Related papers: Spectral multiplier theorems and averaged R-bounde…

Let $\Omega$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator…

We show that a bilinear radial Fourier multiplier operator with symbol $\sigma$ is $L^2(\R^n)\times L^2(\R^n) \to L^1(\R^n)$ bounded, $n\in \mathbb N,$ if the function $\sigma$ satisfies the smoothness condition $\sigma(2^j\cdot)\Phi\in…

We study the lacunary analogue of the $\delta$-discretised spherical maximal operators introduced by Hickman and Jan\v{c}ar, for $\delta \in (0, 1/2)$, and establish the boundedness on $L^p$ for all $1 < p < \infty$, along with the endpoint…

The paper is devoted to operators given formally by the expression \begin{equation*} -\partial_x^2+\big(\alpha-\frac14\big)x^{-2}. \end{equation*} This expression is homogeneous of degree minus 2. However, when we try to realize it as a…

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(\lambda)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq…

In this work, we prove a bound on multiplicity of the singular spectrum for certain class of Anderson Hamiltonians. The class of operator is $H^\omega=\Delta+\sum_{n\in\mathbb{Z}^d}\omega_n P_n$ on the Hilbert space $\ell^2(\mathbb{Z}^d)$,…

We establish the $L^p$-boundedness of the local covariant Riesz transform for differential forms on manifold $M$ with bounded $\|Rm\|$. Let $\Delta_j$ be the Hodge Laplace operator on $j$-forms. For any $p \in (1, \infty)$ and…

We study the boundedness of the linear operator $S$ on $L^{p}_{a}(dA_{\alpha})$ $(0<p<\infty)$. In particular, we obtain a sufficient and necessary condition for the compactness of the linear operator $S$ on $L^{p}_{a}(dA_{\alpha})$…

We discuss $L^p(\mathbb R^n)$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the H\"ormander multiplier theorem in terms of an optimal condition that relates the distance $|\frac 1p-\frac12|$ to the smoothness…

We develop a very general operator-valued functional calculus for operators with an $H^{\infty}-$calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an $H^{\infty}$calculus. Using…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Denote by $dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x…

Let $(X, d, \mu)$ be a space of homogeneous type and $\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\Omega) \to L^q(\Omega)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for…

For two dimensional Schr\"odinger operator $H$ with point interactions, We prove that wave operators of scattering for the pair $(H,H_0)$, $H_0$ being the free Schr\"odinger operator, are bounded in the Lebesgue space $L^p(\R^2)$ for…

We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $\Omega\subseteq\mathbb{R}^d$ that are uniformly $C^{2,1}$. Starting with the Hodge Laplacian we etablish a bounded H\"ormander functional calculus…

Let H(f)(x)=\int_{(0,infty)^d} f(v) E_{x}(v) d\nu(v), be the multivariable Hankel transform, where E_{x}(v)=\prod_{k=1}^d (x_k v_k)^{-a_k+1/2} J_{a_k-1/2}(x_k v_k), d\nu(v)=v^a dv, a=(a_1,...,a_d). We give sufficient conditions on a bounded…

We prove the global $L^p$-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical H\"ormander classes $S^{m}_{\rho, \delta}(\mathbb{R}^n)$ for…

Let $X,Y$ be normal bounded operators on a Hilbert space such that $e^X=e^Y$. If the spectra of $X$ and $Y$ are contained in the strip $\s$ of the complex plane defined by $|\Im(z)|\leq \pi$, we show that $|X|=|Y|$. If $Y$ is only assumed…

Let $\Delta$ be the Laplace--Beltrami operator acting on a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$ with $m > n \ge 3$. Let $\frak{h}_t(x,y)$ be the kernels of the semigroup $e^{-t\Delta}$ generated by $\Delta$.…

Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian…

Let $L = \Delta + V$ be Schr{\"o}dinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the conical square functional associated with $L$ is bounded on $L^p$ under different assumptions. This…