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"carried out over" is correct and usable in written English.
You would use this phrase to describe an activity that is taking place for a certain period of time. For example, "The survey was carried out over two months."
Exact(59)
"Restore Rundberg" policing efforts were carried out over 15 months, starting April 2014 thru June 2015.
Separate and simultaneous hydrolyzed were carried out over 72 h.
Nine operations were carried out over the next hours.
Simulations were carried out over two days.
Experiment was carried out over a 22-day period.
Then, PFM measurements were carried out over the area.
The training was carried out over 1000 epochs.
The study was carried out over 4 consecutive days.
The region of interest analysis was carried out over the entire viewing period.
where summation is carried out over all the burrows in the given depth interval.
"Safer Union" was carried out over a 90-day period starting in October 2010.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com