Decoupling inequalities for quadratic forms

We prove sharp ${\mathrm{\ell }}^q{L}^p$ decoupling inequalities for $p,q\in [2,\mathrm{\infty })$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp–Lieb inequalities.