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Let $X$ be an abstract orientable not necessarily compact CR manifold ofdimension $2n+1$, $n\geq1$, and let $L^k$ be the $k$-th tensor power of a CRcomplex line bundle $L$ over $X$. Suppose that condition $Y(q)$ holds at eachpoint of $X$, we establish asymptotics of the heat kernel of Kohn Laplacianwith values in $L^k$. As an application, we give a heat kernel proof of Morseinequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbbR$-action, we also establish asymptotics of the $\mathbb R$-equivariant heatkernel of Kohn Laplacian with values in $L^k$. As an application, we get$\mathbb R$-equivariant Morse inequalities on compact CR manifolds withtransversal CR $\mathbb R$-action.

Heat kernel asymptotics for Kohn Laplacians on CR manifolds