On Fourier exchange theorems for temperate distributions

Abstract The Fourier exchange theorem according to Y. Hirata and H. Ogata says that if S , T are two temperate distributions that are $$\mathcal {S}'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:math> -convolvable, then their Fourier transforms are multiplicable and fulfill $$\mathcal {F}(S*T)=\mathcal {F}S\cdot \mathcal {F} T.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mo>·</mml:mo> <mml:mi>F</mml:mi> <mml:mi>T</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> This implication is sharpened to an equivalence by modifying the concepts of convolvability and multiplicability.