Related papers: On oscillatory integrals with H\"older phases

We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on $(x_1 \partial_{x_1} f(x) ,..., x_n \partial_{x_n}f(x))$, where $f(x)$ is the phase function. We prove a theorem which provides estimates…

We study an operator analogue of the classical problem of finding the rate of decay of an oscillatory integral on the real line. This particular problem arose in the analysis of oscillatory Riemann-Hilbert problems associated with partial…

Oscillatory integrals arise in many situations where it is important to obtain decay estimates which are stable under certain perturbations of the phase. Examining the structural problems underpinning these estimates leads one to consider…

In this paper, we study time-asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of…

We study weighted sum processes associated to elements in a Wiener chaos with fixed order. More precisely, we show H\"older estimates and a functional limit theorem for them. Main tools we use are the integration by parts formula in…

We consider the oscillatory integrals with parameter-dependent phases. We decompose the integrals into a leading term and a remainder term. Instead of the pointwise estimate, we use some $L^p$-estimate for the remainder term and get various…

The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly H\"older continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the…

Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form $F(t)=\sum_{n=1}^\infty f(n)e^{2\pi i nt}/n$, for a large…

These notes arose from my Cambridge Part III course on Additive Combinatorics, given in Lent Term 2009. The aim was to understand the simplest proof of the Bourgain-Glibichuk-Konyagin bounds for exponential sums over subgroups. As a…

We use recent advances in the theory of Furstenberg sets to prove new incidence results of Szemer\'edi--Trotter strength for $\delta$-discretized structures with Cartesian product flavor. We use these results to make progress on a number of…

In this paper, we propose a numerical method of computing an integral whose integrand is a slowly decaying oscillatory function. In the proposed method, we consider a complex analytic function in the upper-half complex plane, which is…

Using the birational map between a smooth toric variety (adapted to the phase function of the oscillatory integral) and $\mathbb{R}^n\textbackslash\{0\}$, we can effectively carry out the van der Corput-type analysis in higher dimensions.…

Let $H\subset \R^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $\sigma $ on $H$. Let $\kappa$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $(\kappa^{1/2}…

In this paper we show how to apply classical probabilistic tools for partial sums $\sum_{j=0}^{n-1}\varphi\circ\tau^j$ generated by a skew product $\tau$, built over a sufficiently well mixing base map and a random expanding dynamical…

Consider an ensemble of $N\times N$ non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded…

We consider stochastic integration with respect to fractional Brownian motion (fBm) with $H < 1/2$. The integral is constructed as the limit, where it exists, of a sequence of Riemann sums. A theorem by Gradinaru, Nourdin, Russo & Vallois…

In this paper we establish an estimate for the rate of convergence of the Krasnosel'ski\v{\i}-Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon-Bruck conjecture [3] on the asymptotic…

We utilise the recent work of Orponen to yield a sum-product result for Ahlfors-regular sets. As a corollary, we obtain the fractal analogue of Solymosi's $4/3$-bound for finite subsets of $\mathbb{R}.$

In this article, $q$-regular sequences in the sense of Allouche and Shallit are analysed asymptotically. It is shown that the summatory function of a regular sequence can asymptotically be decomposed as a finite sum of periodic fluctuations…

In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm and Carbery and Wright on any product $\prod_{j=1}^d L^{p_j}(\mathbb R)$ with each…