In this work we study the skew group algebra $\Lambda[G]$ when $G$ is a finite group acting on $\Lambda$ whose order is invertible in $\Lambda$. Here, we assume that $\Lambda$ is a finite-dimensional algebra over an algebraically closed field $k$. The aim is to describe all possible actions of a finite abelian group on an hereditary algebra of finite or tame representation type, to give a description of the resulting skew group algebra for each action and finally to determinate their representation type.

This paper deals with the compact trace theorem in domains $\Omega \subset \mathbb{R}^3$ with external cusps. We show that if the power sharpness of the cusp is bellow a critical exponent, then the trace operator $\gamma : H^1 (\Omega) \rightarrow L^2 (\partial\Omega)$ exists and it is compact.

In this paper, we study some direct results in simultaneous ap- proximation for a new sequence of linear positive operators $M_n(f(t); x)$ of Szãsz-Beta type operators. First, we establish the basic pointwise convergence theorem and then proceed to discuss the Voronovaskaja-type asymptotic formula. Finally, we obtain an error estimate in terms of modulus of continuity of the function being approximated.

In this paper we will give suitable notions of Amalgamation and Super-amalgamation properties for the class of quasi-modal algebras introduced by the author in his paper Quasi-Modal algebras.

The variety $\mathcal{QH}$ of Heyting algebras with a quantifier [14] corresponds to the algebraic study of the modal intuitionistic propositional calculus without the necessity operator. This paper is concerned with the subvariety $\mathcal{C}$ of $\mathcal{QH}$ generated by chains. We prove that this subvariety is characterized within $\mathcal{QH}$ by the equations $\nabla(x \wedge y) \approx \nabla x \wedge \nabla y$ and $(x \rightarrow y) \vee (y \rightarrow x) \approx 1$. We investigate free objects in $\mathcal{C}$.

Very recently Jain et al. [4] proposed generalized integrated Baskakov operators $V_{n,\alpha} (f,x)$, $\alpha > 0$ and estimated some approximation properties in simultaneous approximation. In the present paper we establish the rate of convergence of these operators and its Bezier variant, for functions which have derivatives of bounded variation.

For some natural families of elliptic curves we show that “on average” the exponent of the point group of their reductions modulo a prime $p$ grows as $p^{1+o(1)}$.

A graph $G$ is coordinated if, for every induced subgraph $H$ of $G$, the minimum number of colors that can be assigned to the cliques of $H$ in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of $H$ with a common vertex. In a previous work, coordinated graphs were characterized by minimal forbidden induced subgraphs within some classes of graphs. In this note, we present families of minimally non-coordinated graphs whose cardinality grows exponentially on the number of vertices and edges. Furthermore, we describe some ideas to generate similar families. Based on these results, it seems difficult to find a general characterization of coordinated graphs by minimal forbidden induced subgraphs.

Many functional versions of the Caristi-Kirk fixed point theorem are nothing but logical equivalents of the result in question.

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham cohomology of locally symmetric spaces, and the cohomology of symmetric spaces with coefficients in a system of local groups. We construct canonical isomorphisms among such cohomology groups and discuss the compatibility of the Hecke operators with respect to those canonical isomorphisms.

A discretization method for the study of the weak type (1, 1) for the maximal of a sequence of convolution operators on $\mathbb{R}^n$ has been introduced by Miguel de Guzmán and Teresa Carrillo, by replacing the integrable functions by finite sums of Dirac deltas. Trying to extend the above mentioned result to integral operators defined on metric measure spaces, a general setting containing at once continuous, discrete and mixed contexts, a caveat comes from the result in On restricted weak type (1, 1); the discrete case (Akcoglu M.; Baxter J.; Bellow A.; Jones R., Israel J. Math. 124 (2001), 285-297). There a sequence of convolution operators in $\ell^1(\mathbb{Z})$ is constructed such that the maximal operator is of restricted weak type (1, 1), or equivalently of weak type (1, 1) over finite sums of Dirac deltas, but not of weak type (1, 1). The purpose of this note is twofold. First we prove that, in a general metric measure space with a measure that is absolutely continuous with respect to some doubling measure, the weak type (1, 1) of the maximal operator associated to a given sequence of integral operators is equivalent to the weak type (1, 1) over linear combinations of Dirac deltas with positive integer coefficients. Second, for the non-atomic case we obtain as a corollary that any of these weak type properties is equivalent to the weak type (1, 1) over finite sums of Dirac deltas supported at different points.

