Convolution operators with homogeneous kernels and applications

Martin, Mircea; Szeptycki, P.

doi:10.1512/iumj.1997.46.1405

Cited by 8 publications

(4 citation statements)

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“…For a complete proof of (5.1) and for some other more general related results we refer to Martin [20-24, 26, 27], and [30].…”

Section: Further Results and Commentsmentioning

confidence: 99%

“…and 30) referred to as the Bochner-Kodaira-Nakano identities. The two remainders R 0,1 and R 1,0 are operators of order zero on E, and likewise R in (2.25) they can be computed using the curvature operator on E. The Bochner-Weitzenböck identity (2.25) is just the average of the two Bochner-Kodaira-Nakano identities (2.29) and (2.30).…”

Section: Dirac and Laplace Operators On Dirac Vector Bundlesmentioning

confidence: 99%

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Seminormal systems of operators in Clifford environments

Martin

2017

OpMath

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Abstract. The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex C * -algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a C * -algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.

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“…For a complete proof of (5.1) and for some other more general related results we refer to Martin [20-24, 26, 27], and [30].…”

Section: Further Results and Commentsmentioning

confidence: 99%

Section: Dirac and Laplace Operators On Dirac Vector Bundlesmentioning

confidence: 99%

Seminormal systems of operators in Clifford environments

Martin

2017

OpMath

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“…Section 3 presents a technical result that provides sharp pointwise estimates for convolution transforms associated with kernels in the weak Lebesgue spaces L κ weak (R m ), where m ≥ 1 and κ > 1. It is a refinement of an inequality proved in Martin and Szeptycki [14] for homogeneous kernels on the Euclidean spaces R m , m ≥ 1, that better serves our specific purposes.…”

Section: Theorem Cmentioning

confidence: 90%

Uniform Approximation by Closed Forms in Several Complex Variables

Martin

2009

AACA

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The first main result in this article provides uniform estimates for solid Bochner-Martinelli-Koppelman transforms of type (p, n − 1), 0 ≤ p ≤ n, of continuous forms on a compact set Ω in the complex space C n , in terms of the Euclidean volume of Ω. In the single variable case this result generalizes a classical inequality for the Cauchy kernel due to Ahlfors and Beurling. The second main result is a quantitative Hartogs-Rosenthal theorem which points out that the uniform distance in the space of continuous (p, n − 1)-forms, 0 ≤ p ≤ n, on a compact set Ω in C n from a smooth form to the subspace of ∂-closed forms on Ω is controlled by the Euclidean volume of Ω. This theorem as well as a third main result are generalizations to higher dimensions of an inequality in single variable complex analysis due to Alexander.

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“…Before proceeding with a proof of Theorem 3.4 we want to mention an inequality proved in [20] that can be used to estimate the constant Λ(Ω). To formulate it, we suppose that k : R n 0 → R is a continuous function satisfying the homogeneity condition k(tx…”

Section: Theorem the Riesz Transform Model Satisfies The Inequalitymentioning

confidence: 99%