arXiv subjects
Publications and source records attributed to Maria Carmen Reguera.
Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective property with power savings (b) characterize uniform distribution, (c) establish polynomial mean ergodic theorems and (d) study metric pair correlation of Cantor sets.
Let $\mathcal C= \{k_1 0. $$ This is an extension of the IP Ergodic Theorem of Furstenberg and Katznelson, and a partial extension of recent work of Kra and Shalom. In particular, this implies that for any subset of integers $A$ of positive upper Banach density, there is a set $B$ of integers $n$ of positive lower Banach density such that $A$ contains an $\ell+1$ term progression, with step size $k_n$, where $n\in B$. This is a complement to recent results of Kra and Shalom, for IP Sets of integers, and Burgin, concerning Sarkozy's Theorem for Primes with restricted digits.
We provide quantitative weighted weak type estimates for non-integral square functions in the critical case $p=2$ in terms of the $A_p$ and reverse Hölder constants associated to the weight. The method of proof uses a decoupling of the role of the weights via a quantitative version of Gehring's lemma. The results can be extended to other $p$ in the range of boundedness of the square function at hand.
We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob's maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension.
We prove that, for totally irregular measures $μ$ on $\mathbb{R}^{d}$ with $d\geq3$, the $(d-1)$-dimensional Riesz transform $$ T_{A,μ}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \, d μ(y) $$ adapted to the Schrödinger operator $L_{A}^{V} = -\mathrm{div} A \nabla + V$ with fundamental solution $\mathcal{E}_{A}^{V}$ is not bounded on $L^{2}(μ)$. This generalises recent results obtained by Conde-Alonso, Mourgoglou and Tolsa for free-space elliptic operators with Hölder continuous coefficients $A$ since it allows for the presence of potentials $V$ in the reverse Hölder class $RH_{d}$. We achieve this by obtaining new exponential decay estimates for the kernel $\nabla_{1} \mathcal{E}_{A}^{V}$ as well as Hölder regularity estimates at local scales determined by the potential's critical radius function.
We prove a quadratic sparse domination result for general non-integral square functions $S$. That is, we prove an estimate of the form \begin{equation*} \int_{M} (S f)^{2} g \, \mathrm{d}μ\le c \sum_{P \in \mathcal{S}} \left(\frac{1}{\lvert 5P \rvert}\int_{5 P} \lvert f\rvert^{p_{0}} \, \mathrm{d}μ\right)^{2/p_{0}} \left(\frac{1}{\lvert 5P \rvert} \int_{5 P} \lvert g\rvert^{q_{0}^*}\,\mathrm{d}μ\right)^{1/q_{0}^*} \lvert P\rvert, \end{equation*} where $q_{0}^{*}$ is the Hölder conjugate of $q_{0}/2$, $M$ is the underlying doubling space and $\mathcal{S}$ is a sparse collection of cubes on $M$. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace-Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space $L^{p}(w)$.
The Bochner-Riesz multipliers $ B_{δ}$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< δ< \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ δ$ increases to the critical value, $ δ=\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $ n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.
Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $μ$ in $\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(μ)$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Δ)^{α/2}$, $α\in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
In this paper we obtain precise estimates for the $L^2$ norm of the $s$-dimensional Riesz transforms on very general measures supported on Cantor sets in $\mathbb R^d$, with $d-1<s<d$. From these estimates we infer that, for the so called uniformly disconnected compact sets, the capacity $γ_s$ associated with the Riesz kernel $x/|x|^{s+1}$ is comparable to the capacity $\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$ from non-linear potential theory.
We obtain a sharp norm estimate for Hankel operators with anti-analytic symbol for weighted Bergman spaces. For the classical Bergman space, the estimate improves the corresponding classical Putnam inequality for commutators of Toeplitz operators with analytic symbol by a factor of $1/2$, answering a recent conjecture by Bell, Ferguson and Lundberg. As an application, this yields a new proof of the de St. Venant inequality, which relates the torsional rigidity of a domain with its area.
