I have a range of IP addresses but my ISP says the range belongs to them. Who owns the IP addresses I use?

IP addresses are a shared public resource.

The LIRs assign IP address space to meet the needs of network operators, including themselves. The hierarchical manner in which the public Internet addresses are distributed permits the aggregation of routing information. Aggregation ensures proper operation of Internet routing.

A clear contractual arrangement that specifies the validity and duration of the address assignment is strongly recommended for every address assignment.