The tune In Your Own Sweet Way was composed by Dave Brubeck, but most people have heard it first by the Miles Davis Quartet. In 1956 Miles recorded a bunch of Standards and contemporary tunes for Prestige, resulting in his superb collection (Workin, Steamin, Cookin, Relaxin). The band had the "young" Coltrane on board, as well as Red Garland.
Nevertheless, the tune is great. The melody is cool and the changes are interesting.
For the I chord, at the end of each A-section, Brubeck suggests the use of Bb-lydian. Miles ignores this mostly, but you hear Coltrane hinting at the lydian sound.
For me, personally, I like Bb played as ionian. The melody wants to resolve, but, hey, I'm not a virtuoso, so, don't take my word for it.
As always, you can download the tune HERE. If you load it and you don't have the appropriate Jazzstylezz style, it will say "style missing". But you can always substitute it with any other style of your liking. Or better: get a copy of the Standards Pack 1.