Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data

(Localize coefficients without convexity.) Cover $K$ by finitely many balls whose doubled closed balls lie in $O$. In each doubled ball choose a smooth bump equal to $1$ on the inner ball and zero outside the doubled ball; such a radial bump is constructed from $\eta (r)=e^{-1\mathrm{/}r}$ for $r>0$ and $0$ otherwise, whose derivatives at zero vanish since exponential decay dominates every polynomial. If these bumps are ${\varphi }_l$, put $\chi =1-{\prod }_l(1-{\varphi }_l)$. Then $\chi =1$ on a neighborhood of $K$ and has compact support in $O$. Extend $\chi F$ and $\chi A$ by zero in the value variable outside $O$. They are smooth and equal to the original coefficients near $K$. On $I_0$ times the torus times the compact convex hull of any fixed compact neighborhood of $K$, all their required derivatives are bounded. This permits line-segment arguments even when $O$ is nonconvex.

(Composition (a).) For a smooth $u$, the order-$j$ chain expansion consists of bounded coefficient derivatives times products ${\prod }_l{D}^{{\beta }_l}u$ with total order at most $j\le s$. At most one factor has order at least $s-1$, since $2(s-1)>s$; all others have order at most $s-2$ and are bounded in $L^{\mathrm{\infty }}$ by [L1]. The possible exceptional factor has $L^2$ norm bounded by $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$; if there is no exceptional factor, use finite volume. Thus each term is at most $C_R(1+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s})$, proving the asserted bound. For a general Sobolev field, Fourier truncations have bounded $H^s$ norms and converge uniformly to $u$, with values eventually in a fixed compact neighborhood of $K$. Apply the smooth bound to the extension of step 1.1. Fourier coefficients of the compositions converge by uniform convergence; finite weighted Fourier sums give the same $H^s$ bound for the limit. This also identifies the weak derivative expansion by testing the smooth approximations.

(Differences (b).) With the extended coefficient and $u_{\theta }=v+\theta (u-v)$, the pointwise identity $F(u)-F(v)={\int }_0^1{F}_u(u_{\theta })(u-v)d\theta$ holds. The maps $u_{\theta }$ have $H^s$ norm at most $R$ and take values in the compact convex hull of $K$. Step 2.1 applied to the extended derivative bounds $F_u(u_{\theta })$ in $H^{s-1}$ uniformly. The algebra estimate [L2] gives the $H^{s-1}$ difference bound after integration. At order zero bounded $F_u$ gives the pointwise Lipschitz estimate and hence its $L^2$ version.

(Homogeneous commutators (c).) Expand $[D^{\alpha },A(u)]w={\sum }_{0<\gamma \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\gamma }\right)D^{\gamma }(A(u))D^{\alpha -\gamma }w$. For $\mathrm{\mid }\gamma \mathrm{\mid }\le s-2$, step 2.1 and [L1] bound the first factor in $L^{\mathrm{\infty }}$; the second has at most $j-1$ derivatives. For $j\le s-1$ the only remaining case is $j=\mathrm{\mid }\gamma \mathrm{\mid }=s-1$, where the first factor is $L^2$ bounded and $w$ is $L^{\mathrm{\infty }}$ bounded by its $H^{s-2}$ norm. For $j=s$ and $\mathrm{\mid }\gamma \mathrm{\mid }\ge s-1$, at most one derivative falls on $w$, which is $L^{\mathrm{\infty }}$ bounded by its $H^{s-1}$ norm. These cases give precisely (c), without an additive constant. Density gives the distributional identity and bounds for general $w$.

(Multipliers (d).) In the Leibniz expansion at order $j\le s-1$, coefficient derivatives at most $s-2$ are $L^{\mathrm{\infty }}$ bounded. The only remaining case is a coefficient derivative of order $s-1$, with no derivative on $w$ and $j=s-1$; its $L^2$ norm is bounded and $w$ has the required $L^{\mathrm{\infty }}$ bound. Summation gives (d). Smooth approximation proves the claimed product membership for general Sobolev $w$.

(Lower-norm-controlled high estimates (e).) With $\mathrm{\parallel }z{\mathrm{\parallel }}_{H^{n-1}}\le \rho$, derivatives of $z$ of order at most $n-3$ are $L^{\mathrm{\infty }}$ bounded. In the order-$n$ chain expansion at most one factor has order at least $n-2$, since $2(n-2)>n$. Its $L^2$ norm is bounded by $\rho$ unless its order is $n$, in which case it is bounded by $\mathrm{\parallel }z{\mathrm{\parallel }}_{H^n}$. This proves the high composition bound. In a commutator with $\mathrm{\mid }\alpha \mathrm{\mid }\le n$, a coefficient derivative of order at most $n-3$ is bounded in $L^{\mathrm{\infty }}$, and the remaining derivative of $w$ has order at most $n-1$. A coefficient derivative of order at least $n-2$ has $L^2$ bound $C_{\rho }(1+\mathrm{\parallel }z{\mathrm{\parallel }}_{H^n})$, and at most two derivatives land on $w$; their $L^{\mathrm{\infty }}$ bound follows from $w\in H^{n-2}$, since $n-4\ge 2$. Sum the finite expansion. Density extends the bound as in step 2.1. All smooth derived coefficients are covered by the same argument. This proves (e).