In the US, there are very few restrictions on recreational flight due to Congressional mandate. You can indeed fly within controlled airspace, BUT you are legally required to NOT interfere with manned aircraft and to fly in a safe manner.
"Notify" is the right word. The airport or tower cannot prohibit your flight legally, but not every employee knows this, and may interpret your notification as a request for permission.
My advice - if you are going to fly near any airport, US or elsewhere:
1. Stay as low as possible.
2. Stay away from the ends of the runways for a good mile or so. You can see the layout on any chart and most maps.
3. Observe the manned aircraft as they enter the landing pattern over the airport, and stay well away from that area. Note: It can change over time as wind conditions change.
4. Do not overfly the field.
5. Give the airport/tower as much information as you can about your intended operating area and maximum altitude, and stick to it.
Bottom line: Be as safe as possible and as respectful for the legitimate concerns that the larger aviation community has about drones. Don't give them a reason to be any more apprehensive about sharing the airspace than they are right now.