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We provide a complete solution to the problem of infinite quantum signal processing for the class of Szeg\H{o} functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a quantum signal processing representation. We do so by introducing a new algorithm called the Riemann-Hilbert-Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szeg\H{o} function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory., Comment: 45 pages, 5 figures. Small updates to main text, and ArXiv title/abstract now renders correctly

We prove a new generalization of a theorem of Carleson, namely bounds for a generalized Carleson operator on doubling metric measure spaces. Additionally, we explicitly reduce Carleson's classical result on pointwise convergence of Fourier series to this new theorem. Both proofs are presented in great detail, suitable as a blueprint for computer verification using the current capabilities of the software package Lean. Note that even Carleson's classical result has not yet been computer-verified., Comment: 130 pages

We prove bounds in the strict local $L^{2}(\mathbb{R}^{d})$ range for trilinear Fourier multiplier forms with a $d$-dimensional singular subspace. Given a fixed parameter $K \ge 1$, we treat multipliers with non-degenerate singularity that are push-forwards by $K$-quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local $L^{2}$ range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.

Elucidating a connection with nonlinear Fourier analysis, we extend a well known algorithm in quantum signal processing to represent measurable signals by square summable sequences. Each coefficient of the sequence is Lipschitz continuous as a function of the signal., Comment: 32 pages, 1 figure. To appear in Revista Matematica Complutense. Thanks to Jiasu Wang for pointing out typos in the previous version of this article

We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all $r\ge 2$ as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any $r<\infty$ for more than three commuting transformations., Comment: 48 pages; v2: introduction expanded, figures added

We construct phase space localizing operators in all dimensions. These are frequency localized variants of the conditional expectation operator related to a dyadic stopping time. Our construction is an improvement over the so-called phase plane projections of Muscalu, Tao and the third author in one dimension. The motivation for such operators comes from time-frequency analysis. They are used in particular to prove uniform estimates for multilinear modulation invariant operators., Comment: v3: added a reference to the application of the result. "Outside of the tree" item of the main theorem now has a slightly stronger form (contained in the old proof but not in the old statement)

In the past decade, much effort has gone into understanding maximizers for Fourier restriction and extension inequalities. Nearly all of the cases in which maximizers for inequalities involving the restriction or extension operator have been successfully identified can be seen as partial answers to the question in the title. In this survey, we focus on recent developments in sharp restriction theory relevant to this question. We present results in the algebraic case for spherical and hyperbolic extension inequalities. We also discuss the use of the Penrose transform leading to some negative answers in the case of the cone., Comment: 33 pages, 4 figures, 1 table, survey paper

We prove a range of $L^p$ bounds for singular Brascamp-Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy-Schwarz inequality. We allow $2^{m-1}

We prove bounds in the local $ L^2 $ range for exotic paraproducts motivated by bilinear multipliers associated with convex sets. One result assumes an exponential boundary curve. Another one assumes a higher order lacunarity condition., Comment: v2: authors now in right alphabetical order

Among the class of functions on the circle with Fourier modes up to degree 120, constant functions are the unique real-valued maximizers for the endpoint Tomas-Stein inequality., Comment: for v5: simplified and improved supporting information. Minor amendment to introduction reflecting improved calculation performance. No changes to results or proof

We present an overview of results on multi-linear singular integrals in the broader context of Brascamp-Lieb inequalities. This elaborates a lecture given at the inspiring conference on Geometric Aspects of Harmonic Analysis at Cortona 2018 in honor of Fulvio Ricci., Comment: 21 pages

We prove a singular Brascamp-Lieb inequality, stated in Theorem 1, with a large group of involutive symmetries., Comment: 16 pages

Among the class of functions with Fourier modes up to degree 30, constant functions are the unique real-valued maximizers for the endpoint Tomas-Stein inequality on the circle., Comment: 10 pages, 3 figures, 1 table, 1 ancillary file (Mathematica notebook)

We discuss a formal system of mathematics. We use it to construct the natural numbers., Comment: Updated from "On the first chapter" and "Definitions in mathematics" 39 pages, many tree figures

First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direction fields with finitely many directions., Comment: 31 p

We study the linearized maximal operator associated with dilates of the hyperbolic cross multiplier in dimension two. Assuming a Lipschitz condition and a lower bound on the linearizing function, we obtain $L^{p}(\mathbb{R}^{2}) \to L^{p}(\mathbb{R}^{2})$ bounds for all $1

We prove $\mathrm{L}^p$ bounds for the truncated simplex Hilbert transform which grow with a power less than one of the truncation range in the logarithmic scale., Comment: 11 pages, 3 figures; v2: minor corrections, accepted for publication in J. Anal. Math

We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations., Comment: 48 pages

