Paris Hilton -- I'm APOLOGIZING To Lindsay

Paris Hilton feels so bad about the JAB word, she took to Lindsay Lohan ... she is now apologizing to full

I apologize to Lindsay for the comment I made​​. I was joking around, but he was reckless and unnecessary. "

We broke the story ... Hilton zinged Lohan on her reality show, "The world according to Paris"- a shot Lohan on her for theft. Lohan told friends she was hurt by the comment.

But Hilton insists she's serious about repairing the relationship ... saying: "I am reaching out to Lindsay that I can tell him myself."