arXiv subjects
Publications and source records attributed to Sheldy Ombrosi.
We obtain an improved version of the pointwise sparse domination principle established by the first author in [19]. This allows us to determine nearly minimal assumptions on a singular integral operator $T$ for which it admits a sparse domination.
In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes $A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators. Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range and extend the estimates to the quasi-Banach range. As a direct consequence, we obtain weighted vector-valued inequalities reproving some of the results by Benea and Muscalu. We also extend recent results of Carando et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators. Finally, our last application gives new weighted estimates for the commutators of multilinear Calderón-Zygmund operators and for the bilinear Hilbert transform with BMO functions using ideas from Bényi et al.
We construct an example showing that the upper bound $[w]_{A_1}\log({\rm{e}}+[w]_{A_1})$ for the $L^1(w)\to L^{1,\infty}(w)$ norm of the Hilbert transform cannot be improved in general.
We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.
We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on $\ell^r$-valued extensions of linear operators. We show that for certain $1 \leq p, q_1, \dots, q_m, r \leq \infty$, there is a constant $C\geq 0$ such that for every bounded multilinear operator $T\colon L^{q_1}(μ_1) \times \cdots \times L^{q_m}(μ_m) \to L^p(ν)$ and functions $\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(μ_1), \dots, \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(μ_m)$, the following inequality holds \begin{equation}\label{MZ ineq abstract} (1) \quad \quad \left\Vert \left(\sum_{k_1, \dots, k_m} |T(f_{k_1}^1, \dots, f_{k_m}^m)|^r\right)^{1/r} \right\Vert_{L^p(ν)} \leq C \|T\| \prod_{i=1}^m \left\| \left(\sum_{k_i=1}^{n_i} |f_{k_i}^i|^r\right)^{1/r} \right\|_{L^{q_i}(μ_i)}. \end{equation} In some cases we also calculate the best constant $C\geq 0$ satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.
We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\, \|f\|_{L^1(uv)},$$ where $T$ can be the Hardy-Littlewood maximal function or any Calderón-Zygmund operator. This result was conjectured in [IMRN, (30)2005, 1849--1871] and constitutes the most singular case of some extensions of several problems proposed by E. Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.
In recent years, it has been well understood that a Calderón-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a locally integrable function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.
We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.
We improve on several mixed weak type inequalities both for the Hardy-Littlewood maximal function and for Calderón-Zygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by Sawyer estimating the $L^{1, \infty}(uv)$ norm of $v^{-1}T(fv)$ for special cases. The emphasis is made in proving new and more precise quantitative estimates involving the $A_p$ or $A_\infty$ constants of the weights involved.
We show that the Hilbert transform does not map $L^1(M_Φw)$ to $L^{1,\infty}(w)$ for every Young function $Φ$ growing more slowly than $t\log\log ({\rm e}^{\rm e}+t)$. Our proof is based on a construction of M.C. Reguera and C. Thiele.