This thesis collects various linear and nonlinear techniques developed by the author and his collaborators to attack problems in Function Space Theory, Phase Retrieval and PDE. The thesis begins with an analysis of the \emph{generalized derivative nonlinear Schr\"odinger equation}\begin{equation}\label{gDNLSi}\tag{gDNLS}
\begin{cases}
&i\partial_t u+\partial_x^2u=i|u|^{2\sigma}\partial_xu,
\
&u(0)=u_0.
\end{cases}
\end{equation}
This is a canonical model of a quasilinear dispersive PDE, i.e., a dispersive PDE where one expects continuity but not uniform continuity of the data-to-solution map. As opposed to the semilinear case where Strichartz and contraction mapping arguments are directly applicable, the well-posedness theory for such quasilinear PDE is largely open. In Chapter 2, we study \eqref{gDNLSi} when $\sigma<1$, which is the regime where local well-posedness is hardest to establish. Our main result establishes global well-posedness in the energy space $H^1$, as long as $\sigma$ is not too small.
In Chapter 3 we transition to the water waves problem. That is, we consider the motion of water when the interface between the water and the air is free to move. In this case, we do not consider the well-posedness problem, but rather the existence of special solutions. Our primary interest is in solitary waves, which are waves that travel across the ocean's surface at constant speed while never changing shape. When modelling water waves, the fundamental physical parameters are the gravity, surface tension, and fluid depth. It is then an interesting question to identify which combinations of parameters lead to a given physical phenomenon. For solitary waves in two dimensions, we discuss the complete solution to the existence/non-existence problem. More specifically, we prove non-existence of solitary waves when surface tension and depth are arbitrary but gravity is zero, which was the only case that had not yet yielded a solution.
Chapter 4 is dedicated to the phase retrieval problem; that is, the determination of a function $f$ up to unavoidable ambiguity from $|f|$. In a recent article, Calderbank, Daubechies, Freeman and Freeman dispelled of the prevailing belief that phase retrieval in infinite dimensions is inherently unstable. Motivated by this, Chapter 4 contains an extensive study of the stability of phase retrieval, for both real and complex scalars. In particular, we give the first construction of an infinite-dimensional subspace
$E\subseteq \lt(\mu; \mathbb{C})$ with the property that for any $f,g\in E$,
if $|f|$ is approximately equal to $|g|$ with respect to
the $\lt$ norm, then there exists a unimodular scalar $\lambda$
such that $f$ is approximately equal to $\lambda g$.
Recall that a basis of a Banach space $E$ is a sequence $(f_n)$ in $E$ such that every $f\in E$ admits a unique sequence of scalars $(a_n)$ satisfying $f=\sum_{n=1}^\infty a_nf_n.$ The goal of Chapter 5 is to study bases $(f_n)$ of $L^p(\mathbb{R})$ consisting entirely of non-negative functions. Such non-negative coordinate systems are of relevance in both Functional Analysis and Applied Mathematics. However, constructing them is notoriously difficult, as can be extrapolated from the following fact: For any non-negative basis $(f_n)$ of $L^p(\mathbb{R})$ there exists a permutation $\sigma:\mathbb{N}\to\mathbb{N}$ such that $(f_{\sigma(n)})$ is \emph{not} a basis of $L^p(\mathbb{R})$.
Overcoming this issue, in Chapter 5 we give the first construction of a non-negative basis of $L^2(\mathbb{R})$.
Chapter 6 is devoted to free Banach lattices. Given a Banach space $E$, one can generate a Banach lattice $\fbl[E]$ so that every operator $T:E\to X$ into a Banach lattice $X$ uniquely extends to $\fbl[E]$ as a lattice homomorphism of the same norm. The correspondence $E\mapsto \fbl[E]$ provides an indispensable link between Banach space theory and Banach lattice theory. In Chapter 6, we give a convenient functional representation of $\fbl[E]$ and its $p$-convex variants, and then deeply study these spaces. In particular, we study how properties of an operator $T:E\rightarrow F$ between Banach spaces transfer to the associated lattice homomorphism $\overline{T}:\fbl[E]\rightarrow \fbl[F]$. Special consideration is devoted to the case when the operator $T$ is an isomorphic embedding, which leads us to examine extension properties of operators into $\ell_p$, and several classical Banach space properties such as being a G.T. space. A detailed investigation of basic sequences and sublattices of free Banach lattices is also provided. Among other things, this allows us to settle an a priori unrelated question, providing the first instance of a subspace of a Banach lattice without bibasic sequences. Along the way, a dictionary between Banach space properties of $E$ and Banach lattice properties of $\fbl[E]$ is assembled. For example, we characterize the existence of lattice copies of $\ell_1$ in $\fbl[E]$ and show that $\fbl[E]$ has an upper $p$-estimate if and only if $id_{E^*}$ is $(q,1)$-summing ($\frac{1}{p}+\frac{1}{q}=1$).
