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Loredana Lanzani

Publications and source records attributed to Loredana Lanzani.

At least 19 recordsLinked to original sources
arXiv2025

The Unified Transform Method: beyond circular or convex domains

A new transform-based approach is presented that can be used to solve mixed boundary value problems for Laplace's equation in non-convex and other planar domains, specifically the so-called Lipschitz domains. This work complements Crowdy (2015, CMFT, 15, 655--687), where new transform-based techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the present method is the exploitation of the properties of the Szegő kernel and its connection with the Cauchy kernel to obtain transform pairs for analytic functions in such domains. Several examples are solved in detail and are numerically implemented to illustrate the application of the new transform pairs.

math.CV↗
arXiv2025

The Cauchy--Szegö Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted theory

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani \& Stein states that the Cauchy--Szegö projection $S_ω$ defined with respect to a bounded, positive continuous multiple $ω$ of induced Lebesgue measure, {maps $L^p(bD, ω)$ to $L^p(bD, ω)$ continuously} for any $1<p<\infty$. Here we show that $S_ω$ satisfies explicit quantitative bounds in $L^p(bD, Ω)$, for any $1<p<\infty$ and for any $Ω$ in the maximal class of \textit{$A_p$}-measures, that is for $Ω_p = ψ_pσ$ where $ψ_p$ is a Muckenhoupt $A_p$-weight and $σ$ is the induced Lebesgue measure (with $ω$'s as above being a sub-class). Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy--Szegö kernel, but these are unavailable in our setting of minimal regularity {of $bD$}; at the same time, more recent techniques that allow to handle domains with minimal regularity (Lanzani--Stein 2017) are not applicable to $A_p$-measures. It turns out that the method of {quantitative} extrapolation is an appropriate replacement for the missing tools. To finish, we identify a class of holomorphic Hardy spaces defined with respect to $A_p$-measures for which a meaningful notion of Cauchy--Szegö projection can be defined when $p=2$.

math.CV↗
arXiv2025

The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$, and let $S_ω$ denote the Cauchy--Szegő projection defined with respect to (any) positive continuous multiple $ω$ of induced Lebesgue measure for the boundary of $D$. We characterize compactness and boundedness (the latter with explicit bounds) of the commutator $[b, S_ω]$ in the Lebesgue space $L^p(bD, Ω_p)$ where $Ω_p$ is any measure in the Muckenhoupt class $A_p(bD)$, $1<p<\infty$. We next fix $p =2$ and we let $S_{Ω_2}$ denote the Cauchy--Szegő projection defined with respect to (any) measure $Ω_2 \in A_2(bD)$, which is the largest class of reference measures for which a meaningful notion of Cauchy-Leray measure may be defined. We characterize boundedness and compactness in $L^2(bD, Ω_2)$ of the commutator $\displaystyle{[b,S_{Ω_2}]}$.

math.CV↗
arXiv2025

A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains

We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $Ω\subset \mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in $Ω$ while the other is holomorphic in $\mathbb C \setminus \overlineΩ$ and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain. Uniqueness of the decomposition is elementary in the smooth case, but extending it to the $L^p$ setting relies upon a classical albeit little-known regularity theorem for the holomorphic Hardy space $h^p(bΩ)$ of planar domains for which we provide a new proof that is valid also in higher dimensions. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on $L^p(bΩ)$. These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.

math.CV↗
arXiv2024

Boundary value problems for holomorphic functions on Lipschitz planar domains

We study the $\bar\partial$ equation subject to various boundary value conditions on bounded simply connected Lipschitz domains $D\subset\mathbb C$: for the Dirichlet problem with datum in $L^p(bD, σ)$, this is simply a restatement of the fact that members of the holomorphic Hardy spaces are uniquely and completely determined by their boundary values. Here we identify the maximal data spaces and obtain estimates in the maximal $p$-range for the Dirichlet, Regularity-for-Dirichlet, Neumann, and Robin boundary conditions for $\bar\partial$.

math.CV↗
arXiv2024

New Properties of Holomorphic Sobolev-Hardy Spaces

We give new characterizations of the optimal data space for the $L^p(bD,σ)$-Neumann boundary value problem for the $\bar{\partial}$ operator associated to a bounded, Lipschitz domain $D\subset\mathbb{C}$. We show that the solution space is embedded (as a Banach space) in the Dirichlet space and that for $p=2$, the solution space is a reproducing kernel Hilbert space.

math.CV↗
arXiv2023

A transform pair for bounded convex planar domains

A new transform pair which can be used to solve mixed boundary value problems for Laplace's equation and the complex Helmholtz equation in bounded convex planar domains is presented. This work is an extension of Crowdy (2015, CMFT, 15, 655--687) where new transform techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the method is the analysis of the so called global relation which provides a coupling of integral transforms of the given boundary data and of the unknown boundary values. Three problems which involve mixed boundary conditions are solved in detail, as well as numerically implemented, to illustrate how to apply the new approach.

