arXiv subjects
Publications and source records attributed to Israel P. Rivera-Ríos.
We provide convex body domination results for the generalized vector-valued commutator of those operators that admit specific forms of convex body domination themselves. We also prove some strong type estimates and other consequences of these results, and we study the BMO spaces that appear naturally in this context.
In this paper matrix quantitative weighted estimates on spaces of homogeneous type, such as endpoint estimates, strong type estimates are provided. To that end we extend some earlier results on convex body domination due to Nazarov, Petermichl, Treil and Volberg to this setting. We also provide a $T(1)$ alike convex body domination result analogous to the one provided by Lerner and Ombrosi, and an application to vector valued extensions of Petermichl operators.
In this paper, an approach to the one sided maximal function in the spirit of the Christ-Fefferman proof for the strong type weighted estimates of the maximal function is provided. As applications of that approach, we provide an alternative proof of the sharp weighted estimate for the one sided maximal function that was settled by one of us and de la Torre, a one sided two weight bumps counterpart of a result of Pérez and Rela, and also one sided counterparts of some very recent mixed weak type results due to Sweeting.
In this paper one sided counterparts of compactness extrapolation results of Hytönen and Lappas are provided. As a consequence of those results, compactness results for one sided singular integrals, commutators of one sided fractional integrals and a certain class of $L^{r}$-Hörmander operators are provided. The results for commutators of $L^{r}$-Hörmander operators seem new even in the classical setting.
In a recent work by Cruz-Uribe et al. was obtained that \[|\{x\in{\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>α\}|\lesssim\frac{[w]_{A_1}^2}α\int_{{\mathbb{R}^d}}|f|dx\] both in the matrix and scalar settings, where $G$ is either the Hardy-Littlewood maximal function or any Calderón-Zygmund operator. In this note we show that the quadratic dependence on $[w]_{A_1}$ is sharp. This is done by constructing a sequence of scalar-valued weights with blowing up characteristics so that the corresponding bounds for the Hilbert transform and maximal function are exactly quadratic.
In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calderón-Zygmund operators with a quantity of operators beyond the multilinear Calderón-Zygmund theory. We prove that such multilinear operators and corresponding commutators are locally pointwise dominated by two sparse dyadic operators, respectively. We also establish three kinds of typical estimates: local exponential decay estimates, mixed weak type estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de Francia extrapolation for abstract multilinear compact operators, we obtain weighted compactness for commutators of specific multilinear operators on spaces of homogeneous type. A compact extrapolation allows us to get full range of exponents, while weighted interpolation for multilinear compact operators is crucial to the compact extrapolation. These are due to a weighted Fréchet-Kolmogorov theorem in the quasi-Banach range, which gives a characterization of relative compactness of subsets in weighted Lebesgue spaces. As applications, we illustrate multilinear bounded oscillation operators with examples including multilinear Hardy-Littlewood maximal operators on measure spaces, multilinear $ω$-Calderón-Zygmund operators on spaces of homogeneous type, multilinear Littlewood-Paley square operators, multilinear Fourier integral operators, higher order Calderón commutators, maximally modulated multilinear singular integrals, and $q$-variation of $ω$-Calderón-Zygmund operators.
In this paper we obtain the sharp quantitative matrix weighted weak type bounds for the Christ--Goldberg maximal operator $M_{W,p}$ in the case $1<p<2$, improving a recent result by Cruz-Uribe and Sweeting. Also, in the scalar setting, we improve a weak type bound obtained in the aforementioned work for Calderón--Zygmund operators.
The purpose of this note is to extend the extrapolation result by by Cruz-Uribe Martell and Pérez as follows. Given a family $\mathcal{F}$ of pairs of functions suppose that for some $0 0$, then, for every $u\in A_{1}$ and every $v\in A_{\infty}$ we have that \[ \left\Vert \frac{f}{v}\right\Vert_{L^{A,\infty}(uv)}\lesssim\left\Vert \frac{g}{v}\right\Vert_{L^{A,\infty}(uv)}, \] where \[ L^{A,\infty}(uv)=\inf\left\{ λ>0:\sup_{t>0}A(t)w\left(\left\{ x\in\mathbb{R}:|f(x)|>λt\right\} \right)\leq1\right\} \] is the weak Orlicz type introduced by Iaffei. As a corollary of this extrapolation result we derive a mixed weak type inequality for Coifman-Rochberg-Weiss commutators.
In this note we revisit the upper endpoint estimates for commutators following the line by Harboure, Segovia and Torrea. Relying upon the suitable BMO subspace suited for the commutator that was introduced by Accomazzo, we obtain a counterpart for commutators of the upper endpoint extrapolation result by Harboure, Macías and Segovia. Multilinear counterparts are provided as well.
