A Lower Estimate for the Norm of the Kerzman-Stein Operator

Bolt, Michael

doi:10.1216/jiea/1192628618

Cited by 5 publications

(6 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An intimate relationship between Szegő projections and "Cauchy-like" integral operators C (the Leray transform is just one such example) was noticed by Kerzman and Stein in [24,25]. They observed that detailed information about the Szegő projection can be extracted from the operator A := C * − C. See [10] for an expository treatment of these ideas in the complex plane and [13,12,9,28] for more recent developments.…”

Section: Introductionmentioning

confidence: 93%

See 1 more Smart Citation

High frequency behavior of the Leray transform: model hypersurfaces and projective duality

Barrett¹,

Edholm²

2021

Preprint

View full text Add to dashboard Cite

The Leray transform L is studied on a family Mγ of unbounded hypersurfaces in two complex dimensions. For a large class of measures, we obtain necessary and sufficient conditions for the L 2 -boundedness of L, along with an exact spectral description of L * L. This yields both the norm and high-frequency norm of L, the latter giving an affirmative answer to an unbounded analogue of an open conjecture relating the essential norm of L to a projective invariant on a bounded hypersurface. L is also shown to play a central role in bridging the function theoretic and projective geometric notions of duality. Our work leads to the construction of projectively invariant Hardy spaces on the Mγ , along with the realization of their duals as invariant Hardy spaces on the dual hypersurfaces.

show abstract

Section: Introductionmentioning

confidence: 93%

“…The operator A stems from work of Kerzman and Stein [24,25] examining the relation between certain particular Cauchy-Fantappiè projections and the self-adjoint Szegő projection S (corresponding to K = 0). See [13,12,5,16,10] for results on A in the one-dimensional setting and [9] for results on Reinhardt domains.…”

Section: Related Operatorsmentioning

confidence: 99%

High frequency behavior of the Leray transform: model hypersurfaces and projective duality

Barrett¹,

Edholm²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Subsequent work on the problem was concerned with giving a complete description of the spectrum for model domains [3], asymptotics of eigenvalues for ellipses with small eccentricity [5], and norm estimates that are invariant with respect to Möbius transformation [1,4]. For a disc or halfplane, there is complete cancellation of singularities and the Kerzman-Stein operator is trivial [13].…”

Section: Theoremmentioning

confidence: 99%

The Kerzman–Stein operator for piecewise continuously differentiable regions

Bolt¹,

Raich²

2014

Complex Variables and Elliptic Equations

View full text Add to dashboard Cite

Abstract. The Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman-Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman-Stein operator has large norm, and we demonstrate the variation in norm with respect to a continuously differentiable perturbation.

show abstract

“…(Recall that the essential norm -see [13] -measures the distance to the set of compact operators; it often occurs in localized analysis.) See [5,9,10] for related results about C.…”

Section: Introductionmentioning

confidence: 99%

“…However, it can also be extracted from this work that the essential L 2 -norm C e = 1 for every smooth Ω ⊂ C. (Recall that the essential norm -see [13] -measures the distance to the set of compact operators; it often occurs in localized analysis.) See [5,9,10] for related results about C.…”

Section: Introductionmentioning

confidence: 99%

The Leray transform: Factorization, dual CR structures, and model hypersurfaces in CP2

Barrett

Edholm

2020

Advances in Mathematics

View full text Add to dashboard Cite

We compute the exact norms of the Leray transforms for a family S β of unbounded hypersurfaces in two complex dimensions. The S β generalize the Heisenberg group, and provide local projective approximations to any smooth, strongly C-convex hypersurface S to two orders of tangency. This work is then examined in the context of projective dual CR-structures and the corresponding pair of canonical dual Hardy spaces associated to S, leading to a universal description of the Leray transform and a factorization of the transform through orthogonal projection onto the conjugate dual Hardy space.

show abstract

A Lower Estimate for the Norm of the Kerzman-Stein Operator

Cited by 5 publications

References 15 publications

High frequency behavior of the Leray transform: model hypersurfaces and projective duality

High frequency behavior of the Leray transform: model hypersurfaces and projective duality

The Kerzman–Stein operator for piecewise continuously differentiable regions

The Leray transform: Factorization, dual CR structures, and model hypersurfaces in CP2

Contact Info

Product

Resources

About