Let $L$ be a non-negative self-adjoint operator acting on $L^2\left({ℝ}^n\right)$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_r$ weight on ${ℝ}^n\times {ℝ}^n$, $1<r<\infty$. In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L,w}^p({ℝ}^n\times {ℝ}^n)$, $0<p\le 1$ associated to $L$. Based on the atomic decomposition, we show the dual relationship between $H_{L,w}^1({ℝ}^n\times {ℝ}^n)$ and ${\mathrm{BMO}}_{L,w}({ℝ}^n\times {ℝ}^n)$.

Let A be a pseudodifferential operator on ${ℝ}^N$ whose Weyl symbol a is a strictly positive smooth function on $W={ℝ}^N\times {ℝ}^N$ such that $|{\partial }^{\alpha }a|\le C_{\alpha }{a}^{1-\varrho }$ for some ϱ>0 and all |α|>0, ${\partial }^{\alpha }a$ is bounded for large |α|, and $li{m}_{w\to \infty }a\left(w\right)=\infty$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.

The aim of this paper is to use an abstract realization of the Weyl correspondence to define functions of pseudo-differential operators. We consider operators that form a self-adjoint Banach algebra. We construct on this algebra a functional calculus with respect to functions which are defined on the Euclidean space and have a finite number of derivatives.

Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into ${ℝ}^d$. The paper deals with Y-weak cluster points ϕ̅ of the sequence $\varphi (·,z_j\left(·\right))$ in X, where $z_j:\Omega \to {ℝ}^m$ is measurable for j ∈ ℕ and $\varphi :\Omega \times {ℝ}^m\to {ℝ}^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{\varphi }$, the integral $I(\varphi ,{\nu }_x):={\int }_{{ℝ}^m}\varphi (x,\lambda )d{\nu }_x\left(\lambda \right)$ exists for $x\in \Omega \setminus A_{\varphi }$ and $\varphi ̅\left(x\right)=I(\varphi ,{\nu }_x)$ on $\Omega \setminus A_{\varphi }$, where $\nu ={{\nu }_x}_{x\in \Omega }$ is a measurable-dependent family of Radon probability measures on ${ℝ}^m$.