Related papers: Higher decay inequalities for multilinear oscillat…

We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_\Omega$ associated with an integrable function…

The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local…

This text contains an alternative presentation, and in certain cases an improvement, of the "hyperbolic dispersive estimate" that was proved by Anantharaman and Nonnenmacher and used to make progress towards the quantum unique ergodicity…

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the…

We proof pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_\rho$ and phases $\varphi$ such that $\varphi(x,\xi) -…

This thesis is devoted to asymptotic norm estimates for oscillatory integral operators acting on the L^2 space of functions of one real variable. The operators in question have compact support and an oscillatory kernel of the form exp(i…

We provide estimates for weighted Fourier sums of integrable functions defined on the sphere when the weights originate from a multiplier operator acting on the space where the function belongs. That implies refined estimates for weighted…

We consider a class of Fourier integral operators, globally defined on $\mathbb{R}^{d}$, with symbols and phases satisfying product type estimates (the so-called $SG$ or scattering classes). We prove a sharp continuity result for such…

We establish endpoint bounds on a Hardy space $H^1$ for a natural class of multiparameter singular integral operators which do not decay away from the support of rectangular atoms. Hence the usual argument via a Journ\'e-type covering lemma…

We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz exponent in a slightly weaker form than the assertion in…

This paper investigates higher order wave-type equations of the form $\partial_{tt}u+P(D_{x})u=0$, where the symbol $P(\xi)$ is a real, non-degenerate elliptic polynomial of the order $m\ge4$ on ${\bf R}^n$. Using methods from harmonic…

Equilibrium measures are special invariant measures of chaotic dynamical systems and iterated function systems, commonly studied as salient examples of fractal measures. While useful analytic expressions are rare, computational exploration…

In this paper we consider the uniform estimates for oscillatory integrals with a two-order homogeneous polynomial phase. The estimate is sharp and the result is an analogue of the more general theorem of V. N. Karpushkin…

In this paper, we study the degenerate or singular fully nonlinear dead-core systems coupled with strong absorption terms. We establish several properties, including improved regularity of viscosity solutions along the free boundary,…

We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a H\"older-type singularity at the origin. We prove boundedness in $L^1$ with a…

In this paper we present a multidimensional version of the van der Corput lemma where the decay of the oscillatory integral is gained with respect to all space variables, connecting the standard one-dimensional van der Corput lemma with the…

We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…

We study a family of Fourier integral operators, by allowing their symbols to satisfy a multi-parameter differential inequality. We extend the sharp L^p-result obtained by Seeger, Sogge and Stein to product spaces.

In this work we obtain weighted boundedness results for singular integral operators with kernels exhibiting exponential decay. We also show that the classes of weights are characterized by a suitable maximal operator. Additionally, we study…

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright (1999) and Phong, Stein, and Sturm (2001) from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous,…