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We develop a set of techniques that enable us to effectively recover Besov rough analysis from p-variation rough analysis. Central to our approach are new metric groups, in which some objects in rough path theory that have been previously viewed as two-parameter can be considered as path increments. Furthermore, we develop highly precise Lipschitz estimates for Young and rough differential equations, both in the variation and Besov scale.

We present a few techniques for proving $L^p$ estimates for martingales. Basic applications to It\^o integration and rough paths are included., Comment: 45 pages. University of Bonn, Winter term 2021/22

We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\widehat{f}$ is supported on a small perturbation of the moment curve in $\mathbb{R}^k$, then $f$ is identically zero. This improves the more general result of Agranovsky and Narayanan, and the exponents are sharp in all dimensions. In the process, we develop a mechanism that should lead to further progress on related problems.

We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti (arXiv:2005.09287) to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel-Lizorkin type spaces., Comment: v3: 11 pages, with small corrections and additional references. To appear in Rev. Mat. Iberoam

For a Hilbert space valued martingale $(f_n)$ and an adapted sequence of positive random variables $(w_n)$, we show the weighted Davis type inequality \[ \mathbb{E} \Bigl( |f_0| w_0 + \frac{1}{4} \sum_{n=1}^{N} \frac{|df_n|^2}{f^*_n} w_n \Bigr) \leq \mathbb{E} ( f^*_N w^*_N). \] This inequality is sharp and implies several results about the martingale square function. We also obtain a variant of this inequality for martingales with values in uniformly convex Banach spaces., Comment: 9 pages

Let $f$ be a martingale with values in a uniformly $p$-smooth Banach space and $w$ any positive weight. We show that $\mathbb{E} (f^* \cdot w) \lesssim \mathbb{E}(S_p f \cdot w^*)$, where $\cdot^*$ is the martingale maximal operator and $S_p$ is the $\ell^p$ sum of martingale increments., Comment: v2: 16 pages, corrected following referee report

We propose a new method of estimating oscillatory integrals, which we call a stationary set method. We use it to obtain the sharp convergence exponents of Tarry's problems in dimension two for every degree $k\ge 2$. As a consequence, we obtain sharp Fourier extension estimates for a family of monomial surfaces., Comment: Added an argument showing sharpness of the main result in an average sense; shared to us by Wright

Logarithmic potentials and many other potentials satisfy maximum principle. The dyadic version of logarithmic potential can be easily introduced, it lives on dyadic tree and also satisfies maximum principle. But its analog on bi-tree does not have this property. We prove here that "on average" we can still have something like maximum principle on bi-tree. We use the surrogate maximum principle to prove embedding theorems of Carleson type on bi-disc., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1906.11145

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities., Comment: v2: corrected following referee reports, 37 pages

We show a variable coefficient version of Bourgain's multi-frequency lemma. It can be used to obtain major arc estimates for a discrete Stein--Wainger type operator considered by Krause arXiv:1803.09431 and Roos arXiv:1907.00405., Comment: 7 pages

We prove sharp weighted estimates for $r$-variations of averages and truncated singular integrals on suitable spaces of homogeneous type, including homogeneous nilpotent Lie groups., Comment: This note, written in 2017, can be seen as an expanded version of arXiv:1409.7120 and arXiv:1604.05506

We establish a new scale of $p$-variation estimates for martingale paraproducts, martingale transforms, and It\^o integrals, of relevance in rough paths theory, stochastic, and harmonic analysis. As an application, we introduce rough semimartingales, a common generalization of classical semimartingales and (controlled) rough paths, and their integration theory., Comment: v2: 40 pages, corrected following referee reports

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 prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces in bi-disc and tri-disc this proves the embedding theorem of those Dirichlet spaces of holomorphic function on bi- and tri-disc. We completely describe the Carleson measures for such embeddings. The result below generalizes embedding result of \cite{AMPVZ} from bi-tree to tri-tree. One of our embedding description is similar to Carleson--Chang--Fefferman condition and involves dyadic open sets. On the other hand, the unusual feature of \cite{AMPVZ} was that embedding on bi-tree turned out to be equivalent to one box Carleson condition. This is in striking difference to works of Chang--Fefferman and well known Carleson quilt counterexample. We prove here the same unexpected result for the tri-tree. Finally, we explain the obstacle that prevents us from proving our results on polydiscs of dimension four and higher., Comment: 33 pages, We are grateful to Sergei Treil for indicating a faulty reasoning in Section 2.4, we corrected this reasoning

