The Laplace convolution theorem states that $$\mathcal{L}f\cdot\mathcal{L}g = \mathcal{L}(f*g),$$ where $f*g := \int_0^tf(\tau)g(t-\tau)\mathrm{d}\tau$.
My question is: Is there a function $K$ that satisfies $$\mathcal{M}f\cdot\mathcal{M}g = \mathcal{M}\left(\int_{0}^{t}K(t,\tau)\mathrm{d}\tau\right)?$$