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Papers by Adrian Infante

Revista Bases de la Ciencia. e-ISSN 2588-0764, 2019

Estudiamos los subconjuntos de números reales con la propiedad de que todas las sumas de dos elem... more Estudiamos los subconjuntos de números reales con la propiedad de que todas las sumas de dos elementos son distintos, es decir que si 𝑎𝑖 + 𝑎𝑗= 𝑎𝑖′+ 𝑎𝑗′ entonces se verifica la igualdad {𝑎′}. A estos conjuntos los llamaremos conjuntos de Sidon. El problema es saber cuál es el mayor número de elementos que puede tener un conjunto de Sidon 𝑖, 𝑎𝑗} = {𝑎𝑖′, 𝑎𝑗 en el intervalo [1, 𝑁]. Presentamos ejemplos que evidencian la necesidad de conocer el tamaño del intervalo [1, 𝑁] donde se va a ubicar el conjunto de Sidon para saber el tamaño 𝐹(𝑁) del conjunto de Sidon. Ruzsa I. Z. (1998) demostró la existencia de una sucesión infinita de Sidon tal que su tamaño 𝐵(𝑁)> 𝑁√2−1+𝑜(1). En este trabajo rehacemos detalladamente la demostración de Ruzsa, introduciendo en la prueba una modificación sustancial, al sustituir las sucesiones {log 𝑝} por la sucesión de los argumentos de los enteros de Gauss 𝑎 + 𝑖𝑏 = 𝑝 con 0 < 𝑎 < 𝑏, 𝑎 y 𝑏 enteros y 𝑝 primo. Palabras clave: Conjuntos de Sidon, Sumas...

Boletín de la Sociedad Matemática Mexicana

We study the behavior of the constant appearing in the weak type inequality (1,1), for the center... more We study the behavior of the constant appearing in the weak type inequality (1,1), for the centered maximal operator acting on radial functions and associated with monotone radial measures, with respect to the dimension of the underlying space euclidean. In the case of a monotone increasing radial measure, we prove that this constant is indeed independent of the dimension. We also see that the techniques used in this case cannot be extended to the case of the maximal operator associated with decreasing radial measures.

Advances in Applied Clifford Algebras, 2013

In recent years, the integral representation problems have been studied in many context and gener... more In recent years, the integral representation problems have been studied in many context and generalities. For example, for the monogenic and meta functions in some Clifford type algebras, see [10, 11].In this paper we construct a Cauchy-Pompeiu type formula for meta-monogenic operator of order {n, (D-\lambda)^n, \lambda \in \mathbb{R}}$$n,(D-λ)n,λ∈R, and its conjugate {(\overline{D} - \lambda)^n}$$(D¯-λ)n in a Clifford algebra depending on parameters {\mathcal{A}_n(2, \alpha_j, \gamma_{ij})}$$An(2,αj,γij). Using these explicit representation formula of Cauchy-Pompeiu type we will show some applications.

Acta Mathematica Sinica, English Series, 2010

The aim of this work is to investigate the integrability properties of the maximal operator M µ, ... more The aim of this work is to investigate the integrability properties of the maximal operator M µ, associated with a non-doubling measure µ defined on ℝ n . We start by establishing for a wide class of radial and increasing measures µ, that M µ is bounded on all the spaces L µ p (ℝ n ), p &amp;gt; 1.

Advances in Applied Clifford Algebras, 2012

In the present paper we use the piecewise constant structure relations of a Clifford algebra in o... more In the present paper we use the piecewise constant structure relations of a Clifford algebra in order to obtain a Cauchy-Pompeiu representation for D − λ and D + λ operators, with these formulas we construct a distributional solutions for the equations that involves these operators with arbitrary right hand side. We also present an example where we build an integral representation for combinations of these operators.

Advances in Applied Clifford Algebras, 2014

We show fundamental solutions for some second order elliptic operators in the framework of parame... more We show fundamental solutions for some second order elliptic operators in the framework of parameter-depending Clifford-type algebras. Then we show integral representation formulae for sufficiently smooth functions defined in a domain of R n+1. This leads to integral formulae for solutions of certain partial differential equations of higher order.

