Decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations

We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations. The achievements of this paper are two folds. One is improved decay rates of $u_{\th}$ and $\na {\bf u}$, especially we show that $|u_{\th}(r,z)|\leq c\left(\f{\log r}{r}\right)^{\f 12}$ for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of $\om_{\th}$, integral representations of ${\bf u}$ in terms of $\bm{\om}=\textit{curl }{\bf u}$ and $A_p$ weight for singular integral operators, which yields good decay estimates for $(\na u_r, \na u_z)$ and $(\om_r, \om_{z})$, where $\bm{\om}= \om_r {\bf e}_r + \om_{\th} {\bf e}_{\th}+ \om_z {\bf e}_z$. Another is the first decay rate estimates in the $Oz$-direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.