user1@debian:~$ sudo -u user2 -H ulimit -S -c unlimited
sudo: ulimit: command not found
user1@debian:~$


user1@debian:/home/user1$ ulimit -S -c unlimited
user1@debian:/home/user1$


user2@debian:/home/user2$ ulimit -S -c unlimited
user2@debian:/home/user2$

# /etc/sudoers
#
# This file MUST be edited with the 'visudo' command as root.
#
# See the man page for details on how to write a sudoers file.
#

Defaults        env_reset,timestamp_timeout=0

# Host alias specification

# User alias specification

# Cmnd alias specification

# User privilege specification
root    ALL=(ALL) ALL
user1 debian=(user2) NOPASSWD: ALL

# Uncomment to allow members of group sudo to not need a password
# (Note that later entries override this, so you might need to move
# it further down)
# %sudo ALL=NOPASSWD: ALL

user1@debian:~$ sudo -u user2 -H -c "ulimit -S -c unlimited"
sudo: illegal option `-c'
usage: sudo -h | -K | -k | -L | -l | -V | -v
usage: sudo [-bEHPS] [-p prompt] [-u username|#uid] [VAR=value]
            {-i | -s | <command>}
usage: sudo -e [-S] [-p prompt] [-u username|#uid] file ...
user1@debian:~$

user1@debian:~$ sudo -u user2 -H bash -c "ulimit -S -c unlimited"
user1@debian:~$