# Bilinear wave estimates

### Bilinear estimates

• Let ${\displaystyle d>1}$. If ${\displaystyle \phi }$, ${\displaystyle \psi }$ are free ${\displaystyle {\dot {H}}^{s_{1}}}$ and ${\displaystyle {\dot {H}}^{s_{2}}}$ solutions respectively, then one can control ${\displaystyle \phi \psi }$ in ${\displaystyle {\dot {X}}^{s,b}}$ if and only if
• (Scaling) ${\displaystyle s+b=s_{1}+s_{2}-(d-1)/2}$
• (Parallel interactions) ${\displaystyle b\geq (3-d)/4}$
• (Lack of smoothing) ${\displaystyle s\leq s_{1},s_{2}}$
• (Frequency cancellation) ${\displaystyle s_{1}+s_{2}\geq 1/2}$
• (No double endpoints) ${\displaystyle (s_{1},b),(s_{2},b)\neq ((d+1)/4,-(d-3)/4);(s_{1}+s_{2},b)\neq (1/2,-(d-3)/4)}$.

See FcKl2000. Null forms can also be handled by identities such as

${\displaystyle 2Q_{0}(\phi ,\psi )=\Box (\phi ,\psi ).}$
• Some bilinear Strichartz estimates are also known. For instance, if ${\displaystyle s}$, ${\displaystyle q}$, ${\displaystyle r}$ are as in the linear Strichartz estimates ${\displaystyle \phi }$, ${\displaystyle \psi }$ are ${\displaystyle {\dot {H}}^{s-a}}$ solutions, then
${\displaystyle D^{-2a}(\phi \psi )\in L_{t}^{q/2}L_{x}^{r/2}}$

as long as ${\displaystyle 0\leq a\leq d/2-2/q-d/r}$ FcKl2000. Similar estimates for null forms also exist Pl2002; see also TaVa2000b, Ta2001b.