Proceedings of the American Mathematical Society, 2011

The purpose of this paper is to prove that there exist measures d μ ( x ) = γ ( x ) d x d\mu (x)=... more The purpose of this paper is to prove that there exist measures d μ ( x ) = γ ( x ) d x d\mu (x)=\gamma (x)dx , with γ ( x ) = γ 0 ( | x | ) \gamma (x)=\gamma _{0}(|x|) and γ 0 \gamma _{0} being a decreasing and positive function, such that the Hardy-Littlewood maximal operator, M μ \mathcal {M}_{\mu } , associated to the measure μ \mu does not map L μ p ( R n ) L^{p}_{\mu }(\mathbb {R}^{n}) into weak L μ p ( R n ) L^{p}_{\mu }(\mathbb {R}^{n}) , for every p > ∞ p>\infty . This result answers an open question of P. Sjögren and F. Soria.

In this paper we show the boundedness of the non-centered Hardy-Littlewood maximal operator M µ a... more In this paper we show the boundedness of the non-centered Hardy-Littlewood maximal operator M µ associated to certain rotational invariant measures. We prove that this maximal operator satisfies the modular inequality µ x ∈ R n : M µ f (x) > λ ≤ C R n | f | λ 1 + log + | f | λ m dµ, for λ > 0 and m > 0. We prove the modular inequality for the maximal operator associated to rotated squares from R 2 and with a radial and decreasing measure. The technique used in the proof for cubes suggests extending this result to cones whose axes of symmetry pass through the origin. In both cases it is proved that the exponent of the modular inequality is m = n, which we prove is sharp.

Advances in Applied Clifford Algebras, 2013

In recent years, the integral representation problems have been studied in many context and gener... more In recent years, the integral representation problems have been studied in many context and generalities. For example, for the monogenic and meta functions in some Clifford type algebras, see [10, 11].In this paper we construct a Cauchy-Pompeiu type formula for meta-monogenic operator of order {n, (D-\lambda)^n, \lambda \in \mathbb{R}}$$n,(D-λ)n,λ∈R, and its conjugate {(\overline{D} - \lambda)^n}$$(D¯-λ)n in a Clifford algebra depending on parameters {\mathcal{A}_n(2, \alpha_j, \gamma_{ij})}$$An(2,αj,γij). Using these explicit representation formula of Cauchy-Pompeiu type we will show some applications.

Revista Bases de la Ciencia. e-ISSN 2588-0764, 2019

Estudiamos los subconjuntos de números reales con la propiedad de que todas las sumas de dos elem... more Estudiamos los subconjuntos de números reales con la propiedad de que todas las sumas de dos elementos son distintos, es decir que si 𝑎𝑖 + 𝑎𝑗= 𝑎𝑖′+ 𝑎𝑗′ entonces se verifica la igualdad {𝑎′}. A estos conjuntos los llamaremos conjuntos de Sidon. El problema es saber cuál es el mayor número de elementos que puede tener un conjunto de Sidon 𝑖, 𝑎𝑗} = {𝑎𝑖′, 𝑎𝑗 en el intervalo [1, 𝑁]. Presentamos ejemplos que evidencian la necesidad de conocer el tamaño del intervalo [1, 𝑁] donde se va a ubicar el conjunto de Sidon para saber el tamaño 𝐹(𝑁) del conjunto de Sidon. Ruzsa I. Z. (1998) demostró la existencia de una sucesión infinita de Sidon tal que su tamaño 𝐵(𝑁)> 𝑁√2−1+𝑜(1). En este trabajo rehacemos detalladamente la demostración de Ruzsa, introduciendo en la prueba una modificación sustancial, al sustituir las sucesiones {log 𝑝} por la sucesión de los argumentos de los enteros de Gauss 𝑎 + 𝑖𝑏 = 𝑝 con 0 < 𝑎 < 𝑏, 𝑎 y 𝑏 enteros y 𝑝 primo. Palabras clave: Conjuntos de Sidon, Sumas...

Boletín de la Sociedad Matemática Mexicana

We study the behavior of the constant appearing in the weak type inequality (1,1), for the center... more We study the behavior of the constant appearing in the weak type inequality (1,1), for the centered maximal operator acting on radial functions and associated with monotone radial measures, with respect to the dimension of the underlying space euclidean. In the case of a monotone increasing radial measure, we prove that this constant is indeed independent of the dimension. We also see that the techniques used in this case cannot be extended to the case of the maximal operator associated with decreasing radial measures.

