The Hardy spaces for Fourier integral operators H p FIO (R n ), for 1 ≤ p ≤ ∞, were introduced by H. Smith in [16] and A. Hassell et al. in [10]. In this article, we give several equivalent characterizations of H 1 FIO (R n ), for example in terms of Littlewood-Paley g functions and maximal functions. This answers a question from [13].

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Let L be a non-negative self-adjoint operator acting on L 2 (X) where X is a space of homogeneous type with a dimension n. In this paper, we study sharp endpoint L p -Sobolev estimates for the solution of the initial value problem for the Schrödinger equation i∂ t u + Lu = 0 and show that for all f ∈ L p (X), 1 < p < ∞,where the semigroup e −tL generated by L satisfies a Poisson type upper bound. This extends the previous result in [8] in which the semigroup e −tL generated by L satisfies the exponential decay.

In this paper, we study the L p boundedness and L p (w) boundedness (1 < p < ∞ and w a Muckenhoupt A p weight) of fractional maximal singular integral operators T # Ω,α with homogeneous convolution kernel Ω(x) on an arbitrary homogeneous group H of dimension Q. We show that if 0 < α < Q, Ω ∈ L 1 (Σ) and satisfies the cancellation condition of order [α], then for any 1 < p < ∞,, where for the case α = 0, the L p boundedness of rough singular integral operator and its maximal operator were studied by Tao ([36]) and Sato ([32]), respectively.We also obtain a quantitative weighted bound for these operators. To be specific, if 0 ≤ α < Q and Ω satisfies the same cancellation condition but a stronger condition that Ω ∈ L q (Σ) for some q > Q/α, then for any 1 < p < ∞ and w ∈ A p , T # Ω,α f L p (w) Ω L q (Σ) {w} A p (w) A p f L p α (w) , 1 < p < ∞.

Let L be a non-negative self-adjoint operator acting on L 2 (X), where X is a space of homogeneous type with a dimension n. Suppose that the heat operator e −tL satisfies the generalized Gaussian (p 0 , p ′ 0 )-estimates of order m for some 1 ≤ p 0 < 2. It is known that the operator (I + L) −s e itL is bounded on L p (X) for s ≥ n|1/2 − 1/p| and p ∈ (p 0 , p ′ 0 ) (see for example, [6,8,10,12,15,31]). In this paper we study the endpoint case p = p 0 and show that for s 0 = n 1 2 − 1 p 0 , the operator (I + L) −s 0 e itL is of weak type (p 0 , p 0 ), that is, there is a constant C > 0, independent of t and f so thatfor α > 0 when µ(X) = ∞, and α > f p 0 /µ(X) p 0 when µ(X) < ∞.Our results can be applied to Schrödinger operators with rough potentials and higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds.