In　cognitive　radio　networks　(CRN),　underlay　spectrum　sharing　allows　secondary　users　(SUs)　to　utilize　the　spectrum　on　which　the　primary　users　(PUs)　in　primary　radio　networks　(PRN)　are　working　at　the　same　time,　without　introducing　intolerant　interferences.　In　this　paper,　we　study　the　joint　transmission　power　and　time　allocation　in　CRN　with　underlay　spectrum　sharing　technology.　To　maximize　the　total　capacity　of　CRN　and　maintain　the　fairness　for　SUs,　the　system　is　modeled　as　an　optimization　problem　with　the　constraints　of　interference　and　transmission　power　and　time.　Furthermore,　we　prove　that　the　problem　of　joint　optimization　resource　allocation　under　the　constraints　condition　can　be　implemented　independently　from　the　time-frequency　domain.　Based　on　this　study,　we　propose　the　optimal　joint　transmission　power　and　time　allocation　method.　Extensive　simulation　results　show　that　the　proposed　optimal　method　can　significantly　improve　the　system　capacity　and　maintain　the　fairness　compared　to　the　existing　methods.　©2010　IEEE.