We give several integral representations for the Euler-Mascheroni constant using a combinatorial identity for $\sum_{n=1}^N \frac{1}{(n+x)(n+y)}$. The derivation of this combinatorial identity is done in an elemental way.

Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators.

This paper is essentially the second authors' lecture at the CIMPA-UNESCO Argentina School 2008, Real Analysis and its Applications. It summarises large parts of the three authors' paper [MSV]. Only one proof is given. In the setting of a Euclidean space, we consider operators defined and uniformly bounded on atoms of a Hardy space $H^p$. The question discussed is whether such an operator must be bounded on $H^p$. This leads to a study of the difference between countable and finite atomic decompositions in Hardy spaces.

[MSV] S. Meda, P. Sjögren and M. Vallarino, On the $H^1$-$L^1$ boundedness of operators, Proc. Amer. Math. Soc. 136 (2008), 2921-2931.

A locally integrable function $m(\xi, \eta)$ defined on $\mathbb{R}^n \times \mathbb{R}^n$ is said to be a bilinear multiplier on $\mathbb{R}^n$ of type $(p_1, p_2, p_3)$ if $$ B_m(f,g)(x) = \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \hat{f}(\xi) \hat{g}(\eta) m(\xi,\eta) e^{2\pi i\langle\xi+\eta, x\rangle} \, d\xi d\eta $$ defines a bounded bilinear operator from $L^{p_1}(\mathbb{R}^n) \times L^{p_2}(\mathbb{R}^n)$ to $L^{p_3}(\mathbb{R}^n)$. The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where $m(\xi, \eta) = M(\xi - \eta)$ for a given $M$ defined on $\mathbb{R}^n$ is also addressed.

In this paper we discuss a transference method of $L^p$-boundedness properties for harmonic analysis operators in the Hermite setting to the corresponding operators in the Laguerre context. As a byproduct of our procedure we obtain new characterizations of certain classes of Banach spaces and Köethe spaces.

This paper is essentially the talk I addressed in the CIMPA-UNESCO Argentina School 2008. It is about mixed weak type inequalities and it is based on a joint paper with S. Ombrosi: Francisco J. Martín-Reyes and Sheldy J. Ombrosi, Mixed weak type inequalities for one-sided operators, Q. J. Math. 60 (2009), no. 1, 63-73.

In this article we collect some known results concerning (generalized) Gelfand pairs $(K, N)$, where $N$ is a group of Heisenberg type and $K$ is a subgroup of automorphisms of $N$. We also give new examples.

In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold $\Gamma\backslash \mathbb{H}^n$. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of $L^2(\Gamma\backslash \mathbb{H}^n)$ into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and characterise the image of $L^2(\Gamma\backslash \mathbb{H}^n)$ in terms of twisted Bergman and Hermite Bergman spaces.

We review some recent results on construction of entire solutions to the classical semilinear elliptic equation $\Delta u + u - u^3 = 0$ in $\mathbb{R}^N$. In various cases, large dilations of an embedded, complete minimal surface approximate the transition set of a solution that connects the equilibria $\pm 1$. In particular, our construction answers negatively a celebrated conjecture by E. De Giorgi in dimensions $N \geq 9$.

We give a detailed proof, in the case of one space dimension, of a pointwise upper estimate for the space gradient of a temperature. The operators involved are a one-sided Hardy-Littlewood maximal in time and the Calderón sharp maximal operator in space.

We survey on some recent results concerning weak and strong weighted $L^p$ estimates for Calderón-Zygmund operators with sharp bounds when the weight satisfies the $A_1$ condition. These questions are related to a problem posed by Muckenhoupt and Wheeden in the seventies.

Let $m$ have compact support in $(0, \infty)$. For $1 < p < 2d/(d + 1)$, we give a necessary and sufficient condition for the $L^p_\mathrm{rad}(\mathbb{R}^d)$-boundedness of the maximal operator associated with the radial multiplier $m(|\xi|)$. More generally we prove a similar result for maximal operators associated with multipliers of modified Hankel transforms. The result is obtained by modifying the proof of the characterization of Hankel multipliers given by the authors in Math. Ann. 342 (2008), 31-68.

