We　study　solutions　of　the　initial　value　problem　for　the　2D　regularized　surface　quasi-geostrophic　(RSQG)　equation.　For　(H)　over　dot(1)(Omega)　boolean　AND　L-q(Omega)(q　＜　2　＜　infinity)　initial　data,　we　prove　the　global　existence　and　uniqueness　of　weak　solution　for　RSQG　equation　with　subcritical　powers.　For　RSQG　equation,　we　establish　some　regularization　results　and　prove　the　inviscid　limit　of　the　RSQG　equation　to　the　classical　quasi-geostrophic　equation.