Looking on the internet deeply has found these results:

in the first week is the most popular phrase on the web.

on the first week 1161,000 results on the web	in the first week 160,400,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
May 21, 2014 ... up vote 3 down vote favorite. 1. Do you say I�ll arrive in the first week of July? or on the first week of July? prepositions�... On the first week I did 500 thousand. Gold in the spring, and diamond in the fall � And then a world tour just to top it all off � And let's pretend like they call me the�... "Things to do IN the first week of June, 2011" OR "Things to do ON the first week of June, 2011" Thank you very much for your prompt response. Cheers, RENAN. Thoughts on the first week overall. by Doug V on April 13, 2015. Just thought I'd give some brief thoughts on both the Cards and baseball and general. This type�...	It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

on the first week

1161,000 results on the web

in the first week

160,400,000 results on the web

More popular!

May 21, 2014 ... up vote 3 down vote favorite. 1. Do you say I�ll arrive in the first week of July? or on the first week of July? prepositions�...
On the first week I did 500 thousand. Gold in the spring, and diamond in the fall � And then a world tour just to top it all off � And let's pretend like they call me the�...
"Things to do IN the first week of June, 2011" OR "Things to do ON the first week of June, 2011" Thank you very much for your prompt response. Cheers, RENAN.
Thoughts on the first week overall. by Doug V on April 13, 2015. Just thought I'd give some brief thoughts on both the Cards and baseball and general. This type�...

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.