You can use the function get_the_author_meta() to get the emailaddress of the author of the current post and the wp_mail()function to send an e-mail
<?php
$timelimit=1 * 604800; //1 week * seconds per day
$post_age = date('U') - get_post_time('U');
if ($post_age < $timelimit) : ?>
// Current
<?php elseif ($post_age > $timelimit) :
$email = get_the_author_meta( 'user_email' );
wp_mail( $email, $subject, $message );
endif;
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