Abstract. Let X be a metric space with doubling measure, and L be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a non-negative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, we define Hardy spaces H p L (X) for p > 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L (X) spaces by the complex method.

In this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical BMO space. We show that the John-Nirenberg inequality holds on these spaces and they interpolate with L p spaces by the complex interpolation method. We then give applications on L pboundedness of singular integrals whose kernels do not satisfy the Hörmander condition.

Let L L be the infinitesimal generator of an analytic semigroup on L 2 ( R n ) L^2({\mathbb R}^n) with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space H L 1 H_L^1 by means of an area integral function associated with the operator L L . By using a variant of the maximal function associated with the semigroup { e − t L } t ≥ 0 \{e^{-tL}\}_{t\geq 0} , a space BMO L \textrm {BMO}_L of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if L L has a bounded holomorphic functional calculus on L 2 ( R n ) L^2({\mathbb R}^n) , then the dual space of H L 1 H_L^1 is BMO L ∗ \textrm {BMO}_{L^{\ast }} where L ∗ L^{\ast } is the adjoint operator of L L . We then obtain a characterization of the space BMO L \textrm {BMO}_L in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces K L {\mathcal K}_L of BMO L _{ L} when L L is a second-order elliptic operator of divergence form and when L L is a Schrödinger operator, and study the inclusion between the classical BMO space and BMO L \textrm {BMO}_L spaces associated with operators.

Abstract. We consider abstract non-negative self-adjoint operators on L 2 (X) which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the BochnerRiesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on R n , the harmonic oscillator, elliptic operators on compact manifolds and Schrödinger operators on asymptotically conic manifolds.