arXiv subjects

Teresa Luque

Publications and source records attributed to Teresa Luque.

10 recordsLinked to original sources

arXiv2024

Endpoint multilinear restricted weak type extrapolation theorem

In this paper we present a generalization in the context of multilinear Muckenhoupt classes of the endpoint extrapolation theorem on restricted weights due to Carro, Grafakos and Soria. Moreover, our main result is obtained on limited ranges of boundedness and to this aim we introduce a new limited range, off-diagonal extrapolation theorem in the context of restricted weights. In addition, as one of the applications, we prove endpoint estimates of certain bi-sublinear maximal functions associated with the study of return time theorems in ergodic theory.

math.CA↗

arXiv2019

Uniqueness for the inverse fixed angle scattering problem

We present a uniqueness result in dimensions $2$ and $3$ for the inverse fixed angle scattering problem associated to the Schrödinger operator $-Δ+q$, where $q$ is a small real valued potential with compact support in the Sobolev space $W^{β,2}$ with $β>0.$ This result improves the known result, due to Stefanov, in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension $4$, but for more regular potentials in $W^{β,2}$ with $β>2/3$.

math.AP↗

arXiv2018

Reproducing kernel for elastic Herglotz functions

We study the elastic Herglotz wave functions, which are entire solutions of the spectral Navier equation appearing in the linearized elasticity theory with $L^2-$far-field patterns. We characterize in three-dimensions the set of these functions $\mathcal{W},$ as a close subspace of a Hilbert space $\mathcal{H}$ of vector valued functions such that they and their spherical gradients belong to a certain weighted $L^2$ space. This allows us to prove that $\mathcal{W}$ is a reproducing kernel Hilbert space and to calculate the reproducing kernel. Finally, we outline the proof for the two-dimensional case and give the corresponding reproducing kernel.

math.CA↗

arXiv2018

A new convergent algorithm to approximate potentials from fixed angle scattering data

We introduce a new iterative method to recover a real compact supported potential of the Schrödinger operator from their fixed angle scattering data. The method combines a fixed point argument with a suitable approximation of the resolvent of the Schrödinger operator by partial sums associated to its Born series. Convergence is established for potentials with small norm in certain Sobolev spaces. As an application we show some numerical experiments that illustrate this convergence.

math.AP↗

arXiv2016

Sparse bilinear forms for Bochner Riesz multipliers and applications

We use the very recent approach developed by Lacey in [23] and extended by Bernicot-Frey-Petermichl in [3], in order to control Bochner-Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector-valued inequalities are deduced.

math.CA↗

arXiv2015

Reverse Hölder Property for strong weights and general measures

We present dimension-free reverse Hölder inequalities for strong $A^*_p$ weights, $1\le p < \infty$. We also provide a proof for the full range of local integrability of $A_1^*$ weights. The common ingredient is a multidimensional version of Riesz's "rising sun" lemma. Our results are valid for any nonnegative Radon measure with no atoms. For $p=\infty$, we also provide a reverse Hölder inequality for certain product measures. As a corollary we derive mixed $A_p^*-A_\infty^*$ weighted estimates.

math.CA↗

arXiv2013

Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

We give an alternative characterization of the class of Muckenhoupt weights $A_{\infty, \mathfrak B}$ for homothecy invariant Muckenhoupt bases $\mathfrak B$ consisting of convex sets. In particular we show that $w\in A_{\infty, \mathfrak B}$ if and only if there exists a constant $c>0$ such that for all measurable sets $E\subset \mathbb R^n$ we have $$ w({x\in \mathbb R^n: M_{\mathfrak B} (\mathbf {1}_E)(x)>1/2}) < c w(E).$$ This applies for example to the collection $\mathfrak R$ of rectangles with sides parallel to the coordinate axes, giving a new characterization of strong (multiparameter) Muckenhoupt weights. We also show versions of these results under the presence of a doubling measure. Thus the strong maximal function $M_{\mathfrak R,μ}$, defined with respect to a product-doubling measure $μ$, is bounded on $L^p(μ)$ for some $p>1$ if and only if $$μ({x\in \mathbb R^n: M_{\mathfrak R,μ} (\mathbf{1}_E)(x)>1/2}) < c μ(E)$$ for all measurable sets $E\subset \mathbb R^n$. Finally we discuss applications in differentiation theory, proving among other things that Tauberian conditions as above imply that the corresponding bases differentiate $L^\infty(μ)$, with respect to the measure $μ$.

math.CA↗

arXiv2013

Optimal exponents in weighted estimates without examples

We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ satisfies a bound like $$ \|T\|_{L^{p}(w)}\le c\, [w]^β_{A_p} \qquad w \in A_{p}, $$ then the optimal lower bound for $β$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm $\|T\|_{L^p(\mathbb{R}^n)}$ as $p$ goes to 1 and $+\infty$, which is related to Yano's classical extrapolation theorem. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximal-type, Calderón--Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner-Riesz multipliers. We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.

math.CA↗

arXiv2013

The endpoint Fefferman-Stein inequality for the strong maximal function

Let Mf denote the strong maximal function of f on R^n, that is the maximal average of f with respect to n-dimensional rectangles with sides parallel to the coordinate axes. For any dimension n>1 we prove the natural endpoint Fefferman-Stein inequality for M and any strong Muckenhoupt weight w: w({x \in R^n: M f (x) > t}) \lesssim_{w,n} \int_{R^n} |f|/t [1 + (log^+ |f|/t)^{n-1}] Mw. This extends the corresponding two-dimensional result of T. Mitsis.

math.CA↗

arXiv2012

A $B_p$ condition for the strong maximal function

A strong version of the Orlicz maximal operator is introduced and a natural $B_p$ condition for the rectangle case is defined to characterize its boundedness. This fact let us to describe a sufficient condition for the two weight inequalities of the strong maximal function in terms of power and logarithmic bumps. Results for the multilinear version of this operator and for others multi(sub)linear maximal functions associated with bases of open sets are also studied.

math.CA↗