Absolute Boundedness

Abstract

It is natural to ask “when is a kernel bounded?” Everyone does, and, sooner or later, everyone gives up. There is no good answer, and, in a sense, there cannot be one. Intuition seems to suggest that boundedness is a question of size: to be bounded is to be “small”, or in any event not too large, and every kernel that is smaller than a bounded one is itself bounded. Since kernels are complex-valued functions, “size” presumably refers to absolute value. These vague and heuristic comments lead to at least one specific and precise question: is it true that if k and k’ are kernels, k’ is bounded, and |k(x, y)|≦|k’(x, y)| almost everywhere, then k is bounded?