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Let      A p , α  ( B ) , A q , α  ( B ) {A^{p,\alpha }}(B),{A^{q,\alpha }}(B)       and      A l , α  ( B ) {A^{l,\alpha }}(B)       be weighted Bergman spaces on the unit ball of      C n ( n &gt; 1 ) {{\text {C}}^{n}}\,(n &gt; 1)      . We prove: Theorem 1.  If       1 / l = 1 / p + 1 / q 1/l = 1/p + 1/q        then       A p , α  ( B ) ⋅  A q , α  ( B ) {A^{p,\alpha }}(B) \cdot {A^{q,\alpha }}(B)        is of first category in       A l , α  ( B ) {A^{l,\alpha }}(B)      . Theorem 2.  Theorem 1 holds for Hardy spaces in place of weighted Bergman spaces . We also show that Theorems 1 and 2 hold for the polydisc      U n {U^n}       in place of      B B      .

Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of 𝐶ⁿ (𝑛>1)