Weighted differentiation and multiplication estimates

Put $A=\{1<\mathrm{\mid }y\mathrm{\mid }<2\}$ and $v_R(y)=R^{-\gamma }v(Ry)$. Applying the chain rule to dilation repeatedly gives ${\mathrm{\partial }}_y^I{v}_R(y)=R^{\mathrm{\mid }I\mathrm{\mid }-\gamma }({\mathrm{\partial }}_x^{I}v)(Ry)$. Changing variables in the difference quotient also gives $[{\mathrm{\partial }}_y^I{v}_R{]}_{\alpha ;A}=R^{\mathrm{\mid }I\mathrm{\mid }+\alpha -\gamma }[{\mathrm{\partial }}_x^{I}v{]}_{\alpha ;A_R}$. Thus the supremum of the unweighted order-$k$ norms of $v_R$ on $A$ equals $E_{\gamma }^{k,\alpha }(v)$, up to the fixed finite sum-versus-maximum convention for top derivatives. Each derivative of ${\mathrm{\partial }}_{x^i}v$ appearing in its order-$(k-1)$ norm is a derivative of $v$ of order at most $k$, with precisely the same powers of $R$ after changing the weight to $\gamma -1$. This proves the annular derivative estimate.

On any coordinate set $B$, the identity $a(p)b(p)-a(q)b(q)=a(p)(b(p)-b(q))+b(q)(a(p)-a(q))$ gives $[ab{]}_{\alpha ;B}\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty };B}[b{]}_{\alpha ;B}+\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty };B}[a{]}_{\alpha ;B}$ by division by $\mathrm{\mid }p-q{\mathrm{\mid }}^{\alpha }$ and taking suprema. This inequality is proved here directly from the seminorm definition. Repeated use of the one-variable product rule on coordinate lines expands each derivative of $fg$ through order $k$ as a finite sum of products of derivatives of $f$ and $g$ whose orders add to that derivative order. The number and coefficients of the terms depend only on $k$ and the dimension.

For the core product estimate, first control lower-order Hölder seminorms on each fixed compact chart subset $K_i$. When $k\ge 1$, cover $K_i$ by finitely many smaller coordinate balls or boundary half-balls whose larger concentric closed balls or half-balls lie inside that chart. On their union $L$, all coordinate derivatives through order $k$ are bounded by $C_{\gamma }\mathrm{\parallel }f{\mathrm{\parallel }}_{C_{\gamma }^{k,\alpha }}$. Indeed, $L\cap K$ is covered by the fixed $K_j$; on each compact overlap the repeated chain and product rules express derivatives in chart $i$ using derivatives in chart $j$ and bounded smooth transition coefficients. On $L\setminus K$, its closure lies in the end with $2r_0\le r\le M$ for some fixed finite $M$, so finitely many bounded-radius annuli give the same control, again with bounded transition coefficients. Continuous transition coefficients are bounded on each of these compact sets: continuity gives local bounds and compactness gives a finite subcover. All constants depend only on the fixed data and weight. A sufficiently close pair of points in $K_i$ lies together in one larger convex ball or half-ball. The chain rule and the mean value theorem along its segment bound increments of each derivative of order less than $k$ by its next derivative supremum times the distance. At boundary points use continuity from the interior. Pairs separated by at least a fixed positive distance are controlled by twice the derivative supremum divided by that distance to the power $\alpha$. Hence every lower-order Hölder seminorm on $K_i$ is at most $C_{\gamma }\mathrm{\parallel }f{\mathrm{\parallel }}_{C_{\gamma }^{k,\alpha }}$. The same reasoning applies to $g$ with weight $\eta$.

The fixed annulus $A$ has paths of length at most $(1+\pi )\mathrm{\mid }p-q\mathrm{\mid }$ joining any $p,q\in A$: move radially from the larger radius to the smaller, then along the shorter spherical arc at that smaller radius. The radial length is at most $\mathrm{\mid }p-q\mathrm{\mid }$; the arc length is at most $\pi \mathrm{\mid }p-q\mathrm{\mid }$, since the arc-to-chord ratio is at most $\pi \mathrm{/}2$ and the chord at the smaller radius is at most $2\mathrm{\mid }p-q\mathrm{\mid }$. Both pieces stay in $A$. The chain rule and the mean value theorem on each constant-speed path piece therefore bound the increment of a $C^1$ function by a fixed constant times its gradient supremum times $\mathrm{\mid }p-q\mathrm{\mid }$. Applied to derivatives of order less than $k$, this bounds their Hölder seminorms by the order-$k$ norm, since $A$ has finite diameter. The top-order seminorms are already part of that norm. Step 1.2 now gives the order-$k$ product estimate on $A$; for $k=0$ its difference inequality applies directly. Use $f_R=R^{-\gamma }f(R\cdot )$ and $g_R=R^{-\eta }g(R\cdot )$ and step 1.1 to obtain $E_{\gamma +\eta }^{k,\alpha }(fg)\le C{E}_{\gamma }^{k,\alpha }(f)E_{\eta }^{k,\alpha }(g)$.

For $k\ge 1$, in a chart $z$ define a linear functional at $p$ by $(df{)}_p({\sum }_i{v}^i{\mathrm{\partial }}_{z^i}{\mathrm{\mid }}_p)={\sum }_i{v}^i{\mathrm{\partial }}_{z^i}(f\circ z^{-1})(z(p))$. This defines an element of $T_p^{\ast }{\mathrm{\Sigma }}_1$ because the coordinate tangent vectors form a basis, including at boundary points. On overlapping chart interiors, the chain rule gives ${\mathrm{\partial }}_{w^j}(f\circ w^{-1})={\sum }_i({\mathrm{\partial }}_{z^i}(f\circ z^{-1}))\circ (z\circ w^{-1}){\mathrm{\partial }}_{w^j}(z^i\circ w^{-1})$. Thus the two coordinate functionals agree by the dual transition law in F8. Continuity of the first derivatives and smooth transition matrices gives the same identity at the boundary. These compatible functionals therefore define a single global covector field $df$ with the asserted coordinate components. In each fixed core chart, every derivative through order $k-1$ of a component of $df$ is a derivative through order $k$ of $f$, and its top-order Hölder seminorm is one already in the order-$k$ core norm. Permuting derivatives is valid under the given $C^k$ regularity. Thus $N_K(df)\le C{N}_K(f)$ with the respective orders understood. Together with step 1.1 this proves the global derivative estimate, with a well-defined output on the core as well as on the end.

The top-order core seminorms are already controlled by $N_K$. Apply the finite derivative expansion and product difference inequality of step 1.2 on each $K_i$, using step 1.3 for the lower-order seminorms. Sum over the fixed finite chart collection to get $N_K(fg)\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{C_{\gamma }^{k,\alpha }}\mathrm{\parallel }g{\mathrm{\parallel }}_{C_{\eta }^{k,\alpha }}$. For $k=0$, the core estimate follows immediately from step 1.2 without the lower-order argument. Products have the required local regularity by the same calculation on compact coordinate balls and half-balls. Combining the core bound with step 2.1 proves the global multiplication estimate.

For $\gamma \le \eta$, the annular expression with weight $\eta$ is the one with weight $\gamma$ multiplied by $R^{\gamma -\eta }\le r_0^{\gamma -\eta }$; the core norm is unchanged. This gives the inclusion estimate with $C=\max \{1,r_0^{\gamma -\eta }\}$. Finite component sums and the scalar multiplication estimate control tensor products and contractions with their fixed numerical coefficients. Apply step 1.1 separately to each end component for tensor component differentiation. This proves all the assertions.