Harmonic Analysishttp://hdl.handle.net/20.500.11824/32020-09-02T05:37:43Z2020-09-02T05:37:43ZThe observational limit of wave packets with noisy measurementsCaro P.Meroño C.http://hdl.handle.net/20.500.11824/11442020-09-02T01:00:13Z2019-01-01T00:00:00ZThe observational limit of wave packets with noisy measurements
Caro P.; Meroño C.
The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian white noise. The authors show how wave packets can be used to partially recover the observable from the measurements almost surely. Furthermore, they point out the limitation of wave packets to recover the remaining part of the observable, and show how the errors hide the signal coming from the observable. The recovery results are based on an ergodicity property of the errors produced by wave packets.
2019-01-01T00:00:00ZScattering with critically-singular and δ-shell potentialsCaro P.García A.http://hdl.handle.net/20.500.11824/11432020-09-02T01:00:11Z2019-01-01T00:00:00ZScattering with critically-singular and δ-shell potentials
Caro P.; García A.
The authors consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz hypersurface. They study direct and inverse point-source scattering under the assumptions that the potentials are real-valued and compactly supported. To solve the direct scattering problem, the authors introduce two functional spaces ---sort of Bourgain type spaces--- that allow to refine the classical resolvent estimates of Agmon and Hörmander, and Kenig, Ruiz and Sogge. These spaces seem to be very useful to deal with the critically-singular and δ-shell components of the potentials at the same time. Furthermore, these spaces and their corresponding resolvent estimates turn out to have a strong connection with the estimates for the conjugated Laplacian used in the context of the inverse Calderón problem. In fact, the authors derive the classical estimates by Sylvester and Uhlmann, and the more recent ones by Haberman and Tataru after some embedding properties of these new spaces. Regarding the inverse scattering problem,the authors prove uniqueness for the potentials from point-source scattering data at fix energy. To address the question of uniqueness the authors combine some of the most advanced techniques in the construction of complex geometrical optics solutions.
2019-01-01T00:00:00ZTopics in Harmonic Analysis; commutators and directional singular integralsAccomazzo N.http://hdl.handle.net/20.500.11824/11422020-09-01T01:00:12Z2020-03-01T00:00:00ZTopics in Harmonic Analysis; commutators and directional singular integrals
Accomazzo N.
This dissertation focuses on two main topics: commutators and maximal directional operators.
Our first topic will also distinguish between two cases: commutators of singular integral operators
and BMO functions and commutators of fractional integral operators and a BMO class that comes
from changing the underlying measure. Commutators are not only interesting for its own sake,
but they have been broadly studied because of their connection to PDEs.
Our first result gives us a new way of characterizing the class BMO. Assuming that the commutator
of the Hilbert transform in dimension 1 (or a Riesz transform in dimensions 2 and higher) and the
symbol b satisfy an Llog L-type of modular inequality on the endpoint with constant B, we can
bound the BMO norm of the symbol by a fixed multiple of B; thus providing an endpoint version
of the classical result of Coifman, Rochberg and Weiss for commutators of Calderón-Zygmund
operators and BMO.
We also studied commutators of fractional integrals and BMO. In this case, we were interested in
finding quantitave two-weights estimates for the iterated version of these operators. We extended
the known sharp inequalities for the commutator of first order to the iterated case and also provided
a new proof of the previous results.
Lastly, we studied maximal directional operators. Specifically, we considered a singular integral
operator that commutes with translations and studied the maximal directional operator that arises
from it. We proved that for any subset of cardinality N of a lacunary set of directions we can
bound the Lp(Rn)-norm of the operator by the sharp bound √log N, thus completing some previous results on the Hilbert transform on low dimensions.
2020-03-01T00:00:00ZSharp reverse Hölder inequality for Cp weights and applicationsCanto J.http://hdl.handle.net/20.500.11824/11072020-06-08T01:00:10Z2020-01-01T00:00:00ZSharp 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).
2020-01-01T00:00:00Z