under

from The American Heritage® Dictionary of the English Language, 5th Edition.

• preposition In a lower position or place than.
• preposition To or into a lower position or place than.
• preposition Beneath the surface of.
• preposition Beneath the assumed surface or guise of.
• preposition Less than; smaller than.
• preposition Less than the required amount or degree of.
• preposition Inferior to in status or rank.
• preposition Subject to the authority, rule, or control of.
• preposition Subject to the supervision, instruction, or influence of.
• preposition Undergoing or receiving the effects of.
• preposition Subject to the restraint or obligation of.
• preposition Within the group or classification of.
• preposition In the process of.
• preposition In view of; because of.
• preposition With the authorization of.
• preposition Sowed or planted with.
• preposition Nautical Powered or propelled by.
• preposition During the time conventionally assigned to (a sign of the zodiac).
• adverb In or into a place below or beneath.
• adverb So as to be covered or enveloped.
• adverb So as to be less than the required amount or degree.
• adverb So as to be rendered unconscious, as by an anesthetic.
• adverb In or into a condition of ruin or death.
• adjective Located or situated on a lower level or beneath something else.
• adjective Lower in rank, power, or authority; subordinate.
• adjective Less than is required or customary.
• idiom (out from under) Having gotten free of worries or difficulties.

from The Century Dictionary.

• Lower in position; situated beneath: opposed to upper: as, the under side; the under mandible.
• Lower in rank or degree. See under, adverb, note .
• Of sounds, lower in pitch.
• Below; beneath: expressing position with reference to that which is above, whether in immediate contact or not, or which towers aloft, surmounts, covers, or overtops: as, all under heaven; under the earth or the sea; under the surface; under the table; to take shelter under a tree; to live under the same roof; to hide a thing under a heap of straw; to hide one's light under a bushel; to overhear a conversation under one's windows.
• In or at a place, point, or position that is lower than; further down than; immediately below: as, to hit a man under the belt; to have pains under the arms.
• In the position or state of, or while bearing, supporting, sustaining, receiving, suffering, undergoing, or the like: as, to sink under a load; to act under great excitement.
• Inferior to in point of rank, dignity, social position, or the like.
• Inferior to or less than, with respect to number, amount, quantity, value, age, etc.; falling short of; in or to a less degree than; hence, at, for, or with less than: as, it cannot be bought under $20.
• Of sounds, inferior to, in pitch.
• Subject to.
• Liable or exposed to: as, under fire; under the penalty of fine or imprisonment.
• Subject to the government, rule, command, direction, orders, guidance, or instruction of: as, to serve under Wellington; I studied under him; to sit under a favorite preacher.
• Subject to the influence or operation of; actuated by.
• In accordance with; in conformity with: as, to sell out under the rule.
• Bound by: as, to be under bonds, or a vow.
• In: with reference to circumstances.
• In: with reference to category, division, section, class, etc.: as, to treat several topics under one head.
• In course of: as, to be under treatment, or under discussion.
• In the form or style of; by the appearance or show of; with the character, designation, pretense, pretext, or cover of.
• During the time or existence of: said especially of rulers and their period of rule: as, Christ suffered under Pontius Pilate; the Armada was destroyed under the reign of Elizabeth; the American revolution broke out under the administration of Lord North.
• With the sanction, authorization, permission, or protection of: as, under favor; under leave; under protection, etc.
• [The preposition under in adverbial phrases often coalesces with its noun to form an adverb, from which the adjective or noun may be derived: as, under ground, ⟩ underground, adverb, ⟩ underground, a.; under hand, ⟩ underhand, adverb, ⟩ underhand, adjective; so underboard, underearth, underfoot, etc. Such forms are not true compounds, but are coalesced phrases, like aground, aboard, afoot, etc.]