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Alberto Torchinsky

Publications and source records attributed to Alberto Torchinsky.

15 recordsLinked to original sources
arXiv2025

Integration of monomials over the unit spere and unit ball in $R^n$

We compute the integral of monomials of the form $x^{2\beta}$ over the unit sphere and the unit ball in $R^n$ where $\beta = (\beta_1,...,\beta_n)$ is a multi-index with real components $\beta_k > -1/2$, $1 \le k \le n$, and discuss their asymptotic behavior as some, or all, $\beta_k \to\infty$. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in $ R^n$. We also consider the Fourier transform of monomials $x^\alpha$ restricted to the unit sphere in $R^n$, where the multi-indices $\alpha$ have integer components, and discuss their behaviour at the origin.

math.CA↗
arXiv2019

A Characterization of the Lorentz space $L(p,r)$ in terms of Orlicz type classes

We describe the Lorentz space $L(p, r), 0 < r < p, p > 1$, in terms of Orlicz type classes of functions L . As a consequence of this result it follows that Stein's characterization of the real functions on $R^n$ that are differentiable at almost all the points in $R^n$, is equivalent to the earlier characterization of those functions given by A. P. Calderon.

math.FA↗
arXiv2013

Weighted local estimates for fractional type operators

In this note we prove the estimate $M^{\sharp}_{0,s}(Tf)(x) \le c\,M_\gamma f(x)$ for general fractional type operators $T$, where $M^{\sharp}_{0,s}$ is the local sharp maximal function and $M_\gamma$ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of $Tf$ by $M_\gamma f$. Similar estimates hold for $T$ replaced by fractional type operators with kernels satisfying H\"{o}rmander-type conditions or integral operators with homogeneous kernels, and $M_\gamma $ replaced by an appropriate maximal function $M_T$. We also prove two-weight, $L^p_v$-$L^q_w$ estimates for the fractional type operators described above for $1<p< q<\infty$ and a range of $q$. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.

math.CA↗
arXiv2013

Weighted Local Estimates for Singular Integral Operators

A local median decomposition is used to prove that a weighted local mean of a function is controlled by a weighted local mean of its local sharp maximal function. Together with (a local version of) the estimate $M^{\sharp}_{0,s}(Tf)(x) \le c\,Mf(x)$ for Calder\'{o}n-Zygmund singular integral operators, this allows us to express the local weighted integral control of $Tf$ by $Mf$. Similar estimates hold for $T$ replaced by singular integrals with kernels satisfying H\"{o}rmander-type conditions or integral operators with homogeneous kernels, and $M$ replaced by an appropriate maximal function $M_T$. Using sharper bounds in the local median decomposition we prove two-weight, $L^p_v$-$L^q_w$ estimates for singular integral operators for $1<p\le q<\infty$. In all cases, the results include weights that are not necessarily $A_{\infty}$. The local nature of these estimates leads to results involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.

math.CA↗
arXiv2012

On a Local Mean Oscillation Decomposition

In this note we generate two local median oscillation decompositions of an arbitrary measurable function and discuss some applications to Calder\'{o}n-Zygmund singular integral operators $T$. These applications rely on the inequality $M^{\sharp}_{0,s}(Tf)(x) \leq c\,Mf(x)$, and we complete the results given here with a discussion of a local version of this estimate.

math.CA↗
arXiv2012

Medians, Continuity, and Oscillation

In this paper we consider properties of medians as they pertain to the continuity and vanishing oscillation of a function. Our approach is based on the observation that medians are related to local sharp maximal functions restricted to a cube of $\R^n$.

math.CA↗
arXiv2007

From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha}$

In this paper we show how to compute the $\Lambda_{\alpha}$ norm, $\alpha\ge 0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(R^N)$ in terms of dyadic and special atoms.

math.CA↗
arXiv2007

The Hardy-Lorentz Spaces $H^{p,q}(R^n)$

In this paper we consider the Hardy-Lorentz spaces $H^{p,q}(R^n)$, with $0<p\le 1$, $0<q\le \infty$. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

math.CA↗
arXiv2006

Characterizations of the Hardy Space $H^1$ and BMO

We describe the spaces $H^1(R)$ and BMO$(R)$ in terms of their closely related, simpler dyadic and two-sided counterparts. As a result of these characterizations we establish when a bounded linear operator defined on dyadic or two-sided $H^1(R)$ into a Banach space has a continuous extension to $H^1(R)$ and when a bounded linear operator that maps a Banach space into dyadic or two-sided BMO$(R)$ actually maps continuously into BMO$(R)$.

math.FA↗
arXiv2005

Spaces between $H^1$ and $L^1$

In this paper we consider the $X_s$ spaces that lie between $H^1(R^n)$ and $L^1(R^n)$. We discuss the interpolation properties of these spaces, and the behavior of maximal functions and singular integrals acting on them.

math.FA↗