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Marco M. Peloso

Publications and source records attributed to Marco M. Peloso.

45 records · Page 3Linked to original sources
arXiv2013

Regularity equivalence of the Szegö projection and the complex Green operator

In this paper we prove that on a CR manifold of hypersurface type that satisfies the weak $Y(q)$ condition, the complex Green operator $G_q$ is exactly (globally) regular if and only if the Szegö projections $S_{q-1}, S_q$ and a third orthogonal projection $S'_{q+1}$ are exactly (globally) regular. The projection $S'_{q+1}$ is closely related to the Szegö projection $S_{q+1}$ and actually coincides with it if the space of harmonic $(0,q+1)$-forms is trivial. This result extends the important and by now classical result by H. Boas and E. Straube on the equivalence of the regularity of the $\bar\partial$-Neumann operator and the Bergman projections on a smoothly bounded pseudoconvex domain. We also prove an extension of this result to the case of bounded smooth domains satisfying the weak $Z(q)$ condition on a Stein manifold.

math.CV↗
arXiv2012

Analysis of the Hodge Laplacian on the Heisenberg group

We consider the Hodge Laplacian $Δ$ on the Heisenberg group $H_n$, endowed with a left-invariant and U(n)-invariant Riemannian metric. For $0\le k\le 2n+1$, let $Δ_k$ denote the Hodge Laplacian restricted to $k$-forms. Our first main result shows that $L^2Λ^k(H_n)$ decomposes into finitely many mutually orthogonal subspaces $\V_ν$ with the properties: {itemize} $\dom Δ_k$ splits along the $\V_ν$'s as $\sum_ν(\domΔ_k\cap \V_ν)$; $Δ_k:(\domΔ_k\cap \V_ν)\longrightarrow \V_ν$ for every $ν$; for each $ν$, there is a Hilbert space $\cH_ν$ of $L^2$-sections of a U(n)-homogeneous vector bundle over $H_n$ such that the restriction of $Δ_k$ to $\V_ν$ is unitarily equivalent to an explicit scalar operator. {itemize} Next, we consider $L^pΛ^k$, $1<p<\infty$, and prove that the same kind of decomposition holds true. More precisely we show that: {itemize} the Riesz transforms $dΔ_k^{-\half}$ are $L^p$-bounded; the orthogonal projection onto $\cV_ν$ extends from $(L^2\cap L^p)Λ^k$ to a bounded operator from $L^pΛ^k$ to the the $L^p$-closure $\cV_

math.FA↗
arXiv2008

L^p-summability of Riesz means for the sublaplacian on complex spheres

In this paper we study the L^p-convergence of the Riesz means for the sublaplacian on the sphere S^{2n-1} in the complex n-dimensional space C^n. We show that the Riesz means of order delta of a function f converge to f in L^p(S^{2n-1}) when delta>delta(p):=(2n-1)|1\2-1\p|. The index delta(p) improves the one found by Alexopoulos and Lohoue', $2n|1\2-1\p|$, and it coincides with the one found by Mauceri and, with different methods, by Mueller in the case of sublaplacian on the Heisenberg group.

math.FA↗
arXiv2007

Analysis and geometry on worm domains

We describe recent work on the Bergman kernel of the (non-smooth) worm domain in several complex variables. An asymptotic expansion is obtained for the Bergman kernel. Mapping properties of the Bergman projection are studied. Irregularity properties of the kernal at the boundary are established. This is an expository paper, and considerable background is provided. Discussion of the smooth worm is also included.

math.CV↗
arXiv2007

The Bergman kernel and projection on non-smooth worm domains

This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.

math.CV↗
arXiv2005

L^p-spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group

We prove that, if Δ_1 is the Hodge Laplacian acting on differential 1-forms on the (2n+1)-dimensional Heisenberg group, and if m is a Mihlin-Hörmander multiplier on the positive half-line, with L^2-order of smoothness greater than n+1/2, then m(Δ_1) is L^p-bounded for 1<p<\infty. Our approach leads to an explicit description of the spectral decomposition of Δ_1 on the space of L^2-forms in terms of the spectral analysis of the sub-Laplacian L and the central derivative T, acting on scalar-valued functions.

math.CA↗
arXiv1996

$\overline{\partial}$-Neumann Problem in the Sobolev Topology

We study the $\overline{\partial}$-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev Hodge theory.

math.CV↗
arXiv1996

Hodge theory in the Sobolev topology for the de Rham complex

The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary-value problem belonging to a class of problems first introduced by Višik and Eskin, and by Boutet de Monvel.

math.DG↗