Related papers: Estimates for oscillatory integrals with phase hav…

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space $H^s({\mathbb{R}})$ and from the space $C^s({\mathbb{R}})$ with an arbitrary integer $s\ge1$. We find tight upper and lower…

Let T be an oscillatory integral operator on L^2(R) with a smooth real phase function S(x,y). We prove that, in all cases but the one described below, after localization to a small neighborhood of the origin the norm of T decays like…

In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author…

Superoscillatory wave forms, i.e., waves that locally oscillate faster than their highest Fourier component, possess unusual properties that make them of great interest from quantum mechanics to signal processing. However, the more…

In this paper, we consider the (1+2)-dimensional oscillatory integral with degenerate cubic homogeneous polynomial phase. We prove that the $L^{2}$ decay rate of 3/8 given in (Archiv der Mathematik, 122: 437-447, 2024) is sharp.

This paper systematically treats the asymptotic behavior of many (linear/nonlinear) classes of higher-order fractional differential equations with multiple terms. To do this, we utilize the characteristics of Caputo fractional…

We study $L^p$ bounds for two kinds of Riesz transforms on $\mathbb{R}^d$ related to the harmonic oscillator. We pursue an explicit estimate of their $L^p$ norms that is independent of the dimension $d$ and linear in $\max(p, p/(p-1))$.

In this paper, we consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calder\'{o}n-Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $\mu$.…

In this work, phase singularities embedded in a wavepacket are shown to act as sources of atypical localized oscillations when the packet interacts with a linear system. We refer to these oscillations as \textit{chirpons}, since they arise…

$(L_p, L_q)$ estimates are obtained for oscillatory potentials $(K^\alphaf)(x)=\int\limits_{R^n}\frac{\exp(i|y|)}{|y|^{n-\alpha}}f(x-y)dy$, $0<\alpha<n$, $n\geq 2$, whose symbol has a singularity on the unit sphere. These potentials are…

We study, by means of a topological approach, the forced oscillations of second order functional retarded differential equations subject to periodic perturbations. We consider a delay-type functional dependence involving a gamma probability…

We investigate symmetric oscillators, and in particular their quantization, by employing semiclassical and quantum phase functions introduced in the context of Liouville-Green transformations of the Schr\"{o}dinger equation. For anharmonic…

We describe an algorithm for the numerical solution of second order linear differential equations in the highly-oscillatory regime. It is founded on the recent observation that the solutions of equations of this type can be accurately…

We develop a general framework for identifying phase reduced equations for finite populations of coupled oscillators that is valid far beyond the weak coupling approximation. This strategy represents a general extension of the theory from…

Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.

In this paper we establish square-function estimates on the double and single layer potentials for divergence-form elliptic operators, of arbitrary even order 2m, with variable t-independent coefficients in the upper half-space. This…

In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in $\mathbb{R}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the…

Introducing a perturbative definition, phase space path integrals can be calculated without slicing. This leads to a short-time expansion of the quantum-mechanical path amplitude, or a high-temperature expansion of the unnormalized density…

A generalized notion of oscillatory integrals that allows for inhomogeneous phase functions of arbitrary positive order is introduced. The wave front set of the resulting distributions is characterized in a way that generalizes the…

We study two classes of radial integrals involving a product of bound and continuum one-electron states. Using a representation of the continuum part with an expansion on complex Gaussian Type Orbitals, such integrals can be performed…