Continuity in the Alexiewicz Norm
نویسنده
چکیده
If f is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of f is ‖f‖ = sup I | ∫ I f | where the supremum is taken over all intervals I ⊂ . Define the translation τx by τxf(y) = f(y − x). Then ‖τxf − f‖ tends to 0 as x tends to 0, i.e., f is continuous in the Alexiewicz norm. For particular functions, ‖τxf − f‖ can tend to 0 arbitrarily slowly. In general, ‖τxf − f‖ > osc f |x| as x → 0, where osc f is the oscillation of f . It is shown that if F is a primitive of f then ‖τxF − F‖ 6 ‖f‖|x|. An example shows that the function y 7→ τxF (y) − F (y) need not be in L . However, if f ∈ L then ‖τxF − F‖1 6 ‖f‖1|x|. For a positive weight function w on the real line, necessary and sufficient conditions on w are given so that ‖(τxf − f)w‖ → 0 as x → 0 whenever fw is Henstock-Kurzweil integrable. Applications are made to the Poisson integral on the disc and half-plane. All of the results also hold with the distributional Denjoy integral, which arises from the completion of the space of Henstock-Kurzweil integrable functions as a subspace of Schwartz distributions.
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