Take a look to window (or analytic) functions. Unlike aggregate functions, window functions preserve resulting rows and facilitate operations related to them. When using order by in over clause, windowing is done from first row to current row according to specified order, which is exactly what you need.

select year, week, sum(number_of_records) over (order by year, week)
from (
  select year(creation_date) as year, weekofyear(creation_date) as week,
  count(id) as number_of_records
  from input group by year, week
) your_sql

I guess you will also need to reset sum for each year, which I leave as exercise for you (hint: partition clause).

What you want is called the cumulative sum, you can do something like:

create table transactions (transactionid int, d date);
insert into transactions (transactionid, d) 
    values (1, '2014-08-04'),(2,'2014-08-05'), (3, '2014-08-18')
         , (4, '2014-08-18'), (5,'2014-08-20');

select x.y, x.w,  count(1) 
from ( 
   select distinct year(d) as y, week(d) as w 
   from transactions
) as x 
join transactions y 
    on year(y.d) < x.y
    or ( year(y.d) = x.y
     and week(y.d) <= x.w ) 
group by x.y, x.w;  

+------+------+----------+
| y    | w    | count(1) |
+------+------+----------+
| 2014 |   31 |        2 |
| 2014 |   33 |        5 |
+------+------+----------+

I did not see your additional request for 2 2 for 2014. You can do that by replacing:

select distinct year(d) as y, week(d) as w 
from transactions 

...with an expression that creates the whole domain for weeks. It is often a good idea to create a calendar table that you can use to join against to get reports for missing values etc.

I'm a little confused by your query:

• You have a left join from items to users as if you expect some items with no valid user id.
• You are using u.id in the select, but that would be NULL with no match.

I would suggest:

select it.owner_id,
       date_trunc('week', it.created::timestamp) as week_start,
       date_trunc('week', it.created::timestamp) + interval '6 day' as week_end,
       count(*) as this_week,
       sum(count(*)) over (partition by uu.id order by min(timestamp)) as running_count
from items it
group by it.owner_id, week_start;

This uses Postgres syntax because your code looks like Postgres.