arXiv subjects
Publications and source records attributed to Pavel Zorin-Kranich.
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.
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.
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.
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.
We prove a sharp common generalization of endpoint multilinear Kakeya and local discrete Brascamp-Lieb inequalities.
Jump inequalities are the $r=2$ endpoint of Lépingle'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.
We obtain mixed $A_p$--$A_\infty$ estimates for a large family of multilinear maximal and sparse operators. Operators from this family are known to control for instance multilinear Calderón--Zygmund operators, square functions, fractional integrals, and the bilinear Hilbert transform. Our results feature a new multilinear version of the Fujii--Wilson $A_\infty$ characteristic that allows us to recover the best known estimates in terms of the $A_p$ characteristic for dependent weights as a special case of the mixed characteristic estimates for general tuples of weights.
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).
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 $C^{1+ε}$ 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.
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 Wróbel (arXiv:1708.04639 and arXiv:1804.07679), and also to operators of Radon type treated by Jones, Seeger, and Wright.
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.
We generalize the IP-polynomial Szemerédi theorem due to Bergelson and McCutcheon and the nilpotent Szemerédi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.
We prove that for any bounded functions $f_1, f_2$ on a measure-preserving dynamical system $(X,T)$ and any distinct integers $a_1, a_2$, for almost every $x$ the sequence $$ f_1(T^{a_1 n}x) f_2(T^{a_2 n}x) $$ is a good weight for the pointwise ergodic theorem.
Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-α}$, $0 < α< 1/2$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ orthogonal to the invariant factor, the modulated, random averages \[ \sup_{b} \Big| \frac{1}{N} \sum_{n = 1}^N b(n) T^{a_{n}} f \Big| \] converge to $0$ pointwise almost everywhere, where the supremum is taken over a set of bounded functions with certain uniform approximation properties.
We consider intrinsic square functions defined using (log-)Dini continuous test functions on spaces of homogeneous type. We prove weighted estimates with optimal (at least in the Euclidean case) dependence on the aperture of the cone used to define the square function and linear dependence on the (log-)Dini modulus of continuity.
We prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+p_i(n)$, with rationally independent $p_i$'s with zero constant term. This is in contrast to the single variable case, in which even double recurrence fails unless the transformations generate a virtually nilpotent group. The proof involves reduction to nilfactors and an equidistribution result on nilmanifolds.
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.
We extend Lerner's recent approach to sparse domination of Calderón--Zygmund operators to upper doubling (but not necessarily doubling), geometrically doubling metric measure spaces. Our domination theorem is different from the one obtained recently by Conde-Alonso and Parcet and yields a weighted estimate with the sharp power $\max(1,1/(p-1))$ of the $A_p$ characteristic of the weight.