In this paper, we will study the Hardy and BMO spaces associated to the generalized Hardy operator $$L_{\alpha }= (-\Delta )^{\alpha /2}+a|x|^{-\alpha }$$ L α = ( - Δ ) α / 2 + a | x | - α . Similarly to the classical Hardy and BMO spaces, we will prove that our new function spaces will enjoy some important results such as molecular decomposition and duality. As applications, we show the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm inequalities involving the generalized Hardy operator.

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