A Nonexistence Theorem for Translation Invariant Operators on Weighted L sub p Spaces

摘要

Translation invariant operators, frequently used in approximation and interpolation theory or in the study of partial differential operators, are considered. On the Lebesgue space, L sub P, translation invariant operators can be represented as convolutions with a tempered distribution. After Fourier transformation, the translation invariant operator appears disguised as a Fourier multiplier. The L sub P theory for translation invariant operators can be extended, at least partly, to weighted L sub P spaces, denoted L sub P (w), for certain weight functions, w, closely connected to positive polynomials. Weight functions, w, with a bad behavior of infinity are treated. As an example, w (x) = exp (+ or - (the absolute value of x) to the alpha-th power), alpha 1, is mentioned. It is proven that there are no translation invariant operators T on L sub P (w) other than the obvious one Tf = cf (c = a constant).