Convolution by means of bilinear maps
نویسندگان
چکیده
Given three complex Banach spaces X,Y, Z and u : X × Y → Z a bounded bilinear map. For f(z) = ∑m n=0 xnz n where xn ∈ X and g(z) = ∑k n=0 ynz n where yn ∈ X, we define the u-convolution of f an g as the polynomial given by f ∗u g(z) = ∑min{m,k} n=0 u(xn, yn)z n. It is shown that whenever X and Y veryfies the vector-valued analogue of certain inequalities due to Littlewood and Paley for Hardy spaces we have that if 1 ≤ p1 ≤ 2, 1 ≤ p2 ≤ ∞ such that 1 p1 + 1 p2 ≥ 1 and 1 ≤ p, q ≤ ∞ are such that 1 p = 1 p1 + 1 p2 − 1 and 1 q = 1 2 + 1 max{p2,2} then there exists a constant C > 0 such that
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