Riesz transforms for Jacobi expansions

We define and study Riesz transforms and conjugate Poisson integrals associated with multi-dimensional Jacobi expansions.Comment: 24 pages; the paper will appear in J. Anal. Math. (2008

Pope Francis and Ignatian Discernment

A chapter written by Peter Bernardi SJ called Pope Francis and Ignatian Discernment in the book New World Pope: Pope Francis and the Future of the Church, edited by Michael L. Budde

Maurice Blondel: Precursor of the Second Vatican Council

The ecclesial renewal promoted by the Second Vatican Council would not have been possible without the intrepid labors of Catholic theologians, philosophers, and pastors in the decades preceding the Council. Among these Catholic visionaries, the French Catholic philosopher Maurice Blonde! (1861-1949) rnerits special recognition. He had a decisive impact on ressourcement theologians who helped to draft the conciliar documents. Blondel\u27s influence is especially evident in their critique of conceptual rationalism that marked the pre-conciliar scholastic manuals, in the articulation of a richer understanding of the Church\u27s tradition, and in the Council\u27s teaching of a unitary human destiny. Blondel\u27s thought contributed to the overcoming of a one-sided siege mentality with which the Church had responded to modern developments. Though Blondel was long suspect among certain scholastics, the Council implicitly vindicated key aspects of his \u27\u27philosophy of action. Blonde! is appropriately called the philosopher of Vatican 11

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 $\Delta_{v} = \Delta + 2v\cdot \nabla$ and the measure $d\mu(x) = e^{2 \langle v, x \rangle} dx$, with respect to which $\Delta_{v}$ is self-adjoint. %Let $d$ and $\nabla$ denote the Euclidean distance and the gradient operator on $\R^n$. Consider the space $(\R^n, d,d\mu)$, which has the property of exponential volume growth. 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

Estimates for operators related to the sub-Laplacian with drift in Heisenberg groups

In the Heisenberg group of dimension 2n+1, we consider the sub-Laplacian witha drift in the horizontal coordinates. There is a related measure for whichthis operator is symmetric.The corresponding Riesz transforms are known to be L^p boundedwith respect to this measure.We prove that the Riesz transforms of order 1 are also of weak type (1,1),and that this is false for order 3 and above. Further, we consider the relatedmaximal Littlewood-Paley-Stein operators and prove the weak type (1,1) forthose of order 1 and disprove it for higher orders

On the variation operator for the Ornstein–Uhlenbeck semigroup in dimension one

Consider the variation seminorm of the Ornstein–Uhlenbeck semigroup Ht in dimension one, taken with respect to t. We show that this seminorm defines an operator of weak type (1,\ua01) for the relevant Gaussian measure. The analogous Lp estimates for 1 < p< ∞ were already known

Riesz transforms of a general Ornstein--Uhlenbeck semigroup

We consider Riesz transforms of any order associated to an Ornstein--Uhlenbeck operator \ue238, 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

On the maximal operator of a general Ornstein–Uhlenbeck semigroup

If Q is a real, symmetric and positive definite n 7 n matrix, and B a real n 7 n matrix whose eigenvalues have negative real parts, we consider the Ornstein–Uhlenbeck semigroup on Rn with covariance Q and drift matrix B. Our main result says that the associated maximal operator is of weak type (1,\ua01) 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

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 (- | x1| - ⋯ - | xd|) dx in Rd. The mean values are taken over Euclidean balls or cubes (ℓ∞ balls) or diamonds (ℓ1 balls). Assuming that d≥ 2 , in the cases of cubes and diamonds we prove the Lp-boundedness for p> 1 and disprove the weak type (1,\ua01) estimate. The same is proved in the case of Euclidean balls, under the restriction d≤ 4 for the positive part

Byggnadsteknik och timmermanskonst: en studie med exempel fr\ue5n n\ue5gra medeltida knuttimrade kyrkor och allmogehus

This survey deals with building techniques based on the use of horizontally laid logs with interlocking corner-joints. The aim is to make an inventory of buildings methods and materials, based on existing medieval buildings, literary sources and current methods of carpentry.Chapter 1. A commentary on research and documentation. Methods of documentation in research on historical building techniques are a central question. These methods must be based on the idea of buildings as historical sources. In Sweden we have a lot to learn from the long traditions of research in Norway and Denmark. Methods of documentation in these countries have developed into effective scientific working models.Chapter 2. This chapter deals with particular buildings which have been researched by the author.Chapter 3. Medieval timber construction with corner-joints. This chapter is mainly based on the authors documentation of wooden buildings and different details is these buildings. The intention has been to understand how and why the different parts of the buildings were made in the way they were. In order to illustrate possible historical working methods the author shows how carpenters work today. These analogies must however be regarded as hypothetical.Chapter 4. Former use of timber for buildings work and its preparation before the introduction of the saw. This chapter deals with questions regarding the material properties of building timber with reference to the choice of timber in the buildings in question. The main sources are however from the 18th, 19th and 20th centuries. In order to form an idea as to how relevant a historical knowledge of timber can be, older beliefs on timber have been compared with modern knowledge. This chapter concludes with a short survey of older methods of making beams, planks, and boards