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Peter Sjögren

Publications and source records attributed to Peter Sjögren.

31 records · Page 2Linked to original sources
arXiv2016

Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions

Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain parameters of type that are usually restricted to a half-line, or a product of half-lines if higher dimensions are considered. Following earlier research done by Hajmirzaahmad, we deal in this paper with Laplacians in the above-mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the classical ones, but are different. This reveals new frameworks related to classical orthogonal expansions and thus new potentially rich research areas, at least from the harmonic analysis perspective. To support the last claim we focus on maximal operators of multi-dimensional Laguerre, Bessel and Jacobi semigroups, with unrestricted type parameters, and prove that they satisfy weak type (1,1) estimates with respect to the appropriate measures. Generally, these measures are not (locally) finite, which makes a contrast with the classical situations and generates new difficulties. An important partial result of the paper is a new proof of the weak type (1,1) estimate for the classical multi-dimensional Laguerre semigroup maximal operator.

math.CA↗
arXiv2015

Wave extension problem for the fractional Laplacian

We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schrödinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.

math.AP↗
arXiv2015

On the global Gaussian Lipschitz space

A Lipschitz space is defined in the Ornstein-Uhlenbeck setting, by means of a bound for the gradient of the Ornstein-Uhlenbeck Poisson integral. This space is then characterized with a Lipschitz-type continuity condition. These functions turn out to have at most logarithmic growth at infinity. The analogous Lipschitz space containing only bounded functions was introduced by Gatto and Urbina and has been characterized by the authors in \cite{LS}.

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arXiv2012

Analysis related to all admissible type parameters in the Jacobi setting

We derive an integral representation for the Jacobi-Poisson kernel valid for all admissible type parameters $α,β$ in the context of Jacobi expansions. This enables us to develop a technique for proving standard estimates in the Jacobi setting, which works for all possible $α$ and $β$. As a consequence, we can prove that several fundamental operators in the harmonic analysis of Jacobi expansions are (vector-valued) Calderón-Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory. The new Jacobi-Poisson kernel representation also leads to sharp estimates of this kernel. The paper generalizes methods and results existing in the literature, but valid or justified only for a restricted range of $α$ and $β$.

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arXiv2011

Bounds for maximal functions associated to rotational invariant measures in high dimensions

In recent articles it was proved that when $μ$ is a finite, radial measure in $\real^n$ with a bounded, radially decreasing density, the $L^p(μ)$ norm of the associated maximal operator $M_μ$ grows to infinity with the dimension for a small range of values of $p$ near 1. We prove that when $μ$ is Lebesgue measure restricted to the unit ball and $p<2$, the $L^p$ operator norms of the maximal operator are unbounded in dimension, even when the action is restricted to radially decreasing functions. In spite of this, this maximal operator admits dimension-free $L^p$ bounds for every $p>2$, when restricted to radially decreasing functions. On the other hand, when $μ$ is the Gaussian measure, the $L^p$ operator norms of the maximal operator grow to infinity with the dimension for any finite $p> 1$, even in the subspace of radially decreasing functions.

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arXiv2011

Sharp estimates of the Jacobi heat kernel

The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal polynomials. We deduce sharp estimates giving the order of magnitude of this kernel, for type parameters $α,β\ge -1/2$. As an application of the upper bound obtained, we show that the maximal operator of the multi-dimensional Jacobi heat semigroup satisfies a weak type $(1,1)$ inequality. We also obtain sharp estimates of the Poisson-Jacobi kernel.

math.CA↗
arXiv2011

Heat maximal function on a Lie group of exponential growth

Let G be the Lie group R^2\rtimes R^+ endowed with the Riemannian symmetric space structure. Let X_0, X_1, X_2 be a distinguished basis of left-invariant vector fields of the Lie algebra of G and define the Laplacian Δ=-(X_0^2+X_1^2+X_2^2). In this paper, we show that the maximal function associated with the heat kernel of the Laplacian Δis bounded from the Hardy space H^1 to L^1. We also prove that the heat maximal function does not provide a maximal characterization of the Hardy space H^1.

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arXiv2010

Calderón-Zygmund operators related to Jacobi expansions

We study several fundamental operators in harmonic analysis related to Jacobi expansions, including Riesz transforms, imaginary powers of the Jacobi operator, the Jacobi-Poisson semigroup maximal operator and Littlewood-Paley-Stein square functions. We show that these are (vector-valued) Calderón-Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory. Our proofs rely on an explicit formula for the Jacobi-Poisson kernel, which we derive from a product formula for Jacobi polynomials.

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arXiv2009

On the boundary convergence of solutions to the Hermite-Schrödinger equation

In the half-space $\mathbb{R}^d \times \mathbb{R}_+$, we consider the Hermite-Schrödinger equation $i\partial u/\partial t = - Δu + |x|^2 u$, with given boundary values on $\mathbb{R}^d$. We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary, by means of this link.

math.AP↗
arXiv2007

Boundedness from H^1 to L^1 of Riesz transforms on a Lie group of exponential growth

Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$ endowed with the Riemannian symmetric space structure. Let $X_0, X_1, X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $Δ=-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_iΔ^{-1/2}$ and $S_i=Δ^{-1/2}X_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second order Riesz transforms $T_{ij}=X_iΔ^{-1}X_j$ are bounded from $H^1$ to $L^1$, while the Riesz transforms $S_{ij}=Δ^{-1}X_iX_j$ and $R_{ij}=X_iX_jΔ^{-1}$ are not.

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