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Peter Sjögren

Publications and source records attributed to Peter Sjögren.

At least 19 recordsLinked to original sources
arXiv2026

Sharp variational, jump and oscillation bounds in a general Gaussian context

We consider a general, nonsymmetric Ornstein--Uhlenbeck semigroup $(\mathcal H_t)_{t>0}$. We prove an $L^p$ bound for the jump quasi-seminorms for $1 < p < \infty$ and a weak type (1,1) oscillation inequality, both with respect to the invariant measure. These results are established for the order $\varrho=2$. To do so, we analyze specific components of $(\mathcal H_t)_{t>0}$, by distinguishing between small and large values of $t$, and between local and global spatial zones. This decomposition allows us to explicitly identify which parts of the semigroup remain bounded and which are responsible for the failure of boundedness, both in a weak and in a strong sense, even with respect to Lebesgue measure.

math.FA↗
arXiv2026

Weak type (1,1) jump inequalities in a nonsymmetric Gaussian setting

We prove that the jump quasi-seminorm of order $\varrho= 2$ for a general Ornstein--Uhlenbeck semigroup $\left(\mathcal H_t\right)_{t>0}$ in $\mathbb R^n$ defines an operator of weak type $(1,1)$ with respect to the invariant measure. This provides an example of a weak-type jump inequality for a nonsymmetric semigroup in a nondoubling measure space. Our result may be seen as an endpoint refinement of the weak type $(1,1)$ inequality for the $\varrho$-th order variation seminorm of $\left(\mathcal H_t\right)_{t>0}$, recently proved by the authors when $\varrho>2$, and disproved for $\varrho=2$.

math.FA↗
arXiv2025

Variational inequalities for the Ornstein--Uhlenbeck semigroup: the higher--dimensional case

We study the $\varrho$-th order variation seminorm of a general Ornstein--Uhlenbeck semigroup $\left(\mathcal H_t\right)_{t>0}$ in $\mathbb R^n$, taken with respect to $t$. We prove that this seminorm defines an operator of weak type $(1,1)$ with respect to the invariant measure when $\varrho> 2$. For large $t$, one has an enhanced version of the standard weak-type $(1,1)$ bound. For small $t$, the proof hinges on vector-valued Calderón--Zygmund techniques in the local region, and on the fact that the $t$ derivative of the integral kernel of $\mathcal H_t$ in the global region has a bounded number of zeros in $(0,1]$. A counterexample is given for $\varrho= 2$; in fact, we prove that the second order variation seminorm of $\left(\mathcal H_t\right)_{t>0}$, and therefore also the $\varrho$-th order variation seminorm for any $\varrho\in [1,2)$, is not of strong nor weak type $(p,p)$ for any $p \in [1,\infty)$ with respect to the invariant measure.

math.FA↗
arXiv2025

Boundedness properties of the maximal operator in a nonsymmetric inverse Gaussian setting

We introduce a generalized inverse Gaussian setting and consider the maximal operator associated with the natural analogue of a nonsymmetric Ornstein--Uhlenbeck semigroup. We prove that it is bounded on $L^{p}$ when $p\in (1,\infty]$ and that it is of weak type $(1,1)$, with respect to the relevant measure. For small values of the time parameter $t$, the proof hinges on the "forbidden zones" method previously introduced in the Gaussian context. But for large times the proof requires new tools.

math.FA↗
arXiv2024

Hardy spaces and Riesz transforms on a Lie group of exponential growth

Let $G$ be the Lie group ${\Bbb{R}}^2\rtimes {\Bbb{R}}^+$ endowed with the Riemannian symmetric space structure. Take a distinguished basis $X_0,\, X_1,\,X_2$ of left-invariant vector fields of the Lie algebra of $G$, and consider the Laplacian $Δ=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_iΔ^{-1/2}$, \hskip3pt $i=0,1,2$. We first show that the atomic Hardy space $H^1$ in $G$ introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms $\mathcal R_i$. It is also proved that two of these Riesz transforms are bounded from $H^1$ to $H^1$.

math.FA↗
arXiv2023

Spectral multipliers in a general Gaussian setting

We investigate a class of spectral multipliers for an Ornstein-Uhlenbeck operator $\mathcal L$ in $\mathbb R^n$, with drift given by a real matrix $B$ whose eigenvalues have negative real parts. We prove that if $m$ is a function of Laplace transform type defined in the right half-plane, then $m(\mathcal L)$ is of weak type $(1, 1)$ with respect to the invariant measure in $\mathbb R^n$. The proof involves many estimates of the relevant integral kernels and also a bound for the number of zeros of the time derivative of the Mehler kernel, as well as an enhanced version of the Ornstein-Uhlenbeck maximal operator theorem.

