Juan Carlos Fariña - Academia.edu (original) (raw)

Papers by Juan Carlos Fariña

arXiv (Cornell University), Sep 2, 2011

In this paper we establish L p-boundedness properties for Laplace type transform spectral multipl... more In this paper we establish L p-boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator L = −∆ + V. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the L p-boundedness of the imaginary powers L iγ , γ ∈ R, of L .

Mathematische Annalen, 2016

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal funct... more In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator ∆ λ f (n) := a λ n f (n + 1) − 2f (n) + a λ n−1 f (n − 1), n ∈ N, λ > 0, where a λ n := {(2λ+n)(n+1)/[(n+λ)(n+1+λ)]} 1/2 , n ∈ N, and a λ −1 := 0. We also prove weighted p-boundedness properties of transplantation operators associated with the system {ϕ λ n } n∈N of ultraspherical functions, a family of eigenfunctions of ∆ λ. In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.

Integral Equations and Operator Theory, Apr 19, 2011

In this paper we represent the k-th Riesz transform in the ultraspherical setting as a principal ... more In this paper we represent the k-th Riesz transform in the ultraspherical setting as a principal value integral operator for every k ∈ N. We also measure the speed of convergence of the limit by proving L p-boundedness properties for the oscillation and variation operators associated with the corresponding truncated operators.

arXiv (Cornell University), Mar 24, 2008

In this paper we investigate L p-boundedness properties for the higher order Riesz transforms ass... more In this paper we investigate L p-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.

Revista Matematica Complutense, Feb 14, 2012

In this paper we prove that the variation operators associated with the heat semigroup and Riesz ... more In this paper we prove that the variation operators associated with the heat semigroup and Riesz transforms related to the Schrödinger operator are bounded on the suitable BM O type space.

Mathematische Nachrichten, Oct 7, 2015

In this paper we define square functions (also called Littlewood-Paley-Stein functions) associate... more In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schrödinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L p-boundedness properties for the square functions to our Banach valued setting by using γ-radonifying operators. We also prove that these L p-boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.

Complex Variables, Oct 1, 1995

Let F be a relatively closed subset of an open set G of the plane. Alice Roth in [10] proved that... more Let F be a relatively closed subset of an open set G of the plane. Alice Roth in [10] proved that if a function f is a uniform limit on F of holomorphic or meromorphic functions on G, it is possible to select the approximating functions m in such a way that the difference function fm can be extended continuously

arXiv (Cornell University), Feb 12, 2022

In this paper we establish L p (R d , γ∞)-boundedness properties for square functions involving t... more In this paper we establish L p (R d , γ∞)-boundedness properties for square functions involving time and spatial derivatives of Ornstein-Uhlenbeck semigroups. Here γ∞ denotes the invariant measure. In order to prove the strong type results for 1 < p < ∞ we use R-boundedness. The weak type (1,1) property is established by studying separately global and local operators defined for the square Littlewood-Paley functions. By the way we prove L p (R d , γ∞)-boundedness properties for maximal and variation operators for Ornstein-Uhlenbeck semigroups.

Revista Tecnica De La Facultad De Ingenieria Universidad Del Zulia, 1988

Banach Journal of Mathematical Analysis, Jul 1, 2019

In this article we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p... more In this article we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p , q)(R d). We characterize H p,q (R d) by using first-order classical Riesz transforms and compositions of first-order Riesz transforms, depending on the values of the exponents p and q. Also, we describe the distributions in H p,q (R d) as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivatives in the time variable. Finally, we characterize the functions in L 2 (R d) ∩ H p,q (R d) by means of Fourier multipliers m θ with symbol θ(•/| • |), where θ ∈ C ∞ (S d−1) and S d−1 denotes the unit sphere in R d .

Israel Journal of Mathematics, 2007

We study g-functions and Riesz transforms related to the Bessel operators ∆µ = −x −µ−1/2 Dx 2µ+1 ... more We study g-functions and Riesz transforms related to the Bessel operators ∆µ = −x −µ−1/2 Dx 2µ+1 Dx −µ−1/2. The method we use allows us to characterize the Banach spaces for which these operators are bounded when acting on-valued functions.

Proceedings of the American Mathematical Society, 2001

In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a dom... more In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on Lipα harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in Lipα harmonic approximation are also studied.

arXiv (Cornell University), Feb 8, 2012

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for... more In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L 2 ((0, ∞), dt/t), γ(H, B) represents the space of γradonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BM O L (R n , B) (respectively, H 1 L (R n , B)) into BM O L (R n , γ(H, B)) (respectively, H 1 L (R n , γ(H, B))), where BM O L and H 1 L denote BM O and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BM O L (R n , B) and H 1 L (R n , B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.

