This formula works by testing each date to see if the date is a Sunday,
and then tests to see if that Sunday's calendar day is less than 8. (8
is the limit because a Sunday that falls on the first, day 1, won't
appear again until day 8. If the first of the month is a Wednesday,
the first Sunday is the 5th of the month; 5 is less than 8.)

=IF(WEEKDAY(A1,1)=1,IF(DAY(A1)<8,"John Doe",""),"")