Rolling and friction

Classical Mechanics

When the velocity of the center $v$ and the angular velocity $\omega$ of a round object are related as $\vec v = \vec \omega \times \vec r,$ where $\vec r$ is the position vector of the particle with respect to the center of the round object, which friction will act on the rolling object at an instant?

The surface is horizontal and rough.
No other force acts in the horizontal direction.