arXiv subjects
Publications and source records attributed to Marco M. Peloso.
We prove a local $L^p$-Poincaré inequality, $1\leq p < \infty$, on noncompact Lie groups endowed with a sub-Riemannian structure. We show that the constant involved grows at most exponentially with respect to the radius of the ball, and that if the group is nondoubling, then its growth is indeed, in general, exponential. We also prove a nonlocal $L^2$-Poincaré inequality with respect to suitable finite measures on the group.
In this paper we study Toeplitz and Cesàro-type operators on holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions for these operators to be continuous or compact, or to belong to suitable Schatten classes.
In this paper we study Carleson and reverse Carleson measures on holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions in order for a measure $μ$ to be Carleson and reverse Carleson on mixed-normed weighted Bergman spaces.
In this paper we study the Bergman kernel and projection on the unbounded worm domain $$ \mathcal{W}_\infty = \big\{(z_1,z_2)\in\mathbb{C}^2 : \big|z_1-e^{i\log|z_2|^2}\big|^2<1, z_2\neq0\big\}. $$ We first show that the Bergman space of $\mathcal{W}_\infty$ is infinite dimensional. Then we study Bergman kernel $K$ and Bergman projection $\mathcal{P}$ for $\mathcal{W}_\infty$. We prove that $K(z,w)$ extends holomorphically in $z$ (and antiholomorphically in $w$) near each point of the boundary except for a specific subset that we study in detail. By means of an appropriate asymptotic expansion for $K$, we prove that the Bergman projection $\mathcal{P}:W^s\not\to W^s$ if $s>0$ and $\mathcal{P}:L^p\not\to L^p$ if $p\neq2$, where $W^s$ denotes the classic Sobolev space, and $L^p$ the Lebesgue space, respectively, on $\mathcal{W}_\infty$.
In this paper we are concerned with the problem of completeness in the Bergman space of the worm domain $\mathcal{W}_μ$ and its truncated version $\mathcal{W}'_μ$. We determine some orthogonal systems and show that they are not complete, while showing that the union of two particular of such systems is complete. In order to prove our completeness result we introduce the Muentz-Szasz problem for the 1-dimensional Bergman space of the disk $\{ζ: |ζ-1|<1\}$ and find a sufficient condition for its solution.
We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley-Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type $a$ whose restriction to the real line belongs to the homogeneous Sobolev space $\dot{W}^{s,p}$ and we call these spaces fractional Paley-Wiener if $p=2$ and fractional Bernstein spaces if $p\in(1,\infty)$, that we denote by $PW^s_a$ and $\mathcal B^{s,p}_a$, respectively. For these spaces we provide a Paley-Wiener type characterization, we remark some facts about the sampling problem in the Hilbert setting and prove generalizations of the classical Bernstein and Plancherel-Pólya inequalities. We conclude by discussing a number of open questions.
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.
In this paper we establish an optimal Lorentz estimate for the Riesz potential in the $L^1$ regime in the setting of a stratified group $G$: Let $Q\geq 2$ be the homogeneous dimension of $G$ and $\mathcal{I}_α$ denote the Riesz potential of order $α$ on $G$. Then, for every $α\in (0,Q)$, there exists a constant $C=C(α,Q)>0$ such that \begin{align} \| \mathcal{I}_αf \|_{L^{Q/(Q-α),1}(G)} \leq C\| X \mathcal{I}_1 f \|_{L^1(G)} \end{align} for distributions $f$ such that $X \mathcal{I}_1 f \in L^1(G)$, where $X$ denotes the horizontal gradient.
In this paper we discuss function spaces on a general noncompact Lie group, namely the scales of Triebel--Lizorkin and Besov spaces, defined in terms of a sub-Laplacian with drift. The sub-Laplacian is written as negative the sum of squares of a collection of left-invariant vector fields satisfying Hörmander's condition. These spaces were recently introduced by the authors. In this paper we prove a norm characterization in terms of finite differences, the density of test functions, and related isomorphism properties.
In this paper we develop a theory of Besov and Triebel--Lizorkin spaces on general noncompact Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.
