Related papers: On the bilinear cone multiplier

We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain…

Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{\alpha_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,\alpha_j+1)$ be the homogeneous dimension…

In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we…

Quantization and spectral properties of Toeplitz operators acting on spaces of pluriharmonic functions over bounded symmetric domains and $\mathbb C^n$ are discussed. Results are presented on the asymptotics \begin{align*} \|…

Given a bilinear (or sub-bilinear) operator $B$, we prove restricted weighted weak type inequalities of the form $$ ||B(f_1, f_2)||_{L^{p, \infty}(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $$…

We introduce a notion of ternary $F$-manifold algebras which is a generalization of $F$-manifold algebras. We study representation theory of ternary $F$-manifold algebras. In particular, we introduce a notion of dual representation which…

For each $\lambda \in \mathbb N^*$, we consider the integral equation: \[ \int_{\lambda y} ^{\lambda x} f(t)\, d t = f(x) - f(y) \mbox{ for every $(x,y)\in {\mathbb R}_+^2$,} \] where $f$ is the concatenation of two continuous functions…

We prove $L^p$ estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$, a sharp…

We exhibit a large class of symbols $m$ on $\R^d$, $d\geq 2$, for which the corresponding Fourier multipliers $T_m$ satisfy the following inequality. If $D$, $E$ are measurable subsets of $\R^d$ with $E\subseteq D$ and $|D|<\infty$, then $$…

In Fourier restriction problems, a cone and a paraboloid are model surfaces. The sharp bilinear cone restriction estimate was first shown by Wolff, and later the endpoint was obtained by Tao. For a paraboloid, the sharp $L^2$ bilinear…

The boundedness of the bilinear fractional integral operator is investigated. This bilinear fractional integral operator goes back to Kenig and Stein. This paper is oriented to the boundedness of this operator on products of Morrey spaces.…

Let $\mathcal{H}=-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and…

Let $A$ be a $\nu$-vector of self-adjoint, pairwise commuting operators and $B$ a bounded operator of class $C^{n_0}(A)$. We prove a Taylor-like expansion of the commutator $[B,f(A)]$ for a large class of functions $f\colon\mathbm{R}^\nu…

Given a local ring $(R, \mathfrak{m})$ and an ideal $\mathfrak{a}$ of positive height, we give a way of computing multiplier module ${J}(\omega_{{T}}, t^{-\lambda})$ for the extended Rees algebra ${T} =R[\mathfrak{a} t, t^{-1}]$ for an…

In this paper we study sharp generalizations of $\dot{F}_p^{0,q}$ multiplier theorem of Mikhlin-H\"ormander type. The class of multipliers that we consider involves Herz spaces $K_u^{s,t}$. Plancherel's theorem proves…

Let $T$ be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of $\mathbb R^n$. We assume that the associated kernel of $T$ satisfies some mild regularity condition which is weaker than the usual…

We show that a continuous bilinear mapping P: C(I) \times C(I) \to C(I) can be presented in the form P(f,g) = B((Af)(Ag)), where A and B are bounded linear operators on C(I) and multiplication is defined pointwise, if and only if for all t…

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class $\Lambda_{f,g}(a, b)$ of mean values where $f, g$ are continuously differentiable convex functions satisfying the relation…

Muscalu, Tao, and Thiele prove $L^p$ estimates for the "Biest" operator defined on Schwartz functions by the map \begin{align*} \hspace{5mm} C^{1,1,1}:& (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2< \xi_3} \left[ \prod_{j=1}^3 \hat{f}_j…

This paper is a successor of \cite{laceyt}. In that paper we considered bilinear operators of the form H_alpha(f_1,f_2)(x) = p.v. \int f_1(x-t) f_2(x + alpha t)/t dt, which are originally defined for f_1, f_2 in the Schwartz class S(R). The…