awk if one liner syntax for equating with a string variable

I would like to have help with syntax for using a string varaibles inside if and else in a awk one liner.
Eg. I would like to say, list all the filenames that have been modified in a particular month(present in a string variable) or list all the filenames whose owner is $owns and owner group is $group in current directory by using awk one liner...
I tried

with '==' but its either taken literally or gives error.. I am using:

ls -ltr | awk '{if ($6 == "${month_value}"){print $9 }' > filestobeprocessed.txt

see this post

ls -ltr | awk -v m="$month_value" '$6==m{print $9}' > filestobeprocessed.txt

try

ls -ltr | awk '$6 == '$month_value' {print $9 }' > filestobeprocessed.tx