  In this paper, our goal is to establish the Sobolev space associated to the
partial harmonic oscillator. Based on its heat kernel estimate, we firstly give
the definition of the fractional powers of the partial harmonic oscillator
$$\AH=-\partial_{\rho}^2-\Delta_x+|x|^2,$$ and show that its negative powers
are well defined on $L^p(\mathbb R^{d+1})$ for $p\in [1,\infty]$. We then
define associated Riesz transforms and show that they are bounded on classical
Sobolev spaces by the calculus of symbols.
  Secondly, by a factorization of the operator $\AH$, we define two families of
Sobolev spaces with positive integer indices, and show the equivalence between
them by the boundedness of Riesz transforms. Moreover, the adapted symbolic
calculus also implies the boundedness of Riesz type transforms on the Sobolev
spaces associated to the partial harmonic oscillator $\AH$.
  Lastly, as applications of our results, we obtain the revised
Hardy-Littlewood-Sobolev inequality, the Gagliardo-Nirenberg-Sobolev
inequality, and Hardy's inequality in the potential space $L_{\AH}^{\alpha,
p}$.