Canadian mathematical bulletin, Mar 1, 1995

Let F be a relatively closed subset of a domain G in the complex plane. Let / be a function that ... more Let F be a relatively closed subset of a domain G in the complex plane. Let / be a function that is the limit, in the Lip a norm on F, of functions which are holomorphic or meromorphic on G (0 < a < 1). We prove that, under the same conditions that give Lip «-approximation (0 < a < 1) on relatively closed subsets of G, it is possible to choose the approximating function m in such a way that/-m can be extended to a function belonging to lip(a, F).

arXiv (Cornell University), Mar 2, 2018

In this article we prove dimension free L p-boundedness of Riesz transforms associated with a Bes... more In this article we prove dimension free L p-boundedness of Riesz transforms associated with a Bessel differential operator. We obtain explicit estimates of the L p-norms for the Bessel-Riesz transforms in terms of p, establishing a linear behaviour with respect to p. We use the Bellman function technique to prove a bilinear dimension free inequality involving Poisson semigroups defined through this Bessel operator.

arXiv (Cornell University), Mar 5, 2012

In this paper we prove that the generalized (in the sense of Caffarelli and Calderón [5]) maximal... more In this paper we prove that the generalized (in the sense of Caffarelli and Calderón [5]) maximal operators associated with heat semigroups for Bessel and Laguerre operators are weak type (1, 1). Our results include other known ones and our proofs are simpler than the ones for the known special cases.

arXiv (Cornell University), May 30, 2018

In this paper we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p ,... more In this paper we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p , q)(R d). We characterize H p,q (R d) by using first order classical Riesz transforms and compositions of first order Riesz transforms depending on the values of the exponents p and q. Also, we describe the distributions in H p,q (R d) as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivative in the time variable. Finally we characterize the functions in L 2 (R d) ∩ H p,q (R d) by means of Fourier multipliers m θ with symbol θ(•/| • |), where θ ∈ C ∞ (S d−1) and S d−1 denotes the unit sphere in R d .

Proceedings of the American Mathematical Society, Feb 9, 2001

In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a dom... more In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on Lipα harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in Lipα harmonic approximation are also studied.

Journal of Approximation Theory, Feb 1, 1994

Annales Academiae Scientiarum Fennicae. Mathematica, Feb 1, 2013

In this paper we establish L p-boundedness properties for Laplace type transform spectral multipl... more In this paper we establish L p-boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator L = −∆ + V. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the L p-boundedness of the imaginary powers L iγ , γ ∈ R, of L.

arXiv (Cornell University), Sep 2, 2011

In this paper we establish L p-boundedness properties for Laplace type transform spectral multipl... more In this paper we establish L p-boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator L = −∆ + V. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the L p-boundedness of the imaginary powers L iγ , γ ∈ R, of L .

Mathematische Annalen, 2016

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal funct... more In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator ∆ λ f (n) := a λ n f (n + 1) − 2f (n) + a λ n−1 f (n − 1), n ∈ N, λ > 0, where a λ n := {(2λ+n)(n+1)/[(n+λ)(n+1+λ)]} 1/2 , n ∈ N, and a λ −1 := 0. We also prove weighted p-boundedness properties of transplantation operators associated with the system {ϕ λ n } n∈N of ultraspherical functions, a family of eigenfunctions of ∆ λ. In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.

Integral Equations and Operator Theory, Apr 19, 2011

In this paper we represent the k-th Riesz transform in the ultraspherical setting as a principal ... more In this paper we represent the k-th Riesz transform in the ultraspherical setting as a principal value integral operator for every k ∈ N. We also measure the speed of convergence of the limit by proving L p-boundedness properties for the oscillation and variation operators associated with the corresponding truncated operators.

arXiv (Cornell University), Mar 24, 2008

In this paper we investigate L p-boundedness properties for the higher order Riesz transforms ass... more In this paper we investigate L p-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.

Revista Matematica Complutense, Feb 14, 2012

In this paper we prove that the variation operators associated with the heat semigroup and Riesz ... more In this paper we prove that the variation operators associated with the heat semigroup and Riesz transforms related to the Schrödinger operator are bounded on the suitable BM O type space.

Mathematische Nachrichten, Oct 7, 2015

In this paper we define square functions (also called Littlewood-Paley-Stein functions) associate... more In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schrödinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L p-boundedness properties for the square functions to our Banach valued setting by using γ-radonifying operators. We also prove that these L p-boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.

