricci curvature bounded below. Then a Schrödinger semigroup with form small negative part of the potential is analytic of angle π 2 on Lp(M), for p from a certain subinterval of [1, ∞) that contains 2 and is determined by the form bound (see Theorem 7