We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform., Comment: 28 pages

We study a trilinear singular integral form acting on two-dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calder\'on commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the ${L}^p$ estimates hold for the considered form., Comment: 12 pages

In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the $L^2 - L^6$ Tomas-Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of sixfold products of Bessel functions developed in a companion paper. We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program., Comment: 19 pages, 2 tables

We establish good numerical estimates for a certain class of integrals involving sixfold products of Bessel functions. We use relatively elementary methods. The estimates will be used in the study of a sharp Fourier restriction inequality on the circle., Comment: 41 pages, 1 figure, 1 table

We prove a multilinear local $T(b)$ theorem that differs from previously considered multilinear local $T(b)$ theorems in using exclusively general testing functions $b$ as opposed to a mix of general testing functions and indicator functions. The main new feature is a set of relations between the various testing functions $b$ that to our knowledge has not been observed in the literature and is necessitated by our approach. For simplicity we restrict attention to the perfect dyadic model., Comment: 24 pages, no figures

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well studied objects of harmonic analysis. We investigate $L^p$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one-dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform., Comment: 20 pages

We study the behavior of the bilinear Hilbert transform $\mathrm{BHT}$ at the boundary of the known boundedness region $\mathcal H$. A sample of our results is the estimate $| \langle\mathrm{BHT}(f_1,f_2),f_3 \rangle | \leq C |F_1|^{\frac34}|F_2| ^{\frac34} |F_3|^{-\frac12} \log\log \Big(\mathrm{e}^{\mathrm{e}} + \frac{|F_3|}{\min\{|F_1|,|F_2|\}} \Big) $ valid for all tuples of sets $F_j \subset \mathbb R $ of finite measure and functions $f_j$ such that $|f_j| \leq \mathbf{1}_{F_j}$, $j=1,2,3$, with the additional restriction that $f_3$ be supported on a major subset $F_3'$ of $F_3$ that depends on $\{F_j:j=1,2,3\}$. The double logarithmic term improves over the single logarithmic term obtained by Bilyk and Grafakos. Whether the double logarithmic term can be removed entirely, as is the case for the quartile operator discussed by Demeter and the first author, remains open. We employ our endpoint results to describe the blow-up rate of weak-type and strong-type estimates for $\mathrm{BHT}$ as the tuple $\vec \alpha$ approaches the boundary of $\mathcal H$. We also discuss bounds on Lorentz-Orlicz spaces near $L^{\frac23}$, improving on results of Carro, Grafakos, Martell and Soria. The main technical novelty in our article is an enhanced version of the multi-frequency Calder\'on-Zygmund decomposition by Nazarov, Oberlin and the second author., Comment: 42 pages, 1 figure, 1 table. Submitted

We develop a theory of Lp spaces based on outer measures rather than measures. This theory includes the classical Lp theory on measure spaces as special case. It also covers parts of potential theory and Carleson embedding theorems. The theory turns out to be an elegant language to describe aspects of classical singular integral theory such as paraproduct estimates and T(1) theorems, and it is particularly useful for generalizations of singular integral theory in time-frequency analysis. We formulate and prove a generalized Carleson embedding theorem and give a relatively short reduction of basic estimates for the bilinear Hilbert transform to this new Carleson embedding theorem., Comment: 47 pages. The August 2014 version further revises previous versions

We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calder\'{o}n-Zygmund operators as several functions can now depend on the same one-dimensional variable. The study of this class is motivated by examples related to the two-dimensional bilinear Hilbert transform and to bilinear ergodic averages. This paper is a sequel to the prior paper arXiv:1108.0917 by the first author., Comment: 20 pages

The nonlinear Fourier transform discussed in these notes is the map from the potential of a one dimensional discrete Dirac operator to the transmission and reflection coefficients thereof. Emphasis is on this being a nonlinear variant of the classical Fourier series, and on nonlinear analogues of classical analytic facts about Fourier series. These notes are a summary of a series of lectures given in 2003 at the Park City Mathematics Institute.

Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, $L^2$ whenever $v$ is Lipschitz. We establish a wide range of $L^p$ estimates for this operator when $v$ is a measurable, non-vanishing, one-variable vector field in $\bbr ^2$. Aside from an $L^2$ estimate following from a simple trick with Carleson's theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author (\cite{B2})., Comment: 25 pages

We prove that the analogue for the Hilbert transform of a classical weighted inequality by Fefferman and Stein for the Hardy Littlewood maximal operator does not hold. This is a sequel to paper arXiv:1008.3943 by the first author, which shows the same for the general Haar multiplier operator., Comment: 5 pages

We generalize a family of variation norm estimates of Lepingle with endpoint estimates of Bourgain and Pisier-Xu to a family of variational estimates for paraproducts, both in the discrete and the continuous setting. This expands on work of Friz and Victoir, our focus being on the continuous case and an expanded range of variation exponents., Comment: 23 pages