Fourier multipliers in Hilbert spaces

This is a survey on a notion of invariant operators, or Fourier multipliers on Hilbert spaces. This concept is defined with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. In particular this notion can be applied to the important case of $L^2(M)$ where $M$ is a compact manifold $M$ endowed with a positive measure. The partition in this case comes from the spectral properties of a a fixed elliptic operator $E$.Comment: These notes are based on our paper arXiv:1404.6479 (to appear in J. Anal. Math.) and have been prepared for the instructional volume associated to the Summer School on Fourier Integral Operators held in Ouagadougou, Burkina Faso, where the authors took part during 14-26 September 201

Kernel and symbol criteria for Schatten classes and rr-nuclearity on compact manifolds

In this note we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of an invariant operator and its full symbol associated to an elliptic operator are introduced. Some applications to the study of $r$-nuclearity on $L^{p}$ spaces are also obtained.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1404.647

Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups

Given a compact Lie group $G$, in this paper we give symbolic criteria for operators to be nuclear and r-nuclear on Lp-spaces, with applications to distribution of eigenvalues and trace formulae. Since criteria in terms of kernels are often not effective in view of Carleman's example, in this paper we adopt the symbolic point of view. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\hat{G}$, where $\hat{G}$ is the unitary dual of $G$.Comment: This is the second half of our paper arXiv:1303.3914 so there is overlap in the preliminaries sectio

Schatten classes on compact manifolds: Kernel conditions

In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.Comment: 22 page

The bounded approximation property of variable Lebesgue spaces and nuclearity

In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the Grothendieck-Lidskii formula. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ in terms of global symbols.Comment: 18 pages. The paper has been updated by adding a remark (on page 6) on a link between bounded and metric approximation properties. This should be the final version to appear in Math Scan

Introducción a la responsabilidad de los profesionales de Enfermería: Revisión y análisis de sentencias sobre responsabilidad profesional sanitaria

En los últimos años el número de demandas contra el personal de enfermería ha experimentado un importante aumento debido en gran parte a la adquisición de nuevas competencias que ha experimentado la profesión, así como a la disminución del temor que en otras épocas han podido tener determinadas profesiones relacionadas con el ámbito de la salud y a un aumento de las exigencias sociales, que han hecho que las profesiones del ámbito de la sanidad estén, en muchos casos, en el punto de mira de la sociedad. El punto de partida de este trabajo es el reconocimiento de que la enfermería es hoy en día una profesión con características propias y con todas las responsabilidades que ello supone y pretende hacer conscientes a estos profesiones de las consecuencias que puede tener el desarrollo de una mala praxis para evitar conductas que puedan desencadenar circunstancias condenables y situaciones difíciles desde el punto de vista profesional. A lo largo de este trabajo, pretendo hacer una introducción sobre de los diferentes tipos de responsabilidad profesional que existen y que pueden afectar a los expertos de la enfermería y analizaré dos sentencias judiciales que enjuician algunas actuaciones de estos profesionales.Departamento de EnfermeríaGrado en Enfermerí

Practical Aspects of Allogeneic Hematopoietic Cell Transplantation for Patients with Poor-risk Chronic Lymphocytic Leukemia

Allogeneic hematopoietic cell transplantation has become a viable option for younger patients with poor-risk chronic lymphocytic leukemia. The results obtained with either conventional or reduced-intensity conditioning regimens have been recently evaluated and compared with alternative nontransplant strategies. This manuscript deals with practical aspects of the procedure, including patient and donor selection, conditioning regimen, GVHD prophylaxis, disease monitoring, infectious and noninfectious complications, and timing of the procedure. Finally, we speculate on how we could improve the results obtained with the procedure and new advances currently in clinical trials

Schatten classes and traces on compact groups

In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential operators criteria in terms of kernels may be less effective (Carleman's example), our criteria are given in terms of the operators' symbols defined on the noncommutative analogue of the phase space $G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and $\hat{G}$ is its unitary dual. We also show results concerning general non-invariant operators as well as Schatten properties on Sobolev spaces. A trace formula is derived for operators in the Schatten class $S_{1}(L^{2}(G))$. Examples are given for Bessel potentials associated to sub-Laplacians (sums of squares) on compact Lie groups, as well as for powers of the sub-Laplacian and for other non-elliptic operators on SU(2)$\simeq\mathbb S^3$ and on SO(3).Comment: 18 pages, updated final version, to appear in Math. Res. Letter

Boundedness of the dyadic maximal function on graded Lie groups

Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C. Calder\'on, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*} \mathcal{M}^{d\sigma}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)d\sigma(y)\right| \end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $d\sigma(y)$ is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure $d\sigma$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function $\mathcal{M}_D^{d\sigma}$ using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform $\widehat{d\sigma}$ of $d\sigma$ with respect to a fixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of $\mathcal{M}_D^{d\sigma}$ on $L^p(G)$ for all $1<p\leq \infty.$Comment: 26 Page