2 Added an EXISTS example (not really worth a separate answer)
source | link

Should be something like this:

DELETE A
FROM 
    table_example1 AS A
    INNER JOIN table_example2 AS B
    ON A.COLUMN1 =B.COLUMN1
    AND A.COLUMN2 = B.COLUMN2
WHERE
    COLUMN_DATETIME > @Period;

Alternatively:

DELETE FROM A
FROM dbo.table_example1 AS A
WHERE EXISTS
(
    SELECT *
    FROM dbo.table_example2 AS B
    WHERE
        B.COLUMN1 = A.COLUMN1
        AND B.COLUMN2 = A.COLUMN2
        AND B.COLUMN_DATETIME > @Period
);

Should be something like this:

DELETE A
FROM 
    table_example1 AS A
    INNER JOIN table_example2 AS B
    ON A.COLUMN1 =B.COLUMN1
    AND A.COLUMN2 = B.COLUMN2
WHERE
    COLUMN_DATETIME > @Period;

Should be something like this:

DELETE A
FROM 
    table_example1 AS A
    INNER JOIN table_example2 AS B
    ON A.COLUMN1 =B.COLUMN1
    AND A.COLUMN2 = B.COLUMN2
WHERE
    COLUMN_DATETIME > @Period;

Alternatively:

DELETE FROM A
FROM dbo.table_example1 AS A
WHERE EXISTS
(
    SELECT *
    FROM dbo.table_example2 AS B
    WHERE
        B.COLUMN1 = A.COLUMN1
        AND B.COLUMN2 = A.COLUMN2
        AND B.COLUMN_DATETIME > @Period
);
1
source | link

Should be something like this:

DELETE A
FROM 
    table_example1 AS A
    INNER JOIN table_example2 AS B
    ON A.COLUMN1 =B.COLUMN1
    AND A.COLUMN2 = B.COLUMN2
WHERE
    COLUMN_DATETIME > @Period;