Vjekoslav Kovač

Boundedness of the twisted paraproduct.

We prove L^p estimates for a two-dimensional bilinear operator of paraproduct type. This result answers a question posed by Demeter and Thiele.We prove L estimates for a two-dimensional bilinear operator of paraproduct type. This result answers a question posed by Demeter and Thiele.

On the combined use of GW approximation and cumulant expansion in the calculations of quasiparticle spectra: The paradigm of Si valence bands

Since the earliest implementations of the various GW approximations and cumulant expansion in the calculations of quasiparticle propagators and spectra, several attempts have been made to combine the advantageous properties and results of these two theoretical approaches. While the GW-plus-cumulant approach has proven successful in interpreting photoemission spectroscopy data in solids, the formal connection between the two methods has not been investigated in detail. By introducing a general bijective integral representation of the cumulants, we can rigorously identify at which point these two approximations can be connected for the paradigmatic model of quasiparticle interaction with the dielectric response of the system that has been extensively exploited in recent interpretations of the satellite structures in photoelectron spectra. We establish a protocol for consistent practical implementation of the thus established GW+cumulant scheme and illustrate it by comprehensive state-of-the-art first-principles calculations of intrinsic angle-resolved photoemission spectra from Si valence bands.

Norm-variation of ergodic averages with respect to two commuting transformations

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.

DYADIC TRIANGULAR HILBERT TRANSFORM OF TWO GENERAL FUNCTIONS AND ONE NOT TOO GENERAL FUNCTION

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

One modification of the martingale transform and its applications to paraproducts and stochastic integrals

L^{p}

Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform

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.

A sharp nonlinear Hausdorff-Young inequality for small potentials

In this paper we introduce a variant of Burkholder’s martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed transformation still satisfies some expected L estimates. Then we apply the obtained inequalities to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. As another application we construct stochastic integrals ∫ t 0 Hsd(XsYs) associated with certain continuous-time martingales (Xt)t≥0 and (Yt)t≥0. The process (XtYt)t≥0 is shown to be a “good integrator”, although it is not necessarily a semimartingale, or even adapted to any convenient filtration.

Sobolev norm estimates for a class of bilinear multipliers

Analogues of Hausdorff-Young inequalities for the Dirac scattering transform (a.k.a. the SU(1, 1) nonlinear Fourier transform) were first established by Christ and Kiselev. Later Muscalu, Tao, and Thiele raised a question whether the constants can be chosen uniformly in 1 ≤ p ≤ 2. Here we give a positive answer to that question when the Euclidean real line is replaced by its Cantor group model.Analogues of Hausdorff-Young inequalities for the Dirac scattering transform (a.k.a. SU(1, 1) nonlinear Fourier transform) were first established by Christ and Kiselev. Later Muscalu, Tao, and Thiele raised a question if the constants can be chosen uniformly in 1≤p≤2. Here we give a positive answer to that question when the Euclidean real line is replaced by its Cantor group model.

On side lengths of corners in positive density subsets of the Euclidean space

The nonlinear Hausdorff-Young inequality follows from the work of Christ and Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently of the exponent. We show that the nonlinear Hausdorff-Young quotient admits an even better upper bound than the linear one, provided that the function is sufficiently small in the L^1-norm. The proof combines perturbative techniques with the sharpened version of the linear Hausdorff-Young inequality due to Christ.

On the share of closed IL formulas which are also in GL

We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing L^p estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential operators of order 0.