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Under the assumption that the underlying measure is a non-negative Radonmeasure which only satisfies some growth condition and may not be doubling, wedefine the product of functions in the regular $BMO$ and the atomic block$\H^{1}$ in the sense of distribution, and show that this product may be splitinto two parts, one in $L^{1}$ and the other in some Hardy-Orlicz space.

Product of functions in $\BMO$ and $\H^{1}$ in non-homogeneous spaces