, Volume 67, Issue 1-2, pp 1-11

Quadratic difference operators in L p -spaces

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Summary.

Let \( (G,+,\Sigma ,\mu ) \) be an abelian complete measurable group with \( \mu(G) \) < \( +\infty \) and let E be a Banach space. For any \( f:G \rightarrow E \) we define the quadratic difference operator Qf by

\( Qf(x,y):= 2f(x) + 2f(y) - f(x + y) - f(x - y),\, x, y \in G. \)

We will prove that if \( Qf \in L_{\mu \times \mu }^{p}(G \times G, E) \) for a certain \( 1 \leq p \leq +\infty \) , then there exists exactly one quadratic function \( K:G \rightarrow E \) , i.e. K satisfying the functional equation

\( 2f(x) + 2f(y) = f(x + y) + f(x - y),\, x, y \in G \)

and a constant C such that

\( \Vert f - K \Vert_{p} \leq C \Vert Qf \Vert_{p}. \)

Moreover, the operator \( Qf : L^{p}(G,E) \rightarrow L^{p}(G \times G, E) \) is linear, continuous and invertible.

Manuscript received: July 19, 1999 and, in final form, May 8, 2003.