Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices

A H\"older type inequality in Besov spaces is established and applied to show that every strong solution $u(t,x)$ on (0,T) of the Navier-Stokes equations can be continued beyond $t>T$ provided that the vorticity $\omega(t,x)\in L^{\frac 2{2-\alpha}}(0,T;\dot{B}^{-\alpha}_{\infty,\infty}(\mr^3))\cap L^{\frac2{1-\alpha}}(0,T;\dot{B}^{-1-\alpha}_{\infty,\infty}(\mr^3))$ for $0<\alpha<1$.