Topics in harmonic analysis related to Rubio de Francia square functions and directional singular integrals
This thesis focuses on the $L^p $-boundedness of maximal directional singular integral operators and Rubio de Francia square functions. Firstly, we study maximal directional singular integral operators in $ \mathbb{R}^n $ defined by a Hörmander--Mihlin multiplier on an $ (n-1)$-dimensional subspace...
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| Tipo de documento: | tese |
| Estado: | Versão publicada |
| Data de publicação: | 2025 |
| País: | España |
| Recursos: | Basque Center for Applied Mathematics (BCAM) |
| Repositório: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/2029 |
| Acesso em linha: | http://hdl.handle.net/20.500.11824/2029 |
| Access Level: | Acceso aberto |
| Palavra-chave: | time-frequency analysis directional singular integrals sparse domination Rubio de Francia square functions Stein's conjecture |
| Resumo: | This thesis focuses on the $L^p $-boundedness of maximal directional singular integral operators and Rubio de Francia square functions. Firstly, we study maximal directional singular integral operators in $ \mathbb{R}^n $ defined by a Hörmander--Mihlin multiplier on an $ (n-1)$-dimensional subspace which act trivially in the perpendicular direction. The choice of subspace depends measurably on the first $ n-1 $ variables of $ \mathbb{R}^n $. Assuming the subspace to be non-degenerate in the sense that it is contained in a subspace of $ \mathbb{R}^n $ away of a cone around $e_n$ and the function $ f $ to be frequency supported in a cone away from $ \mathbb{R}^{n-1} $, we prove $ L^p $ bounds for these operators when $ p > 3/2 $. If we assume, additionally, that $ \widehat{f} $ is supported in a single frequency band, we are able to extend the boundedness range to $ p >1 $. The non-degeneracy assumption cannot, in general, be removed, even in the band-limited case. Secondly, we study one-dimensional square functions in the spirit of Rubio de Francia. Let $P_\omega f$ be the Fourier restriction of $f\in L^2(\mathbb{R})$ to an interval $\omega\subset \mathbb{R}$. If $\Omega$ is an arbitrary collection of pairwise disjoint intervals, the square function of $\{P_\omega f: \omega \in \Omega\}$ is termed the Rubio de Francia square function $T^{\Omega}_{\operatorname{RF}}$. In this thesis we prove a pointwise bound for $T^{\Omega}_{\operatorname{RF}}$ by a sparse operator involving local $L^2$-averages. A pointwise bound for the smooth version of $T^{\Omega}_{\operatorname{RF}}$ by a sparse square function is also proved. These pointwise localization principles lead to quantitative $L^p(w)$, $p>2$, and weighted weak-type $(p,p)$, $p\geq 2$, norm inequalities for $T^{\Omega}_{\operatorname{RF}}$. In particular, the obtained weak $L^p(w)$ norm bounds are new for $p\geq 2$ and sharp for $p>2$. The proofs rely on sparse bounds for abstract balayages of Carleson sequences, oscillation inequalities, local orthogonality and time-frequency analysis discretization techniques. The thesis also contains two results related to the outstanding conjecture that $T^{\Omega}_{\operatorname{RF}}$ is bounded on $L^2(w)$ if and only if $w\in A_1$. The conjecture is verified for radially decreasing, even $A_1$ weights, and in full generality for the Walsh group analogue of $T^{\Omega}_{\operatorname{RF}}$. |
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