adverb
Adverbial use meaning 'to a great extent' or 'very much'. This sense is somewhat literary or old-fashioned; in modern standard Japanese, とても or 大変 are more common.
おおきに感謝しております。
I am greatly obliged to you.
interjection
thanks a lot
Kansai dialect interjection meaning 'thank you very much', abbreviated from おおきにありがとう. Used in casual, friendly situations in the Kansai region.
おおきに、また来てな。
Thanks a lot, come again.
The adverbial sense derives from the adjective 大きい (おおきい, 'big') with the adverbial suffix に. The interjection is a contraction of おおきにありがとう ('thank you greatly'), used in Kansai dialect.