arXiv subjects

Fabio Berra

Publications and source records attributed to Fabio Berra.

23 records · Page 2Linked to original sources

arXiv2020

From $A_1$ to $A_\infty$: New mixed inequalities for certain maximal operators

In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in \cite{Berra}. Concretely, given $r\geq 1$, $u\in A_1$, $v^r\in A_\infty$ and a Young function $Φ$ with certain properties, we have that inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{M_Φv(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. The involved operator $\frac{M_Φ(fv)(x)}{M_Φv(x)}$ seems to be an adequate extension when $v^r\in A_\infty$, since when we assume $v^r\in A_1$ we can replace $M_Φv$ by $v$, yielding a mixed inequality for $M_Φ$ proved in \cite{Berra-Carena-Pradolini(MN)}. As an application, we furthermore exhibe and prove mixed inequalities for the generalized fractional maximal operator $M_{γ,Φ}$, where $0<γ<n$ and $Φ$ is a Young function of $L\log L$ type.

math.CA↗

arXiv2019

Improvements on Sawyer type estimates for generalized maximal functions

In this paper we prove mixed inequalities for the maximal operator $M_Φ$, for general Young functions $Φ$ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given $r\geq 1$, if $u,v^r$ are weights belonging to the $A_1$-Muckenhoupt class and $Φ$ is a Young function as above, then the inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of $M_Φ$. Moreover, it is well-known that for the particular case $Φ(t)=t(1+\log^+t)^m$ with $m\in\mathbb{N}$ these maximal functions control, in some sense, certain operatos in Harmonic Analysis.

math.CA↗

arXiv2018

Mixed weak estimates of Sawyer type for generalized maximal operators

We study mixed weak estimates of Sawyer type for maximal operators associated to the family of Young functions $Φ(t)=t^r(1+\log^+t)^δ$, where $r\geq 1$ and $δ\geq 0$. More precisely, if $u$ and $v^r$ are $A_1$ weights, and $w$ is defined as $w=1/Φ(v^{-1})$ then the following estimate \[uw\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)} > t\right\}\right) \leq C\int_{\mathbb{R}^n} Φ\left(\frac{|f(x)|v(x)}{t}\right)u(x) \,dx\] holds for every positive $t$. This extends mixed estimates to a wider class of maximal operators, since when we put $r=1$ and $δ=0$ we recover a previous result for the Hardy-Littlewood maximal operator. This inequality generalizes some previous results proved by Cruz Uribe, Martell and Pérez in (Int. Math. Res. Not. (30): 1849-1871, 2005). Moreover, it includes estimates for some maximal operators related with commutators of Calderón-Zygmund operators.

math.CA↗

arXiv2017

Mixed weak estimates of Sawyer type for fractional integrals and some related operators

We prove mixed weak estimates of Sawyer type for fractional operators. More precisely, let $\mathcal{T}$ be either the maximal fractional function $M_γ$ or the fractional integral operator $I_γ$, $0<γ t\right\}\right)^{1/q}\leq \frac{C}{t}\left(\int_{\R^n}|f(x)|^pu(x)^{p/q}v(x)\,dx\right)^{1/p}, \end{equation*} holds for every positive $t$ and every bounded function with compact support. As an important application of the results above we further more exhibe mixed weak estimates for commutators of Calderón-Zygmund singular integral and fractional integral operators when the symbol $b$ is in the class Lipschitz-$δ$, $0<δ\leq 1$.

math.AP↗

arXiv2017

Mixed weak estimates of Sawyer type for commutators of singular integrals and related operators

We study mixed weak type inequalities for the commutator $[b,T]$, where $b$ is a BMO function and $T$ is a Calderón-Zygmund operator. More precisely, we prove that for every $t>0$ \begin{equation*}%\label{tesis_teo2.2} uv(\{x\in\R^n: |\frac{[b,T](fv)(x)}{v(x)}|>t\})\leq C\int_{\R^n}ϕ(\frac{|f(x)|}{t})u(x)v(x)\,dx, \end{equation*} where $ϕ(t)=t(1+\log^{+}{t})$, $u\in A_1$ and $v\in A_{\infty}(u)$. Our technique involves the classical Calderón-Zygmund decomposition, which allow us to give a direct proof. We use this result to prove an analogous inequality for higher order commutators. We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $L\log L$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.

math.CA↗