Complex Variables, Oct 1, 1995

Let F be a relatively closed subset of an open set G of the plane. Alice Roth in [10] proved that... more Let F be a relatively closed subset of an open set G of the plane. Alice Roth in [10] proved that if a function f is a uniform limit on F of holomorphic or meromorphic functions on G, it is possible to select the approximating functions m in such a way that the difference function fm can be extended continuously

arXiv (Cornell University), Feb 12, 2022

In this paper we establish L p (R d , γ∞)-boundedness properties for square functions involving t... more In this paper we establish L p (R d , γ∞)-boundedness properties for square functions involving time and spatial derivatives of Ornstein-Uhlenbeck semigroups. Here γ∞ denotes the invariant measure. In order to prove the strong type results for 1 < p < ∞ we use R-boundedness. The weak type (1,1) property is established by studying separately global and local operators defined for the square Littlewood-Paley functions. By the way we prove L p (R d , γ∞)-boundedness properties for maximal and variation operators for Ornstein-Uhlenbeck semigroups.

Revista Tecnica De La Facultad De Ingenieria Universidad Del Zulia, 1988

Banach Journal of Mathematical Analysis, Jul 1, 2019

In this article we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p... more In this article we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p , q)(R d). We characterize H p,q (R d) by using first-order classical Riesz transforms and compositions of first-order Riesz transforms, depending on the values of the exponents p and q. Also, we describe the distributions in H p,q (R d) as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivatives in the time variable. Finally, we characterize the functions in L 2 (R d) ∩ H p,q (R d) by means of Fourier multipliers m θ with symbol θ(•/| • |), where θ ∈ C ∞ (S d−1) and S d−1 denotes the unit sphere in R d .

Israel Journal of Mathematics, 2007

We study g-functions and Riesz transforms related to the Bessel operators ∆µ = −x −µ−1/2 Dx 2µ+1 ... more We study g-functions and Riesz transforms related to the Bessel operators ∆µ = −x −µ−1/2 Dx 2µ+1 Dx −µ−1/2. The method we use allows us to characterize the Banach spaces for which these operators are bounded when acting on-valued functions.

Proceedings of the American Mathematical Society, 2001

In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a dom... more In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on Lipα harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in Lipα harmonic approximation are also studied.

arXiv (Cornell University), Feb 8, 2012

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for... more In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L 2 ((0, ∞), dt/t), γ(H, B) represents the space of γradonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BM O L (R n , B) (respectively, H 1 L (R n , B)) into BM O L (R n , γ(H, B)) (respectively, H 1 L (R n , γ(H, B))), where BM O L and H 1 L denote BM O and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BM O L (R n , B) and H 1 L (R n , B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.

Canadian mathematical bulletin, Mar 1, 1995

Let F be a relatively closed subset of a domain G in the complex plane. Let / be a function that ... more Let F be a relatively closed subset of a domain G in the complex plane. Let / be a function that is the limit, in the Lip a norm on F, of functions which are holomorphic or meromorphic on G (0 < a < 1). We prove that, under the same conditions that give Lip «-approximation (0 < a < 1) on relatively closed subsets of G, it is possible to choose the approximating function m in such a way that/-m can be extended to a function belonging to lip(a, F).

arXiv (Cornell University), Mar 2, 2018

In this article we prove dimension free L p-boundedness of Riesz transforms associated with a Bes... more In this article we prove dimension free L p-boundedness of Riesz transforms associated with a Bessel differential operator. We obtain explicit estimates of the L p-norms for the Bessel-Riesz transforms in terms of p, establishing a linear behaviour with respect to p. We use the Bellman function technique to prove a bilinear dimension free inequality involving Poisson semigroups defined through this Bessel operator.

arXiv (Cornell University), Mar 5, 2012

In this paper we prove that the generalized (in the sense of Caffarelli and Calderón [5]) maximal... more In this paper we prove that the generalized (in the sense of Caffarelli and Calderón [5]) maximal operators associated with heat semigroups for Bessel and Laguerre operators are weak type (1, 1). Our results include other known ones and our proofs are simpler than the ones for the known special cases.

arXiv (Cornell University), May 30, 2018

In this paper we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p ,... more In this paper we study Hardy spaces H p,q (R d), 0 < p, q < ∞, modeled over amalgam spaces (L p , q)(R d). We characterize H p,q (R d) by using first order classical Riesz transforms and compositions of first order Riesz transforms depending on the values of the exponents p and q. Also, we describe the distributions in H p,q (R d) as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivative in the time variable. Finally we characterize the functions in L 2 (R d) ∩ H p,q (R d) by means of Fourier multipliers m θ with symbol θ(•/| • |), where θ ∈ C ∞ (S d−1) and S d−1 denotes the unit sphere in R d .

Proceedings of the American Mathematical Society, Feb 9, 2001

In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a dom... more In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on Lipα harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in Lipα harmonic approximation are also studied.

Journal of Approximation Theory, Feb 1, 1994

Annales Academiae Scientiarum Fennicae. Mathematica, Feb 1, 2013

In this paper we establish L p-boundedness properties for Laplace type transform spectral multipl... more In this paper we establish L p-boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator L = −∆ + V. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the L p-boundedness of the imaginary powers L iγ , γ ∈ R, of L.