1. Under means in a lower position than something else.

Pattern 1: verb + under + noun
We sat under the tree and had a picnic.

Pattern 2: verb + noun + under + noun
Let's put the desk under the window; then wel'll have a great view.

2. Under means covered by something else; underneath.

Pattern 1: verb + under + noun
The children hid under the table, thinking we couldn't see them.

Typical verbs before under:
be, hide, lie, rest, sit, sleep, stand, wait, walk

Pattern 2: verb + noun + under + noun
She stores all her boxes under the bed.

Typical verbs:
bury, find, hide, place, push, put, store, wear

3. Under means less than.

Pattern: under + noun
I'm sure she was driving under the speed limit.
He has three children under age ten.

Nouns commonly used after under:
any number, age, average, height, limit, maximum, minimum, norm weight

Expression:
to be under age—to not be old enough to do something
She can't vote because she is under age.

4. Under can indicate control.

Pattern 1: under + noun
Under this boss we have little freedom to express our own ideas.
You have a lot more benefits under the new insurance policy.

Typical nouns used after under:
boss, coach, contract, dictator, doctor, general, king, mayor, policy, principal, president, professor, supervisor, teacher

Pattern 2: under + the + noun + of + noun
She is under the care of a doctor.

Typical nouns after under the:
administration, care, control, dictatorship, direction, eye, management, presidency

Pattern 3: under + 0 + noun
The children are under supervision at all times.
They were arrested under orders of the chief.