In this article, we will be concerned with questions related to the boundedness of the Riesz transform on manifolds. Since the seminal work of Coulhon and Duong [4], who gave sufficient conditions on the heat kernel so that the Riesz transform is bounded on Lp for 1 < p ≤ 2, several authors have investigated both necessary and sufficient conditions for the boundedness of the Riesz transform on ...

We present here a correction to an error in our paper [5]. We are grateful to Dachun Yang for bringing the error to our attention, and we thank Alan McIntosh for helpful discussions which have led to this correction of the error. In particular, our approach here follows that of Auscher, McIntosh and Russ [1], as explained to us by McIntosh. In [5] we develop a theory of H1 (Hardy type) and BMO ...

Using the Kaczmarz algorithm, we obtain a Fourier series formulation for functions in the L2 space of singular measures on the unit circle. This formula is applied to the problem of finding reproducing kernel Hilbert spaces inside the classical Hardy space, where the norm is instead that of boundary integration with respect to a singular measure. We also give some conditions ensuring that these...

Denote by γ the Gauss measure on R and by L the Ornstein– Uhlenbeck operator. In this paper we introduce a Hardy space h(γ) of Goldberg type and show that for each u in R\{0} and r > 0 the operator (rI+L) is unbounded from h(γ) to L(γ). This result is in sharp contrast both with the fact that (rI+L) is bounded from H(γ) to L(γ), where H(γ) denotes the Hardy type space introduced in [MM], and wi...

Let X be an RD-space with μ(X ) = ∞, which means that X is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this paper, we characterize the atomic Hardy spaces H at(X ) of Coifman and Weiss for p ∈ (n/(n + 1), 1] via the radial maximal function, where n is the “dimension” of X , and the range of index p is the best possible. Thi...

For any simple real group G possessing unitary highest weight representations one can define the Hardy space H(G) . This is a Hilbert space formed by holomorphic functions in a ‘non–commutative’ tube domain Γ satisfying a Hardy–type condition (Γ is the interior of a non– commutative complex semigroup Γ containing the group G ). The space H(G) is identified with the bi–invariant subspace of L(G)...