arXiv subjects
Publications and source records attributed to Alessandro Monguzzi.
In this paper we discuss some recent results concerning the regularity and irregularity of the Bergman and Szegő projections on some weakly pseudoconvex domains that have the common feature to possess a nontrivial Nebenhülle.
We study the fractional Laplacian and the homogeneous Sobolev spaces on R^d , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way we also prove some properties of the fractional Laplacian.
We study properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.
In this paper we characterize the closed invariant subspaces for the ($*$-)multiplier operator of the quaternionic space of slice $L^2$ functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy space on the unit ball and we provide a characterization of quaternionic outer functions in terms of cyclicity.
In this paper we study the regularity of the Szegő projection on Lebesgue and Sobolev spaces on the distinguished boundary of the unbounded model worm domain $D_β$. We denote by $d_b(D_β)$ the distinguished boundary of $D_β$ and define the corresponding Hardy space $\mathscr{H}^2(D_β)$. This can be identified with a closed subspace of $L^2(d_b(D_β),dσ)$, that we denote by $\mathscr{H}^2(d_b(D_β))$, where $dσ$ is the naturally induced measure on $d_b(D_β)$. The orthogonal Hilbert space projection $\mathscr{P}: L^2(d_b(D_β),dσ)\to \mathscr{H}^2(d_b(D_β))$ is called the Szegő projection on the distinguished boundary. We prove that $\mathscr{P}$, initially defined on the dense subspace $L^2(d_b( D_β),dσ)\cap L^p(d_b(D_β), dσ)$ extends to a bounded operator $\mathscr{P}: L^p(d_b(D_β), dσ)\to L^p(d_b(D_β), dσ)$ if and only if $\textstyle{\frac{2}{1+ν_β}} π$. Furthermore, we also prove that $\mathscr{P}$ defines a bounded operator $\mathscr{P}: W^{s,2}(d_b(D_β),dσ)\to W^{s,2}(d_b(D_β), dσ)$ if and only if $0\leq s<\textstyle{\frac{ν_β}{2}}$ where $W^{s.2}(d_b( D_β), dσ)$ denotes the Sobolev space of order $s$ and underlying $L^2$-norm. Finally, we prove a necessary condition for the boundedness of $\mathscr{P}$ on $W^{s,p}(d_b(D_β), dσ)$, $p\in(1,\infty)$, the Sobolev space of order $s$ and underlying $L^p$-norm.
In this paper we study the regularity of the Szegö projection on Lebesgue and Sobolev spaces on the boundary of the unbounded model worm domain $D'_β$. We consider the Hardy space $H^2(D'_β)$. Denoting by $bD'_β$ the boundary of $D'_β$, it is classical that $H^2(D'_β)$ can be identified with the closed subspace of $L^2(bD'_β,dσ)$, denoted by $H^2(bD'_β)$, consisting of the boundary values of functions in $H^2(D'_β)$, where $dσ$ is the induced Lebesgue measure. The orthogonal Hilbert space projection $P: L^2(D'_β,dσ)\to H^2(bD'_β)$ is called the Szegö projection. Let $W^{s,p}(bD'_β)$ denote the Lebesgue--Sobolev space on $bD'_β$. We prove that $P$, initially defined on the dense subspace $W^{s,p}(bD'_β)\cap L^2(bD'_β,dσ)$, extends to a bounded operator $P: W^{s,p}(bD'_β)\to W^{s,p}(bD'_β)$, for $1<p<\infty$ and $s\ge0$.
In this work we provide an asymptotic expansion for the Szegő kernel associated to a suitably defined Hardy space on the the non-smooth worm domain $D'_β$. After describing the singularities of the kernel, we compare it with an asymptotic expansion of the Bergman kernel. In particular, we show that the Bergman kernel has the same singularities of the first derivative of the Szegő kernel with respect to any of the variables. On the side, we prove the boundedness of the Bergman projection operator on Sobolev spaces of integer order.
We define Hardy spaces $H^p(D'_β)$ on the non-smooth worm domain $D'_β=\{(z_1,z_2)\in\mathbb{C}^2:|Im z_1-\log |z_2|^2|<\fracπ{2}, |\log |z_2|^2|<β-\fracπ{2}\}$ and we prove a series of related results such as the existence of boundary values on the distinguished boundary $\partial D'_β$ of the domain and a Fatou-type theorem (i.e. pointwise convergence to the boundary values). Thus, we study the Szegő projection operator $\widetilde{S}$ and the associated Szegő kernel $K_{D'_β}$. More precisely, if $H^p(\partial D'_β)$ denotes the space of functions which are boundary values for functions in $H^p(D'_β)$, we prove that the operator $\widetilde{S}$ extends to a bounded linear operator $$ \widetilde{S}: L^p(\partial D'_β)\to H^p(\partial D'_β) $$ for every $p\in(1,+\infty)$ and $$ \widetilde{S}: W^{k,p}(\partial D'_β)\to W^{k,p}(\partial D'_β) $$ for every $k>0$. Here $W^{k,p}$ denotes the Sobolev space of order $k$ and underlying $L^p$ norm. As a consequence of the $L^p$ boundedness of $\widetilde{S}$, we prove that $H^p(D'_β)\cap\mathcal{C}(\overline{D'_β})$ is a dense subspace of $H^p(D'_β)$.
In this work we prove a new $L^p$ holomorphic extension result for functions defined on product Lipschitz surfaces with small Lipschitz constants in two complex variables. We define biparameter and partial Cauchy integral operators that play the role of boundary values for holomorphic functions on product Lipschitz domain. In the spirit of the application of David-Journé-Semmes and Christ's $Tb$ theorem to the Cauchy integral operator, we prove a biparameter $Tb$ theorem and apply it to prove $L^p$ space bounds for the biparameter Cauchy integral operator. We also prove some new biparameter Littlewood-Paley-Stein estimates and use them to prove the biparameter $Tb$ theorem.