1. Vehicles are prohibited in this area. Smoking is forbidden in the library.

You're marking. Otherwise, the result will be different. For example, no smoking.

In the library. You can omit (is) when marking. Others have also omitted be or even more.

1。 Vehicles are not allowed to enter this area.

All vehicles are prohibited from entering the area.

This area is closed to traffic.

This area is closed to traffic.

Smoking is forbidden in the library.

Smoking is prohibited in the library.

Smoking is not allowed in the library.

Smoking is prohibited in the library.