2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76319We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n<d$.38 pages, no figures, submittedClassical Analysis and ODEs42B10text