Across A: has the same digit sum as B down.
E: The digit sum of E across equals half the digit product of F down.
G: a prime number.
H: This square number equals half the reverse of G across.
I: the third power of the reverse of G across.
K: A across + E across + I across

Down A: A palindrome.
B: A across + C down.
C: F down squared.
D: A square number.
F: To get the reverse of F down, multiply the digit sum of F down with the reverse of H across.
I: this square number equals twice the digit sum of B down.
J: Yet another square number.