Weighted norm inequalities, embedding theorems and integration operators on vector-valued Fock spaces

In this paper, we characterize the d× d matrix-valued weights W on the complex plane ℂ such that the Fock projection P_α is bounded on the vector-valued spaces L^2_α ,W(ℂ^d) induced by W. It is proved that P_α is bounded on L^2_α ,W(ℂ^d) if and only if W satisfies a restricted A_2 -condition. Then we establish some function-theoretic and operator-theoretic properties for the Fock spaces F^2_α ,W(ℂ^d) induced by the d× d matrix-valued weights W satisfying the restricted A_2 -condition: we show that the ℂ^d -valued polynomials are dense in F^2_α ,W(ℂ^d) ; a Littlewood–Paley formula for F^2_α ,W(ℂ^d) is established; the bounded differentiation and integration operators D^(n):F^2_α ,W(ℂ^d)→ L^2(Φ dA;ℂ^d) are characterized, where Φ is a nonnegative matrix-valued function; we also investigate the boundedness of the Volterra type integration operator T_G acting on F^2_α ,W(ℂ^d) , where G is a matrix-valued entire function. In particular, it is shown that for d≥ 2 , T_G may be unbounded on F^2_α ,W(ℂ^d) when G is a linear polynomial, and T_G may be compact on F^2_α ,W(ℂ^d) when G is a polynomial of degree greater than 3. These phenomena are in sharp contrast with the case d=1 , where T_G is bounded (resp. compact) if and only if G is a polynomial of degree not more than 2 (resp. a linear polynomial).