The Hilbert transform has a linear bound in the $A_{2}$ characteristic on weighted $L^{2}$, \begin{equation*} \left\Vert H\right\Vert _{L^{2}(w)\rightarrow L^{2}(w)}\lesssim \left[ w \right] _{A_{2}}, \end{equation*} and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation $M_{w^{\frac{1}{2}}}\mathcal{S }M_{w^{-\frac{1}{2}}}$ induced by the canonical decomposition of a multiplier into paraproducts:% \begin{equation*} M_{f}=P_{f}^{-}+P_{f}^{0}+P_{f}^{+}. \end{equation*} The main tools used are composition of paraproducts, a product formula for Haar coefficients, the Carleson Embedding Theorem, and the linear bound for the square function.
We provide a counterexample to the Sarason Conjecture for the Bergman space and present a characterisation of bounded Toeplitz products on the Bergman space in terms of test functions by means of a dyadic model approach. We also present some results about two-weighted estimates for the Bergman projection. Finally, we introduce the class $B_{\infty}$ and give sharp estimates for the one-weighted Bergman projection.
We obtain necessary and sufficient conditions to characterize the boundedness of the composition of dyadic paraproduct operators.
We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Bekolle constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Perez.
We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of the two operators is equivalent for $A_p$ weights with $1 < p < \infty$, through examples we illustrate this is not the case in more general contexts. In particular, we study weights for which the maximal operator is bounded on the corresponding $L^p$ spaces while the Hilbert transform is not. We focus on weights which take the value zero on sets of non-zero measure and exploit this lack of strict positivity in our constructions. These type of weights and techniques have been explored previously in Reguera \cite{1008.3943} and Reguera-Thiele \cite{1011.1767}.
As a corollary to our main result we deduce sharp A_p$ inequalities for T being either the Hilbert transform in dimension d=1, the Beurling transform in dimension d=2, or a Riesz transform in any dimension d\ge 2. For T_{\ast} the maximal truncations of these operators, we prove the sharp A_p weighted weak and strong-type L ^{p} (w) inequalities, for all 1<p<\infty. Key elements of the proof are (1) extrapolation (2) a recent argument for the A_2 bound in the untruncated case, an argument of Lacey-Petermichl-Reguera. (3) a weak-L^1 estimate for duals of maximal truncations. And (4) recent characterizations of the two-weight inequalities for strong and weak type inequalities, due to Lacey-Sawyer-Uriate-Tuero.
For a Calderon-Zygmund operator T on d-dimensional space, that has a sufficiently smooth kernel, we prove that for any 1< p \le 2, and weight w in A_p, that the maximal truncations T_* of T map L^p(w) to weak-L^p(w), with norm bounded by the A_p characteristic of w to the first power. This result combined with the (deep) recent result of Perez-Treil-Volberg, shows that the strong-type of T on L^2(w) is bounded by A_2 characteristic of w to the first power. (It is well-known that L^2 is the critical case for the strong type estimate.) Both results are sharp, aside from the number of derivatives imposed on the kernel of the operator. The proof uses the full structure theory of Calderon-Zygmund Operators, reduction to testing conditions, and a Corona argument.
For 1<p< \infty, weight w \in A_p, and any L ^2 -bounded Calderón-Zygmund operator T, we show that there is a constant C(T,P) so that we prove the sharp norm dependence on T_#, the maximal truncations of T, in both weak and strong type L^p(w) norms. Namely, for the weak type norm, T_# maps L^p(w) to weak-L^p(w) with norm at most \|w\|_{A_p}. And for the strong type norm, the norm estimate is \|w\|_{A_p}^{\max(1, (p-1) ^{-1})}. These estimates are not improvable in the power of \lVert w\rVert_{A_p}. Our argument follows the outlines of the arguments of Lacey-Petermichl-Reguera (Math.\ Ann.\ 2010) and Hytönen-Pérez-Treil-Volberg (arXiv, 2010) with new ingredients, including a weak-type estimate for certain duals of T_#, and sufficient conditions for two weight inequalities in L ^{p} for T_#. Our proof does not rely upon extrapolation.