Best N term approximation spaces for tensor product wavelet bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure circle times(q) on certain quasi-Banach spaces. We prove that the approximation spaces A(q)(alpha)(L(2)) and A(q)(alpha)(H(1)) equal tensor products of Besov spaces B(q)(alpha)(L(q)), e.g., A(q)(alpha)(L(2)([0,1](d))) = B(q)(alpha)(L(q)([0,1])) circle times(q) (...) circle times(q) B(q)(alpha)(L(q)([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.