Beltrami Caccioppoli translation and Fourier-tail estimates

Choose $\chi \in C_c^{\mathrm{\infty }}(D_3)$ equal to $1$ on a neighborhood of ${\bar{D}}_2$. The function $\chi f$, extended by zero outside $D_3$, belongs to $W^{1,r}(\mathbb{C})$: for each weak derivative, testing against a global compactly supported smooth function and using the local product rule gives the zero extension of $\chi \mathrm{\nabla }f+f\mathrm{\nabla }\chi$, with no boundary term because $\chi$ vanishes near $\mathrm{\partial }{D}_3$. By [F2], choose global $g_n\in C_c^{\mathrm{\infty }}(\mathbb{C})$ converging to this extension in $W^{1,r}(\mathbb{C})$. On a neighborhood of $\mathrm{supp}\varphi$, $g_n\to f$ in $W^{1,r}$, so the ordinary smooth product rule for $\varphi g_n$ passes to the limit and gives $F=\varphi f\in W^{1,r}(\mathbb{C})$ with $\bar{\mathrm{\partial }}F=\varphi \bar{\mathrm{\partial }}f+f\bar{\mathrm{\partial }}\varphi$ and $\mathrm{\partial }F=\varphi \mathrm{\partial }f+f\mathrm{\partial }\varphi$. Since $r>2$ and $F$ has bounded support, $F\in H^1(\mathbb{C})$. In Fourier variables the two Wirtinger multipliers give $\mathrm{\partial }F=\mathcal{B}\bar{\mathrm{\partial }}F$: this holds first in $L^2$ and then in $L^r$ by the compatible Beurling extension. Setting $G=\bar{\mathrm{\partial }}F$ and $H=f\bar{\mathrm{\partial }}\varphi -\mu f\mathrm{\partial }\varphi -\nu \bar{f}\overline{\mathrm{\partial }\varphi }$, the Beltrami equation becomes $(I-\mu \mathcal{B}-\nu \bar{\mathcal{B}})G=H$. Thus [F1] gives $\mathrm{\parallel }G{\mathrm{\parallel }}_r\le C_{\kappa ,r}\mathrm{\parallel }H{\mathrm{\parallel }}_r\le C_{\kappa ,r}\mathrm{\parallel }f\mathrm{\nabla }\varphi {\mathrm{\parallel }}_r$ and $\mathrm{\parallel }\mathrm{\partial }F{\mathrm{\parallel }}_r\le \mathrm{\parallel }\mathcal{B}{\mathrm{\parallel }}_{r\to r}\mathrm{\parallel }G{\mathrm{\parallel }}_r$. This is the first assertion.

Write ${\tau }_{h}g=g(\cdot -h)$. Translating the equation in step 1.1, subtracting it, and using translation commutation gives $(I-\mu \mathcal{B}-\nu \bar{\mathcal{B}})({\tau }_{h}G-G)=({\tau }_h\mu -\mu )\mathcal{B}{\tau }_{h}G+({\tau }_h\nu -\nu )\overline{\mathcal{B}{\tau }_{h}G}+{\tau }_{h}H-H$. Inverse boundedness on $L^2$ and Hölder therefore yield $\mathrm{\parallel }{\tau }_{h}G-G{\mathrm{\parallel }}_2\le C_{\kappa }[(\mathrm{\parallel }{\tau }_h\mu -\mu {\mathrm{\parallel }}_q+\mathrm{\parallel }{\tau }_h\nu -\nu {\mathrm{\parallel }}_q)\mathrm{\parallel }\mathcal{B}G{\mathrm{\parallel }}_r+\mathrm{\parallel }{\tau }_{h}H-H{\mathrm{\parallel }}_2]$. All terms are in $L^2$: the differentiated cutoff has compact support, $G\in L^r$, and the coefficient translation differences are in $L^q$ because the coefficients are bounded with fixed bounded support.

For any smooth cutoff derivative $\psi$ supported in $D_2$, extend $f\psi$ by zero. It belongs to $H^1(\mathbb{C})$ and $\mathrm{\parallel }f\psi {\mathrm{\parallel }}_{H^1}\le C_{r,\psi }U$ by Hölder on the bounded support. The fundamental theorem along line segments gives $\mathrm{\parallel }{\tau }_h(f\psi )-f\psi {\mathrm{\parallel }}_2\le \mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }\mathrm{\nabla }(f\psi ){\mathrm{\parallel }}_2$ for smooth inputs; smooth approximation in [F2] gives the same inequality for this Sobolev function. Expand each coefficient product in $H$ by ${\tau }_h(\mu f\psi )-\mu f\psi =({\tau }_h\mu -\mu ){\tau }_h(f\psi )+\mu ({\tau }_h(f\psi )-f\psi )$, and the analogous expression with $\nu \bar{f}$. Hölder gives $\mathrm{\parallel }{\tau }_{h}H-H{\mathrm{\parallel }}_2\le CU(\mathrm{\parallel }{\tau }_h\mu -\mu {\mathrm{\parallel }}_q+\mathrm{\parallel }{\tau }_h\nu -\nu {\mathrm{\parallel }}_q+\mathrm{\mid }h\mathrm{\mid })$. Step 1.1 also gives $\mathrm{\parallel }\mathcal{B}G{\mathrm{\parallel }}_r\le CU$. Substitution in step 2.1, followed by the supremum over $\mathrm{\mid }h\mathrm{\mid }\le t$, proves the desired bound for $G$. Finally $\varphi \bar{\mathrm{\partial }}f=G-f\bar{\mathrm{\partial }}\varphi$, and the just-proved translation bound for $f\psi$ yields the second bound.

Average $\mathrm{\parallel }{\tau }_{h}G-G{\mathrm{\parallel }}_2^2$ over the square $[-2\mathrm{/}T,2\mathrm{/}T{]}^2$ with normalized Lebesgue measure. Plancherel identifies the resulting multiplier as $2[1-\mathrm{sinc}(2{\xi }_1\mathrm{/}T)\mathrm{sinc}(2{\xi }_2\mathrm{/}T)]$, where $\mathrm{sinc}(x)=\sin (x)\mathrm{/}x$ and its value at zero is $1$. If $\mathrm{\mid }\xi \mathrm{\mid }\ge T$, one coordinate has magnitude at least $T\mathrm{/}\sqrt{2}$, so one sinc factor has absolute value at most $1\mathrm{/}\sqrt{2}$, while the other is at most $1$. The multiplier is therefore at least $2(1-1\mathrm{/}\sqrt{2})>0$ on this frequency set. Every shift in the square has norm at most $2\sqrt{2}\mathrm{/}T<3\mathrm{/}T$. Hence $\mathrm{\parallel }{\mathbf{1}}_{\{\mathrm{\mid }\xi \mathrm{\mid }\ge T\}}\widehat{G}{\mathrm{\parallel }}_2\le C{\omega }_2(G,3\mathrm{/}T)$. Since $3\mathrm{/}T\le 1\mathrm{/}2$, step 3.1 applies and gives the claimed tail bound.