Use the page number to calculate a new date to get the week to fetch:
$page = ( get_query_var( 'paged' ) ) ? get_query_var( 'paged' ) : 1;
$date_to_fetch = time() - ( 604800 * ( $page - 1 ) ) );
$query_string = array(
'post_type' => 'post',
'date_query' => array(
array(
'year' => date( 'o', $date_to_fetch ),
'week' => date( 'W', $date_to_fetch ),
),
),
'post_status' => 'publish'
);
query_posts($query_string);
If you are working with date( 'W' ), you should always be working with date( 'o' ) instead of date( 'Y' ), because at the beginning of a year it is possible that you are still in a week which is accounted to the past year. For this see the documentation of the date function: http://php.net/manual/de/function.date.php
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