You can have your (very) low power LTE base station. Bandwidths ie radio spectrum frequency allocations, including LTE are licensed/managed by respective governments. Carriers pay these governing entities for use of the bandwidth for a period of time. Carriers once secure the licenses, deploy technology (towers, earth stations, satellites, etc.) for the use of purchased bandwidth. They hope they will recoup those costs and profit through selling the use of the equipment and bandwidth use to you.

Put it in a different way, you do not have the LTE infrastructure & licensing, therefore you must rely on a cellphone carrier to provide you the towers and such.