Advances in Applied Clifford Algebras, 2013

In recent years, the integral representation problems have been studied in many context and gener... more In recent years, the integral representation problems have been studied in many context and generalities. For example, for the monogenic and meta functions in some Clifford type algebras, see [10, 11].In this paper we construct a Cauchy-Pompeiu type formula for meta-monogenic operator of order {n, (D-\lambda)^n, \lambda \in \mathbb{R}}$$n,(D-λ)n,λ∈R, and its conjugate {(\overline{D} - \lambda)^n}$$(D¯-λ)n in a Clifford algebra depending on parameters {\mathcal{A}_n(2, \alpha_j, \gamma_{ij})}$$An(2,αj,γij). Using these explicit representation formula of Cauchy-Pompeiu type we will show some applications.

Acta Mathematica Sinica, English Series, 2010

The aim of this work is to investigate the integrability properties of the maximal operator M µ, ... more The aim of this work is to investigate the integrability properties of the maximal operator M µ, associated with a non-doubling measure µ defined on ℝ n . We start by establishing for a wide class of radial and increasing measures µ, that M µ is bounded on all the spaces L µ p (ℝ n ), p &amp;gt; 1.

Advances in Applied Clifford Algebras, 2012

In the present paper we use the piecewise constant structure relations of a Clifford algebra in o... more In the present paper we use the piecewise constant structure relations of a Clifford algebra in order to obtain a Cauchy-Pompeiu representation for D − λ and D + λ operators, with these formulas we construct a distributional solutions for the equations that involves these operators with arbitrary right hand side. We also present an example where we build an integral representation for combinations of these operators.

Advances in Applied Clifford Algebras, 2014

We show fundamental solutions for some second order elliptic operators in the framework of parame... more We show fundamental solutions for some second order elliptic operators in the framework of parameter-depending Clifford-type algebras. Then we show integral representation formulae for sufficiently smooth functions defined in a domain of R n+1. This leads to integral formulae for solutions of certain partial differential equations of higher order.

Proceedings of the American Mathematical Society, 2011

The purpose of this paper is to prove that there exist measures d μ ( x ) = γ ( x ) d x d\mu (x)=... more The purpose of this paper is to prove that there exist measures d μ ( x ) = γ ( x ) d x d\mu (x)=\gamma (x)dx , with γ ( x ) = γ 0 ( | x | ) \gamma (x)=\gamma _{0}(|x|) and γ 0 \gamma _{0} being a decreasing and positive function, such that the Hardy-Littlewood maximal operator, M μ \mathcal {M}_{\mu } , associated to the measure μ \mu does not map L μ p ( R n ) L^{p}_{\mu }(\mathbb {R}^{n}) into weak L μ p ( R n ) L^{p}_{\mu }(\mathbb {R}^{n}) , for every p > ∞ p>\infty . This result answers an open question of P. Sjögren and F. Soria.

In this paper we show the boundedness of the non-centered Hardy-Littlewood maximal operator M µ a... more In this paper we show the boundedness of the non-centered Hardy-Littlewood maximal operator M µ associated to certain rotational invariant measures. We prove that this maximal operator satisfies the modular inequality µ x ∈ R n : M µ f (x) > λ ≤ C R n | f | λ 1 + log + | f | λ m dµ, for λ > 0 and m > 0. We prove the modular inequality for the maximal operator associated to rotated squares from R 2 and with a radial and decreasing measure. The technique used in the proof for cubes suggests extending this result to cones whose axes of symmetry pass through the origin. In both cases it is proved that the exponent of the modular inequality is m = n, which we prove is sharp.

Advances in Applied Clifford Algebras, 2013

In recent years, the integral representation problems have been studied in many context and gener... more In recent years, the integral representation problems have been studied in many context and generalities. For example, for the monogenic and meta functions in some Clifford type algebras, see [10, 11].In this paper we construct a Cauchy-Pompeiu type formula for meta-monogenic operator of order {n, (D-\lambda)^n, \lambda \in \mathbb{R}}$$n,(D-λ)n,λ∈R, and its conjugate {(\overline{D} - \lambda)^n}$$(D¯-λ)n in a Clifford algebra depending on parameters {\mathcal{A}_n(2, \alpha_j, \gamma_{ij})}$$An(2,αj,γij). Using these explicit representation formula of Cauchy-Pompeiu type we will show some applications.