Harmonic Analysis
http://hdl.handle.net/20.500.11824/3
2020-06-09T05:47:27ZSharp reverse Hölder inequality for Cp weights and applications
http://hdl.handle.net/20.500.11824/1107
Sharp reverse Hölder inequality for Cp weights and applications
Canto J.
We prove an appropriate sharp quantitative reverse Hölder inequality for the $C_p$ class of
weights fromwhich we obtain as a limiting case the sharp reverse Hölder inequality for
the $A_\infty$ class of weights (Hytönen in Anal PDE 6:777–818, 2013; Hytönen in J Funct
Anal 12:3883–3899, 2012). We use this result to provide a quantitative weighted norm
inequality between Calderón–Zygmund operators and theHardy–Littlewood maximal
function, precisely
$$|| T f ||_{ L^p(w)} \leq C_{T,n,p,q} [w]_{C_q} (1 + \log^+[w]_{C_q} ) ||Mf ||_{ L^p(w)} ,$$
for $w ∈ C_q$ and $q > p > 1$, quantifying Sawyer’s theorem (StudMath 75(3):753–763,
1983).
2020-01-01T00:00:00ZA Bilinear Strategy for Calderón’s Problem
http://hdl.handle.net/20.500.11824/1105
A Bilinear Strategy for Calderón’s Problem
Ponce Vanegas F.
Electrical Impedance Imaging would suffer a serious obstruction if two different conductivities yielded the same measurements of potential and current at the boundary. The Calderón’s problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In $\mathbb{R}^d$, for $d\ge 5$, we show that uniqueness holds when the conductivity is in $W^{1+\frac{d-5}{2p}+, p}(\Omega)$ for $d\le p<\infty$. This improves on recent results of Haberman, and of Ham, Kwon and Lee. The main novelty of the proof is an extension of Tao’s Bilinear Theorem.
2020-05-01T00:00:00ZA note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities
http://hdl.handle.net/20.500.11824/1101
A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities
Martinez-Perales J.C.
The main result of this paper supports a conjecture by C. P\'erez and E. Rela about the properties of the weight appearing in their recent self-improving result of generalized inequalities of Poincar\'e-type in the Euclidean space. The result we obtain does not need any condition on the weight, but still is not fully satisfactory, even though the result by P\'erez and Rela is obtained as a corollary of ours. Also, we extend the conclusions of their theorem to the range $p<1$.
As an application of our result, we give a unified vision of weighted improved Poincar\'e-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincar\'e inequalities within an approach which avoids any representation formula. We obtain results related to some already existing results in the literature and furthermore we improve them in some aspects.
Finally, we also explore analog inequalities in the context of metric spaces by means of the already known self-improving results in this setting.
2020-01-01T00:00:00ZA Decomposition of Calderón–Zygmund Type and Some Observations on Differentiation of Integrals on the Infinite-Dimensional Torus
http://hdl.handle.net/20.500.11824/1100
A Decomposition of Calderón–Zygmund Type and Some Observations on Differentiation of Integrals on the Infinite-Dimensional Torus
Fernández E.; Roncal L.
In this note we will show a Calder\'on--Zygmund decomposition associated with a function $f\in L^1(\mathbb{T}^{\omega})$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.
2020-02-13T00:00:00Z