We give a short and elementary proof of the $\ell^{2}$ decoupling inequality for the moment curve in $\mathbb{R}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley (arXiv:1708.01220)., Comment: v3: fixed up issues about weights in the published version. Weights in Section 3 are now required to decay a bit faster at infinity, and proofs of properties of weights are now given in an Appendix

We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in $\mathbb{R}^5$ given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case (arXiv:1609.04107), this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables., Comment: v2: 19 pages, small corrections following referee report

We prove an estimate for weighted $p$-th moments of the pathwise $r$-variation of a martingale in terms of the $A_{p}$ characteristic of the weight. The novelty of the proof is that we avoid real interpolation techniques., Comment: v2: 6 pages, added statement of the vector-valued L\'epingle inequality in Remark 3.5

Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the bi-disc is equivalent to a simple ``box'' condition, for product weights on the bi-disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale of Dirichlet spaces includes the classical Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the ``box'' condition does not suffice for the embedding to hold. But this was the embedding of bi-harmonic functions in bi-harmonic Hardy class. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures., Comment: v4: 29 pages, incorporates arXiv:1906.11145

The first three results in this thesis are motivated by a far-reaching conjecture on boundedness of singular Brascamp-Lieb forms. Firstly, we improve over the trivial estimate for their truncations, thus excluding potential trivial counterexamples. Secondly, in a dyadic model we give a partial result in the trilinear, 2-dimensional case when one of the functions depends only on one variable. Thirdly, we estimate the multidimensional version of the polynomial Carleson operator, whose boundedness would also be a consequence of the conejcture. The final pair of results concerns directional square functions. One of them concerns interaction of Lipschitz change of variable on the line with Littlewood--Paley decomposition and is used to extend the conditional result of Lacey and Li for Hilbert transform along $C^{1+\epsilon}$ vector fields to Lipschitz vector fields. The other concerns directional averaging operators and gives an alternative approach to a single scale maximal function with averages in N directions due to Katz away from the endpoint., Comment: Habilitation thesis, University of Bonn, March 2018, v+127 pages. Based on the articles arXiv:1507.02436v1, arXiv:1506.00861v1, arXiv:1711.03524v4, arXiv:1706.07111v1

We prove sharp $\ell^{p}L^{p}$ decoupling inequalities for $2$ quadratic forms in $4$ variables. We also recover several previous results (arXiv:1409.1634, arXiv:1501.07224, arXiv:1609.02022, arXiv:1609.04107) in a unified way., Comment: v2: revised following referee reports, 22 pages

We show that bilinear variational estimates of Do, Muscalu, and Thiele (arXiv:1009.5187) remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder--Davis--Gundy inequality for martingale rough paths by Chevyrev and Friz (arXiv:1704.08053)., Comment: 13 pages, implications for continuous time martingales discussed in more detail

We prove $\ell^{p}L^{p}$ decoupling inequalities for a class of moment manifolds. These inequalities imply optimal mean value estimates for multidimensional Weyl sums of the kind considered by Arkhipov, Chubarikov, and Karatsuba and by Parsell. In our proofs we take a new point of view on the Bourgain--Demeter--Guth induction on scales argument. This point of view substantially simplifies even the proof of $\ell^{2}L^{p}$ decoupling for the moment curve., Comment: v2: 39 pages, revised following the referee report

We prove strong jump inequalities for a large class of operators of Radon type in the discrete and ergodic theoretical settings. These inequalities are the $r=2$ endpoints of the $r$-variational estimates studied in arXiv:1512.07523., Comment: v2: 39 pages, revised following referee report

The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of $r$-variational estimates, previously known for $r>2$, to end-point results for the jump quasi-seminorm corresponding to $r=2$. This is applied to the dimension-free results recently obtained by the first two authors in collaboration with Bourgain and Wr\'obel (arXiv:1708.04639 and arXiv:1804.07679), and also to operators of Radon type treated by Jones, Seeger, and Wright., Comment: 25 pages, small corrections

Jump inequalities are the $r=2$ endpoint of L\'epingle's inequality for $r$-variation of martingales. Extending earlier work by Pisier and Xu we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $\mathbb{R}^{d}$ to $\mathbb{Z}^{d}$ for jump estimates for Fourier multipliers., Comment: v3: corrections following referee report

We prove a sharp common generalization of endpoint multilinear Kakeya and local discrete Brascamp-Lieb inequalities., Comment: v4: revised following referee reports, 18 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)