arXiv subjects

Luke D. Edholm

Publications and source records attributed to Luke D. Edholm.

5 recordsLinked to original sources

arXiv2026

Arithmetic properties and zeros of the Bergman kernel on a class of quotient domains

An effective formula for the Bergman kernel on $\mathbb{H}_γ = \{|z_1|^γ< |z_2| < 1 \}$ is obtained for rational $γ= \frac{m}{n} >1$. The formula depends on arithmetic properties of $γ$, which uncovers new symmetries and clarifies previous results. The formulas are then used to study the Lu Qi-Keng problem. We produce sequences of rationals $γ_j \searrow 1$, where each $\mathbb{H}_{γ_j}$ has a Bergman kernel with zeros (while $\mathbb{H}_1$ is known to have a zero-free kernel), resolving an open question on this domain class.

math.CV↗

arXiv2024

The Leray transform: distinguished measures, symmetries and polygamma inequalities

New symmetries, norm computations and spectral information are obtained for the Leray transform on a class of unbounded hypersurfaces in $\mathbb{C}^2$. Emphasis is placed on certain distinguished measures, with results on operator norm monotonicity established by proving new polygamma inequalities. Classical techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our analysis. Underpinning this work is a projective geometric theory of duality, which manifests here in the form of Hölder invariance.

math.CV↗

arXiv2024

Projections onto $L^p$-Bergman spaces of Reinhardt Domains

For $1<p<\infty$, we emulate the Bergman projection on Reinhardt domains by using a Banach-space basis of $L^p$-Bergman space. The construction gives an integral kernel generalizing the ($L^2$) Bergman kernel. The operator defined by the kernel is shown to be absolutely bounded projection on the $L^p$-Bergman space on a class of domains where the $L^p$-boundedness of the Bergman projection fails for certain $p \neq 2$. As an application, we identify the duals of these $L^p$-Bergman spaces with weighted Bergman spaces.

math.CV↗

arXiv2022

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

The Leray transform $\bf{L}$ is studied on a family $M_γ$ 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 $\bf{L}$, along with an exact spectral description of $\bf{L}^*\bf{L}$. This yields both the norm and high-frequency norm of $\bf{L}$, the latter giving an affirmative answer to an unbounded analogue of an open conjecture relating the essential norm of $\bf{L}$ to a projective invariant on a bounded hypersurface. $\bf{L}$ 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 $M_γ$, along with the realization of their duals as invariant Hardy spaces on the dual hypersurfaces.

math.CV↗

arXiv2020

$L^p$-regularity of the Bergman projection on quotient domains

We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-boundedness on a domain and its quotient by a finite group. The range of $p$ for which the Bergman projection is $L^p$-bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.

math.CV↗