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This paper is inspired by the work of Nagel and Stein in which the L p (1 < p < ∞) theory has been developed in the setting of the product Carnot-Carathéodory spaces M = M 1 × • • • × M n formed by vector fields satisfying Hörmander's finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product BM O space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.

Abstract. In this article we prove Cotlar's inequality for the maximal singular integrals associated with operators whose kernels satisfy regularity conditions weaker than those of the standard m-linear Calderón-Zygmund kernels. The present study is motivated by the fundamental example of the maximal mth order Calderón commutators whose kernels are not regular enough to fall under the scope of the m-linear Calderón-Zygmund theory; the Cotlar inequality is a new result even for these operators.

In this paper, we establish the two weight commutator of Calderón-Zygmund operators in the sense of Coifman-Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for A 2 weight and by proving the sparse operator domination of commutators. The main tool here is the Haar basis and the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutator) for the following Calderón-Zygmund operators: Cauchy integral operator on R, Cauchy-Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (one-dimension and high dimension).to the subspace of functions {F b } that are boundary values of functions F ∈ H 2 (U n ). The associated Cauchy-Szegö kernel is as follows.Then it is natural to study the following question: is there a setting, by which the characterisation of two weight commutators and the related BMO space for Calderón-Zygmund operators T can be obtained, that can be applied to Calderón-Zygmund operators such as the Bessel Riesz transform, the Cauchy-Szegö projection operator on Heisenberg groups, and many other examples?To address this question we work in a general setting: spaces of homogeneous type introduced by Coifman and Weiss in the early 1970s, in [9], see also [10]. We say that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss if d is a quasi-metric on X and µ is a nonzero measure satisfying the doubling condition. A quasi-metric d on a set X is a function d : X × X −→ [0, ∞) satisfying (i) d(x, y) = d(y, x) ≥ 0 for all x, y ∈ X; (ii) d(x, y) = 0 if and only if x = y; and (iii) the quasi-triangle inequality: there is a constant A 0 ∈ [1, ∞) such that for all x, y, z ∈ X,

In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces X = X 1 × X 2 × · · · × X n , where each factor X i is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood-Paley theory on X, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product H p , the dual CMO p of H p with the special case BMO = CMO 1 , and the predual VMO of H 1 . We also use the wavelet expansion to establish the Calderón-Zygmund decomposition for product H p , and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood-Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type. Contents2010 Mathematics Subject Classification. Primary 42B35; Secondary 43A85, 42B25, 42B30.

Abstract. Long lead time flood forecasting is very important for large watershed flood mitigation as it provides more time for flood warning and emergency responses. The latest numerical weather forecast model could provide 1-15-day quantitative precipitation forecasting products in grid format, and by coupling this product with a distributed hydrological model could produce long lead time watershed flood forecasting products. This paper studied the feasibility of coupling the Liuxihe model with the Weather Research and Forecasting quantitative precipitation forecast (WRF QPF) for large watershed flood forecasting in southern China. The QPF of WRF products has three lead times, including 24, 48 and 72 h, with the grid resolution being 20 km × 20 km. The Liuxihe model is set up with freely downloaded terrain property; the model parameters were previously optimized with rain gauge observed precipitation, and re-optimized with the WRF QPF. Results show that the WRF QPF has bias with the rain gauge precipitation, and a post-processing method is proposed to post-process the WRF QPF products, which improves the flood forecasting capability. With model parameter re-optimization, the model's performance improves also. This suggests that the model parameters be optimized with QPF, not the rain gauge precipitation. With the increasing of lead time, the accuracy of the WRF QPF decreases, as does the flood forecasting capability. Flood forecasting products produced by coupling the Liuxihe model with the WRF QPF provide a good reference for large watershed flood warning due to its long lead time and rational results.

We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for L p . Next we focus on spaces X of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces H 1 (X) and BMO(X). In the setting of product spaces X = X 1 × · · · × X n of homogeneous type, we show that the space BMO( X) of functions of bounded mean oscillation on X can be written as the intersection of finitely many dyadic BMO spaces on X, and similarly for A p ( X), reverse-Hölder weights on X, and doubling weights on X. We also establish that the Hardy space H 1 ( X) is a sum of finitely many dyadic Hardy spaces on X, and that the strong maximal function on X is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for BMO and H 1 due to Mei and to Li, Pipher and Ward.