In this paper we provide some quantitative one-sided estimates that recover the dependences in the classical setting. Among them we provide estimates for the one-sided maximal function in Lorentz spaces and we show that the conjugation method for commutators works as well in this setting.
In this paper we provide mixed weak type inequalities generalizing previous results in an earlier work by Caldarelli and the second author and also in the spirit of earlier results by Lorente, Martell, Pérez and Riveros. One of the main novelties is that, besides obtaining estimates in the euclidean setting, results are provided as well in spaces of homogeneous type, being the first mixed weak type estimates that we are aware of in that setting.
We recall that $w\in C_{p}^{+}$ if there exist $\varepsilon>0$ and $C>0$ such that for any $a<b<c$ with $c-b<b-a$ and any measurable set $E\subset(a,b)$, the following holds \[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^{+}χ_{(a,c)}\right)^{p}w<\infty. \] This condition was introduced by Riveros and de la Torre as a one-sided counterpart of the $C_{p}$ condition studied first by Muckenhoupt and Sawyer. In this paper we show that given $1<p<q<\infty$ if $w\in C_{q}^{+}$ then \[ \|M^{+}f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)} \] and conversely if such an inequality holds, then $w\in C_{p}^{+}.$
In this paper endpoint entropy Fefferman-Stein bounds for Calderón-Zygmund operators introduced by Rahm in are extended to iterated Coifman-Rochberg-Weiss commutators.
In this paper, building upon ideas of Naor and Tao and continuing the study initiated in by the authors and Safe, sufficient conditions are provided for weighted weak type and strong type $(p,p)$ estimates with $p>1$ for the centered maximal function on the infinite rooted $k$-ary tree to hold. Consequently a wider class of weights for those strong and weak type $(p,p)$ estimates than the one obtained in by the authors and Safe in a previous work is provided. Examples showing that the Sawyer type testing condition and the $A_p$ condition do not seem precise in this context are supplied as well. We also prove that strong and weak type estimates are not equivalent, highlighting the pathological nature of the theory of weights in this setting. Two weight counterparts of our conditions will be obtained as well.
In this paper quantitative weighted matrix estimates for vector valued extensions of $L^{r'}$-Hörmander operators and rough singular integrals are studied. Strong type $(p,p)$ estimates, endpoint estimates, and some new results on Coifman-Fefferman estimates assuming $A_\infty$ and $C_p$ condition counterparts are provided. To prove the aforementioned estimates we rely upon some suitable convex body domination results that we settle as well in this paper.
Let $S_α$ be the multilinear square function defined on the cone with aperture $α\geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_α$. We first obtain a sharp weighted estimate in terms of aperture $α$ and $\vec{w} \in A_{\vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer's conjecture, for which a Coifman-Fefferman inequality with the precise $A_{\infty}$ norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood-Paley $g^*_λ$ function. Some results are new even in the linear case.
We provide an abstract estimate of the form \[ \|f-f_{Q,μ}\|_{X \left(Q,\frac{\mathrm{d} μ}{Y(Q)}\right)}\leq c(μ,Y)ψ(X)\|f\|_{\mathrm{BMO}(\mathrm{d}μ)} \] for all cubes $Q$ in $\mathbb{R}^n$ and every function $f\in \mathrm{BMO}(\mathrm{d}μ)$, where $μ$ is a doubling measure in $\mathbb{R}^n$, $Y$ is some positive functional defined on cubes, $\|\cdot \|_{X \left(Q,\frac{\mathrm{d} w}{w(Q)}\right)}$ is a sufficiently good quasi-norm and $c(μ,Y)$ and $ψ(X)$ are positive constants depending on $μ$ and $Y$, and $X$, respectively. That abstract scheme allows us to recover the sharp estimate \[ \|f-f_{Q,μ}\|_{L^p \left(Q,\frac{\mathrm{d} μ(x)}{μ(Q)}\right)}\leq c(μ)p\|f\|_{\mathrm{BMO}(\mathrm{d}μ)}, \qquad p\geq1 \] for every cube $Q$ and every $f\in \mathrm{BMO}(\mathrm{d}μ)$, which is known to be equivalent to the John-Nirenberg inequality, and also enables us to obtain quantitative counterparts when $L^p$ is replaced by suitable strong and weak Orlicz spaces and $L^{p(\cdot)}$ spaces. Besides the aforementioned results we also generalize Theorem 1.2 in [OPRRR20] to the setting of doubling measures and obtain a new characterization of Muckenhoupt's $A_\infty$ weights.
In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators $T_b^m$. Our results are new even for the first order commutator $T_b^1$. A new bump type necessary condition for the two-weighted boundedness of $T_b^m$ is obtained as well. We also provide some results related to a converse to Bloom's theorem.