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If you want to register the name in the US you have to do a trademark search. In the US there are common law trademarks which arise from use of a name on products or services in commerce. If a person has been using a name in commerce even though it is not registered, he will be protected in that use against a later registered trademark. You can also register the trademark at the US Patent and Trademark office which provides the strongest protection.
Because of the prevalence of not just registered trademarks but common law marks, a complete search cannot be done on the internet. To get a full search plus the fees for filing the application and following the process through to completion would run between $1,000 to $2,000 dollars.
It is critical that you be selling under the mark in the US otherwise the mark will be useless because you must have product in commerce under the mark to have a protectable interest. You cannot have a mark and not actively use it and protect your rights.
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