Approximations of the Identity and Maximal Functions
The convolution of two functions f and g that are measurable in Rn is defined by
(f ∗ g)(x) = Rn
f (t)g(x − t) dt, x ∈ Rn,
provided the integral exists. In Theorem 6.14, we saw that if f , g ∈ L1(Rn), then f ∗ g exists a.e. and is
measurable in Rn, and ‖f ∗ g‖1 ≤ ‖f‖1‖g‖1. Moreover, according to Corollary 6.16, ‖ f ∗ g‖1 =‖ f‖1‖g‖1 if f and g are nonnegative and measurable. In this section, we will study some additional properties of convolutions, beginning with the following theorem.