math.FA↗
arXiv2023

On non-centered maximal operators related to a non-doubling and non-radial exponential measure

We investigate mapping properties of non-centered Hardy-Littlewood maximal operators related to the exponential measure $dμ(x) = \exp(-|x_1|-\ldots-|x_d|)dx$ in $\mathbb{R}^d$. The mean values are taken over Euclidean balls or cubes ($\ell^{\infty}$ balls) or diamonds ($\ell^1$ balls). Assuming that $d \ge 2$, in the cases of cubes and diamonds we prove the $L^p$-boundedness for $p > 1$ and disprove the weak type $(1,1)$ estimate. The same is proved in the case of Euclidean balls, under the restriction $d \le 4$ for the positive part.

math.CA↗
arXiv2021

Riesz transforms of a general Ornstein--Uhlenbeck semigroup

We consider Riesz transforms of any order associated to an Ornstein--Uhlenbeck operator $\mathcal L$, with covariance $Q$ given by a real, symmetric and positive definite matrix, and with drift $B$ given by a real matrix whose eigenvalues have negative real parts. In this general Gaussian context, we prove that a Riesz transform is of weak type $(1,1)$ with respect to the invariant measure if and only if its order is at most $2$.

math.FA↗
arXiv2021

On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator

We study the orthogonality of the generalized eigenspaces of an Ornstein--Uhlenbeck operator $\mathcal L$ in $\mathbb{R}^N$, with drift given by a real matrix $B$ whose eigenvalues have negative real parts. If $B$ has only one eigenvalue, we prove that any two distinct generalized eigenspaces of $\mathcal L$ are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if $B$ admits distinct eigenvalues, the generalized eigenspaces of $\mathcal L$ may or may not be orthogonal.

math.FA↗
arXiv2021

On the Riesz Transforms for the inverse Gauss measure

Let $γ_{-1}$ be the absolutely continuous measure on $\mathbb{R}^n$ whose density is the reciprocal of a Gaussian function. Let further $\mathscr{A}$ be the natural self-adjoint Laplacian on $L^2(γ_{-1})$. In this paper, we prove that the Riesz transforms associated with $\mathscr{A}$ of order one or two are of weak type $(1,1)$, but that those of higher order are not.

math.FA↗
arXiv2020

On the maximal operator of a general Ornstein-Uhlenbeck semigroup

If $Q$ is a real, symmetric and positive definite $n\times n$ matrix, and $B$ a real $n\times n$ matrix whose eigenvalues have negative real parts, we consider the Ornstein--Uhlenbeck semigroup on $\mathbb{R}^n$ with covariance $Q$ and drift matrix $B$. Our main result says that the associated maximal operator is of weak type $(1,1)$ with respect to the invariant measure. The proof has a geometric gist and hinges on the "forbidden zones method" previously introduced by the third author.

math.FA↗
arXiv2019

Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.

math.CA↗
arXiv2019

The maximal operator of a normal Ornstein--Uhlenbeck semigroup is of weak type $(1,1)$

Consider a normal Ornstein--Uhlenbeck semigroup in $\Bbb{R}^n$, whose covariance is given by a positive definite matrix. The drift matrix is assumed to have eigenvalues only in the left half-plane. We prove that the associated maximal operator is of weak type $(1,1)$ with respect to the invariant measure. This extends earlier work by G. Mauceri and L. Noselli. The proof goes via the special case where the matrix defining the covariance is $I$ and the drift matrix is diagonal.

math.FA↗
arXiv2018

Analysis in the multi-dimensional ball

We study the heat semigroup maximal operator associated with a well-known orthonormal system in the d-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted $L^p$ estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.

math.CA↗
arXiv2017

Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Let $v \ne 0$ be a vector in $\R^n$. Consider the Laplacian on $\R^n$ with drift $Δ_{v} = Δ+ 2v\cdot \nabla$ and the measure $dμ(x) = e^{2 \langle v, x \rangle} dx$, with respect to which $Δ_{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood-Paley-Stein functions associated with the heat and the Poisson semigroups.

math.CA↗