math.CV↗
arXiv2023

The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

Consider a bounded, strongly pseudoconvex domain $D\subset \mathbb C^n$ with minimal smoothness (namely, the class $C^2$) and let $b$ be a locally integrable function on $D$. We characterize boundedness (resp., compactness) in $L^p(D), p > 1$, of the commutator $[b, P]$ of the Bergman projection $P$ in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on $b$. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

math.CV↗
arXiv2021

Symmetrization of a family of Cauchy-Like kernels: Global instability

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its $L^2$ boundedness. In particular, a famous proof of Melnikov-Verdera [18] relies upon an iconic symmetrization identity of Melnikov [17] linking the universal Cauchy kernel $K_0$ to Menger curvature. Analogous identities hold for the real and the imaginary parts of $K_0$ as well. Such connections have been immensely productive in the study of singular integral operators and in geometric measure theory. \vskip0.1in In this article, given any function $h: \mathbb C \rightarrow \mathbb R$, we consider an inhomogeneous variant $K_h$ of $K_0$ which is inspired by complex function theory. While an operator with integration kernel $K_h$ is easily seen to be $L^2$-bounded for all $h$, the symmetrization identities for each of the real and imaginary parts of $K_h$ show a striking lack of robustness in terms of boundedness and positivity, two properties that were critical in [18] and in subsequent works by many authors. Indeed here we show that for any continuous $h$ on $\mathbb C$, the only member of $\{K_h\}_h$ whose symmetrization has the right properties is $K_0$! This global instability complements our previous investigation [12] of symmetrization identities in the restricted setting of a curve, where a sub-family of $\{K_h\}_h$ displays very different behaviour than its global counterparts considered here. Our methods of proof have some overlap with techniques in recent work of Chousionis-Prat [5] and Chunaev [6].

math.CV↗
arXiv2021

Symmetrization of a Cauchy-like kernel on curves

Given a curve $Γ\subset \mathbb C$ with specified regularity, we investigate boundedness and positivity for a certain three-point symmetrization of a Cauchy-like kernel $K_Γ$ whose definition is dictated by the geometry and complex function theory of the domains bounded by $Γ$. Our results show that $\mathtt S[\text{Re} K_Γ]$ and $\mathtt S[\text{Im} K_Γ]$ (namely, the symmetrizations of the real and imaginary parts of $K_Γ$) behave very differently from their counterparts for the Cauchy kernel previously studied in the literature. For instance, the quantities $\mathtt S[\text{Re} K_Γ](\mathbf z)$ and $\mathtt S[\text{Im} K_Γ](\mathbf z)$ can behave like $\frac32c^2(\mathbf z)$ and $-\frac12c^2(\mathbf z)$, where $\mathbf z$ is any three-tuple of points in $Γ$ and $c(\mathbf z)$ is the Menger curvature of $\mathbf z$. For the original Cauchy kernel, an iconic result of M. Melnikov gives that the symmetrized forms of the real and imaginary parts are each equal to $\frac12c^2(\mathbf z)$ for all three-tuples in $\mathbb C$.

math.CV↗
arXiv2021

Hardy Spaces for a Class of Singular Domains

We set a framework for the study of Hardy spaces inherited by complements of analytic hypersurfaces in domains with a prior Hardy space structure. The inherited structure is a filtration, various aspects of which are studied in specific settings. For punctured planar domains, we prove a generalization of a famous rigidity lemma of Kerzman and Stein. A stabilization phenomenon is observed for egg domains. Finally, using proper holomorphic maps, we derive a filtration of Hardy spaces for certain power-generalized Hartogs triangles, although these domains fall outside the scope of the original framework.

math.CV↗
arXiv2021

Regularity of a $\bar\partial$-solution operator for strongly $\mathbf C$-linearly convex domains with minimal smoothness

We prove regularity of solutions of the $\bar\partial$-problem in the Hölder-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the $\bar\partial$-problem on strongly pseudoconvex domains of class $C^2$.

math.CV↗
arXiv2017

The role of an integration identity in the analysis of the Cauchy-Leray transform

The purpose of this paper is to complement the results in [LS-1] by showing the dense definability of the Cauchy-Leray transform for the domains that give the counterexamples of [LS-1], where $L^p$-boundedness is shown to fail when either the "near" $C^2$ boundary regularity, or the strong $\mathbb C$-linear convexity assumption is dropped.

math.CV↗
arXiv2017

The Cauchy-Leray integral: counter-examples to the $L^p$-theory

We prove the optimality of the hypotheses guaranteeing the $L^p$-boundedness for the Cauchy-Leray integral in $\mathbb C^n$, $n\geq 2$, obtained in [LS-4]. Two domains, both elementary in nature, show that the geometric requirement of strong $\mathbb C$-linear convexity, together with regularity of order 2, are both necessary.

math.CV↗
arXiv2015

Harmonic Analysis Techniques in Several Complex Variables

We give a survey of recent joint work with E. M. Stein (Princeton University) concerning the application of suitable versions of the T(1)-theorem technique to the study of orthogonal projections onto the Hardy and Bergman spaces of holomorphic functions for domains with minimal boundary regularity.

math.CV↗
arXiv2015

On Div-Curl for Higher Order

We present new examples of complexes of differential operators of order $k$ (any given positive integer) that satisfy div-curl and/or $L^1$-duality estimates.

math.AP↗