Related papers: Strongly singular integrals on stratified groups

We prove uniform $L^p$ bounds for multilinear operators which are given by multipliers whose symbols are singular on a one dimensional subspace. The novelty is that these bounds are uniform in the choice of the subspace.

In this note, we consider a non-commutative analogy of the classical Fefferman multiplier problem for the ball. More precisely, if $\chi$ is the characteristic function of the unit interval $I=[0,1],$ we investigate a family of differential…

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the H\"ormander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference…

We investigate Laplace type and Laplace-Stieltjes type multipliers in the $d$-dimensional setting of the Dunkl harmonic oscillator with the associated group of reflections isomorphic to $\mathbb{Z}_2^d$ and in the related context of…

We prove restriction type estimates for sub-Laplacians on general two-step stratified Lie groups. The core of our approach is to use spectral cluster estimates to effectively control the eigenvalue distribution of a family of anisotropic…

Let $G$ be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup $K$. Let $\Omega_K$ be minus the radial Casimir operator. Let $\frac{1}{4} \dim(G/K) < S_G < \frac{1}{2} \dim(G/K) , s \in (0, S_G]$ and $p \in…

We prove sharp $L^2$ regularity results for classes of strongly singular Radon transfoms on the Heisenberg group by means of oscillatory integrals. We show that the problem in question can be effectively treated by establishing uniform…

Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for…

For a limited range of indices $p$, we obtain $L^p(\mathbb{R}^n)$ boundedness for singular integral operators whose kernels satisfy a condition weaker than the typical H\"ormander smoothness estimate. These operators are assumed to be…

Given a discrete group $\G$ and an orthogonal action $\gamma: \G \to O(n)$ we study $L_p$ convergence of Fourier integrals which are frequency supported on the semidirect product $\R^n \rtimes_\gamma \G$. Given a unit $u \in \R^n$ and $1 <…

We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(\mathbb{S}^1_\rho) := \mathbb{R}_+ \times…

We determine the conditions under which singular values of multiple $\eta$-quotients of square-free level, not necessarily prime to~6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number…

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an $L_2$-form of H\"ormander's kernel condition suffices for the weak…

We introduce a spectrum for arbitrary varieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for…

We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $\chi \in…

In this paper, we present new proofs for both the sharp $L^p$ estimate and the decoupling theorem for the H\"ormander oscillatory integral operator. The sharp $L^p$ estimate was previously obtained by Stein\;\cite{stein1} and Bourgain-Guth…

We prove an $L^{p}$ estimate $$ \|e^{-itL} \varphi(L)f\|_{p}\lesssim (1+|t|)^s\|f\|_p, \qquad t\in \mathbb{R}, \qquad s=n\left|\frac{1}{2}-\frac{1}{p}\right| $$ for the Schr\"odinger group generated by a semibounded, selfadjoint operator…

In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$. The methods we use are entirely elementary, and rely only on pigeonholing and…

We construct Steinberg summands of $G$-equivariant spectra with $\mathrm{GL}_n(\mathbb{F}_p)$-action. We prove a lemma about their fixed points when $G$ is a $p$-group, and then use this lemma to compute the fixed points of the Steinberg…

We consider the following model of degenerate and singular oscillatory integral operators: \begin{equation*} Tf(x)=\int_{\mathbb{R}} e^{i\lambda S(x,y)}K(x,y)\psi(x,y)f(y)dy, \end{equation*} where the phase functions are homogeneous…