Convolutions with the Continuous Primitive Integral

If ð ¹ is a continuous function on the real line and ð ‘“ = ð ¹ î…ž is its distributional derivative, then the continuous primitive integral of distribution ð ‘“ is ∫ ð ‘ ð ‘Ž ð ‘“ = ð ¹ ( ð ‘ ) − ð ¹ ( ð ‘Ž ) . This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolution ∫ ð ‘“ ∗ ð ‘” ( ð ‘¥ ) = ∞ − ∞ ð ‘“ ( ð ‘¥ − 𠑦 ) ð ‘” ( 𠑦 ) ð ‘‘ 𠑦 for ð ‘“ an integrable distribution and ð ‘” a function of bounded variation or an ð ¿ 1 function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For ð ‘” of bounded variation, ð ‘“ ∗ ð ‘” is uniformly continuous and we have the estimate ‖ ð ‘“ ∗ ð ‘” ‖ ∞ ≤ ‖ ð ‘“ ‖ ‖ ð ‘” ‖ ℬ ð ’± , where ‖ ð ‘“ ‖ = s u p ð ¼ | ∫ ð ¼ ð ‘“ | is the Alexiewicz norm. This supremum is taken over all intervals ð ¼ âŠ‚ â„ . When ð ‘” ∈ ð ¿ 1 , the estimate is ‖ ð ‘“ ∗ ð ‘” ‖ ≤ ‖ ð ‘“ ‖ ‖ ð ‘” ‖ 1 . There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.