Let G be a noncompact connected Lie group and $ρ$ be the right Haar measure of G. Let $X_1,...,X_q$ be a family of left invariant vector fields which satisfy Hörmander's condition, and let $Δ=-\sum_{i=1}^qX_i^2$ be the corresponding subLaplacian. For $1\leq p<\infty$ and $α\geq 0$ we define the Sobolev space $L^p_α(G)={f in L^p(ρ): Δ^{α/2}f\in L^p(ρ) }$, endowed with the norm $\|f\|_{α,p}=\|f\|_{p}+\|Δ^{α/2}f\|_p$, where we denote by $\|f\|_p$ the norm of $f$ in $L^p(ρ)$. In this paper we show that for all $α\geq 0$ and $p\in (1,\infty)$, the space $L^{\infty}\cap L^p_α(G)$ is an algebra under pointwise product. Such result was proved by T. Coulhon, E. Russ and V. Tardivel-Nachef in the case when G is unimodular. We shall prove it on Lie groups, thus extending their result to the nonunimodular case. In order to prove our main result, we need to study the boundedness of local Riesz transforms $R^c_J=X_J(cI+Δ)^{-m/2}$, where c>0, $X_J=X_{j_1}...X_{j_m}$ and $j_\ell \in\{1,\dots,q\}$ for $\ell=1,...,m$. We show that if c is sufficiently large, the Riesz transform R^c_J is bounded on $L^p(ρ)$ for every $p\in (1,\infty)$, and prove also appropriate endpoint results involving Hardy and BMO spaces.
Let $G$ be a noncompact connected Lie group, denote with $ρ$ a right Haar measure and choose a family of linearly independent left-invariant vector fields $\mathbf{X}$ on $G$ satisfying Hörmander's condition. Let $χ$ be a positive character of $G$ and consider the measure $μ_χ$ whose density with respect to $ρ$ is $χ$. In this paper, we introduce Sobolev spaces $L^p_α(μ_χ)$ adapted to $\mathbf{X}$ and $μ_χ$ ($1<p<\infty$, $α\geq 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.
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 paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_ω$, whose growth conditions are given in terms of a translation invariant measure $ω$ on the closed half-plane $\overline\cal R$. Such a measure has the form $ω=ν\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and $ν$ is a regular Borel measure on $[0,+\infty)$. We call these spaces generalized Hardy-Bergman spaces on the half-plane $\cal R$. We study in particular the case of $ν$ purely atomic, with point masses on an arithmetic progression on $[0,+\infty)$. We obtain a Paley-Wiener theorem for $\cal M^2_ω$, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that $\cal M^p_ω$ contains functions of order 1. Moreover, we prove that the orthogonal projection from $L^p(\cal R,dω)$ into $\cal M^p_ω$ is unbounded for $p\neq2$. Furthermore, we compare the spaces $\cal M^p_ω$ with the classical Hardy and Bergman spaces, and some other Hardy-Bergman-type spaces introduced more recently.
We introduce and study some new spaces of holomorphic functions on the right half-plane. In a previous work, S. Krantz, C. Stoppato and the first named author formulated the M"untz--Sz'asz problem for the Bergman space, that is, the problem to characterize the sets of complex powers that form a complete set the unweighted Bergman space of a disc. In this paper, we construct a space of holomorphic functions on the right half-plane, whose sets of uniqueness correspond exactly to the sets of powers that are a complete set in Bergman space. We show that this space is a reproducing kernel Hilbert space and we prove a Paley--Wiener type theorem among several other structural properties. Moreover, we determine a sufficient condition on a set of powers to be a set of uniqueness for this space, thus providing a sufficient condition for the solution of the M"untz--Sz'asz problem for the Bergman space.
We study the nonlinear Schrödinger equation associated with the sublaplacian L on the unit sphere $S^{2n+1}$ in $C^{n+1}$ equipped with its natural CR structure. We first prove Strichartz estimates with fractional loss of derivatives for the solutions of the free Schrödinger equation and we then deduce some local in time well-posedness results. Our results are stated in terms of certain Sobolev-type spaces, that measure the regularity of functions on $S^2n+1$ differently according to their spectral localization. Stronger conclusions are obtained for particular classes of solutions, corresponding to initial data whose spectrum is contained in a proper cone of $N^2$.