Multilinear Maps and uniform boundedness

Abstract

Motivated by a question that naturally arises concerning certain nonlinear integral operators, we give an extension, to multilinear maps, of the Banach-Steinhaus principle of uniform boundedness for linear operators. Applications are considered, and of particular interest to us are operators <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> that, for some positive integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> , have the representation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(H_{x})(t)=\int_{0}^{t} \cdots \int_{0}^{t} k (t, \tau_{1}, \cdots ,\tau_{p})x(\tau_{1})\cdots x(\tau_{p})d\tau_{1} \cdots d \tau_{p}, t \geq 0</tex> for an arbitrary bounded (Lebesgue measurable) complex-valued function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[0, \infty)</tex> , where the kernel <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> has certain very reasonable integrability properties. We show, using the extension mentioned above, that such operators (which play an important role in the theory of representation of nonlinear systems) have the basic property that whenever they take the set of bounded functions into itself, there is a positive constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\parallel Hx \parallel \leq c\parallel x \parallel^{P}</tex> for all bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\parallel \cdot \parallel</tex> denotes the usual sup norm; this had been proved earlier only for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p = 1</tex> . Related results for much more general cases are also given.