The optimal constants of the mixed $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$-Littlewood inequality

In this note, among other results, we find the optimal constants of the generalized Bohnenblust–Hille inequality for $m$-linear forms over $\mathbb{R}$ and with multiple exponents $(1,2,\mathrm{.}\mathrm{.}\mathrm{.},2)$, sometimes called mixed $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$-Littlewood inequality. We show that these optimal constants are precisely ${\left(\sqrt{2}\right)}^{m-1}$ and this is somewhat surprising since a series of recent papers have shown that similar constants have a sublinear growth. This result answers a question raised by Albuquerque $\mathit{\text{et}}\mathit{\text{al.}}$ in a paper published in 2014 in the $\mathit{\text{Journal}}\mathit{\text{of}}\mathit{\text{Functional}}\mathit{\text{Analysis}}$.