Peng Chen

Let $(X,d,μ)$ be a metric space with doubling measure and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. We assume that there exists an $L$-harmonic function $h$ such that the semigroup $\exp(-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. In this paper we apply the Doob transform and some techniques as in Grafakos-Honzík-Seeger \cite{GHS2006} to obtain an optimal $\sqrt{\log(1+N)}$ bound in $L^p$ for the maximal function $\sup_{1\leq i\leq N}|m_i(L)f|$ for multipliers $m_i,1\leq i\leq N,$ with uniform estimates. Based on this, we establish sufficient conditions on the bounded Borel function $m$ such that the maximal function $M_{m,L}f(x) = \sup_{t>0} |m(tL)f(x)|$ is bounded on $L^p(X)$. The applications include Schrödinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators.

For $p\ge 2$, and $λ>\max\{n|\tfrac 1p-\tfrac 12|-\tfrac12, 0\}$, we prove the pointwise convergence of cone multipliers, i.e. $$ \lim_{t\to\infty}T_t^λ(f)\to f \text{ a.e.},$$ where $f\in L^p(\mathbb R^n)$ satisfies $supp\ \widehat f\subset\{ξ\in\mathbb R^n:\ 1<|ξ_n|<2\}$. Our main tools are weighted estimates for maximal cone operators, which are consequences of trace inequalities for cones.

Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. Let $f$ be in the space $ {\rm BMO}_L(X)$ associated with the operator $L$ and we define its distance from the subspace $L^{\infty}(X)$ under the $ {\rm BMO}_L(X)$ norm as follows: $$ {\rm dist} (f, L^{\infty}):= \inf_{g\in L^{\infty}} \|f -g\|_{{\rm BMO}_L(X)}. $$ In this paper we prove that ${\rm dist} (f, L^{\infty})$ is equivalent to the infimum of the constant $\varepsilon$ in the John-Nirenberg inequality for the space ${\rm BMO}_L(X)$: $$ \sup_B { μ\big(\{ x\in B: |f(x)-e^{-{r_B^2}L}f(x)|>λ\}\big) \over μ(B)} \leq e^{-λ/\varepsilon}\ \ \ \ {\rm for\ large\ } λ. $$ This extends the well-known result of Garnett and Jones \cite{GJ1} for the classical ${\rm BMO}$ space (introduced by John and Nirenberg). As an application, we show that a ${\rm BMO}_L(X)$ function with compact support can be decomposed as the summation of an $L^\infty$-function and the integral of the heat kernel (associated with $L$) against a finite Carleson measure on $X\times[0,\infty)$. The key new technique is a geometric construction involving the semigroup $e^{-tL}$. We also resort to several fundamental tools including the stopping time argument and the random dyadic lattice.

We establish a complete theory of the flag Hardy space on the Heisenberg group $\mathbb H^{n}$ with characterisations via atomic decompositions, area functions, square functions, maximal functions and singular integrals. We introduce several new techniques to overcome the difficulties caused by the noncommutative Heisenberg group multiplication, and the lack of a suitable Fourier transformation and Cauchy--Riemann type equations. Applications include the boundedness from the flag Hardy space to $L^1(\mathbb H^n)$ of various singular integral operators that arise in complex analysis, a sharp boundedness result on the flag Hardy space of the Marcinkiewicz-type multipliers introduced by M\"uller, Ricci and Stein, and the decomposition of flag BMO space via singular integrals.

Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$ , where $X$ is a positive measure space. Let ${ L}=\int _0^{\infty } \lambda dE_{ L}({\lambda })$ be the spectral resolution of $L$ and $S_R({ L})f=\int _0^R dE_{ L}(\lambda ) f$ denote the spherical partial sums in terms of the resolution of ${ L}$ . In this article we give a sufficient condition on $L$ such that $\begin{equation*} \lim _{R\rightarrow \infty } S_R({ L})f(x) =f(x),\ \ \text {a.e.} \end{equation*}$ for any $f$ such that $\operatorname {log}(2+L) f\in L^2(X)$ . These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.