In this paper we consider a fixed initial condition $u_0$ and we study the limit as $\varepsilon \rightarrow 0$ of $u_\varepsilon$, the solution to the re-scaled problem $$ u_t(x,y) = \frac{1}{\varepsilon^2}\left(\int_R J\left(\frac{x-y}{\varepsilon u(y,t)}\right) \frac{dy}{\varepsilon} - u(x,t)\right) \quad \text{in } R \times [0,\infty) $$ with initial condition $u_\varepsilon(x, 0) = u_0(x)$. Here $J$ is a smooth non negative even function supported in the interval $[-1, 1]$. Moreover it is assumed that $\int J = 1$ and that $J$ is decreasing on $[0, 1]$.

We prove that, under adequate hypothesis on the initial condition, the limit $$ \lim_{\varepsilon\rightarrow 0} u_\varepsilon = u $$ is the solution to the well known porous medium equation $v_t = D(v^3)_{xx}$, with initial condition $u_\varepsilon(x, 0) = u_0(x)$ for a suitable constant $D$.

Some recent results for bilinear or multilinear singular integrals operators are presented. The focus is on some of the results that can be viewed as natural counterparts of classical theorems in Calderón-Zygmund theory, adding to the already existing extensive literature in the subject. In particular, two different classes of operators that can be seen as bilinear counterparts of linear Calderón-Zygmund operators are considered. Some highlights of the recent progress done for operators with variable coefficients are a modulation invariant bilinear T1-Theorem and some new weighted norm inequalities.

We review recent results proved jointly with B. Di Blasio and F. Astengo. On the Heisenberg group $H_n$, consider the two commuting self-adjoint operators $L$ and $i^{-1}T$, where $L$ is the sublaplacian and $T$ is the central derivative. Their joint $L^2$-spectrum is the so-called Heisenberg fan, contained in $\mathbb{R}^2$. To any bounded Borel function $m$ on the fan, we associate the operator $m(L, i^{-1}T)$. The main result that we describe says that the convolution kernel of $m(L, i^{-1}T)$ is a Schwartz function if and only if $m$ is the restriction of a Schwartz function on $\mathbb{R}^2$. We point out that this result can be interpreted in terms of the spherical transform for the convolution algebra of $U(n)$-invariant functions on $H_n$. We also describe extensions to more general situations.

For $\mathbb{B}_n$ the unit ball of $\mathbb{C}^n$, we consider Hardy-Orlicz spaces of holomorphic functions $\mathcal{H}^\Phi$, which are preduals of spaces of BMOA type with weight. We characterize the symbols of Hankel operators that extend into bounded operators from the Hardy-Orlicz $\mathcal{H}^{\Phi_1}$ into $\mathcal{H}^{\Phi_2}$. We also consider the closely related question of integrability properties of the product of two functions, one in $\mathcal{H}^{\Phi_1}$ and the other one in the dual of $\mathcal{H}^{\Phi_2}$.

Let $L$ be a non-negative, self-adjoint operator on $L^2(\Omega)$, where $(\Omega, d, \mu)$ is a space of homogeneous type. Assume that the semigroup $\{T_t\}_{t > 0}$ generated by $-L$ satisfies Gaussian bounds, or more generally Davies-Gaffney estimates. We say that $f$ belongs to the Hardy space $H^1_L$ if the square function $$ S_h f(x) = \left( \iint_{\Gamma(x)} |t^2 L e^{-t^2 L} f(y)|^2 \frac{d\mu(y)}{\mu(B_d(x,t))} \frac{dt}{t} \right)^{1/2} $$ belongs to $L^1(\Omega, d\mu)$, where $\Gamma(x) = \{(y, t) \in \Omega \times (0, \infty) : d(x, y) < t\}$. We prove spectral multiplier theorems for $L$ on $H^1_L$.

This paper is a survey about recent results on sparse representations and optimal models in different settings. Given a set of functions, we show that there exists an optimal collection of subspaces minimizing the sum of the square of the distances between each function and its closest subspace in the collection. Further, this collection of subspaces gives the best sparse representation for the given data, in a sense defined later, and provides an optimal model for sampling in a union of subspaces.