Planar Cauchy and Beurling transforms and near-two Beltrami inversion

For $f\in C_c^{\mathrm{\infty }}$, [F1] shows that $Sf:=\mathrm{\partial }\mathcal{C}f$ is bounded locally and decays as $O(\mathrm{\mid }z{\mathrm{\mid }}^{-2})$, hence is in $L^2$. Distributional commutation gives $\bar{\mathrm{\partial }}Sf=\mathrm{\partial }f$. Fourier transformation gives $(i{\xi }_1-{\xi }_2)\widehat{Sf}=(i{\xi }_1+{\xi }_2)\widehat{f}$, so almost everywhere off the origin $\widehat{Sf}=(({\xi }_1-i{\xi }_2)\mathrm{/}({\xi }_1+i{\xi }_2))\widehat{f}$. The multiplier has modulus one. [F2] extends $S$ uniquely to the isometry $\mathcal{B}$ on $L^2$. Multiplication by the translation character commutes with this multiplier, giving translation commutation there.

The kernel $K(z)=-1\mathrm{/}(\pi z^2)$ has annular integral $2\log 2$. Its gradient is bounded by $C\mathrm{\mid }z{\mathrm{\mid }}^{-3}$, so for $\mathrm{\mid }z\mathrm{\mid }\ge 2\mathrm{\mid }h\mathrm{\mid }$, the segment estimate gives $\mathrm{\mid }K(z-h)-K(z)\mathrm{\mid }\le C\mathrm{\mid }h\mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^{-3}$. Polar integration from $2\mathrm{\mid }h\mathrm{\mid }$ to infinity bounds its integral by an absolute constant. The off-support representation for compactly supported $L^2$ inputs follows by approximating them by smooth inputs in one fixed compact neighborhood of their support: on any compact set at positive distance from that neighborhood the kernel integrals converge uniformly by Cauchy–Schwarz, while the operator outputs converge in $L^2$ by step 1.1. Exhausting the complement of the essential support by compact sets and shrinking the neighborhoods gives the representation almost everywhere off the essential support. Thus [F3] applies. Let $A=\max \{1,\mathrm{\parallel }\mathcal{B}{\mathrm{\parallel }}_{4\to 4}\}$. Interpolating the norm-one $L^2$ bound and the $L^4$ bound gives $\mathrm{\parallel }\mathcal{B}{\mathrm{\parallel }}_{p\to p}\le A^{\theta }$ with $1\mathrm{/}p=(1-\theta )\mathrm{/}2+\theta \mathrm{/}4$, that is $\theta =2-4\mathrm{/}p$. Density makes all extensions compatible and carries translation commutation to every finite $p$.

Let $p>2$, let $q=p\mathrm{/}(p-1)<2$, and let $f$ have support in $D_R$. Hölder gives absolute convergence of $\mathcal{C}f$ at every $z$: the singularity $\mathrm{\mid }z-w{\mathrm{\mid }}^{-q}$ is locally integrable, and the support is bounded. For $\delta =\mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }>0$, split its kernel difference into the union of the two disks of radius $2\delta$ about $z,z^{\mathrm{\prime }}$ and its complement. On the first region, integrating the two singularities with Hölder bounds their contributions by $C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p{\delta }^{1-2\mathrm{/}p}$. On the complement, $\mathrm{\mid }(z-w{)}^{-1}-(z^{\mathrm{\prime }}-w{)}^{-1}\mathrm{\mid }\le C\delta \mathrm{\mid }z-w{\mathrm{\mid }}^{-2}$; its $L^q$ norm is at most $C_p\delta ({\int }_{2\delta }^{\mathrm{\infty }}{r}^{1-2q}dr{)}^{1\mathrm{/}q}=C_p{\delta }^{1-2\mathrm{/}p}$. This proves the global Hölder bound. On $\mathrm{\mid }z\mathrm{\mid }\le 2R$, the same singular-kernel estimate gives $\mathrm{\mid }\mathcal{C}f(z)\mathrm{\mid }\le C_{p,R}\mathrm{\parallel }f{\mathrm{\parallel }}_p$; on $\mathrm{\mid }z\mathrm{\mid }>2R$, the bound is $C\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{/}\mathrm{\mid }z\mathrm{\mid }\le C_R\mathrm{\parallel }f{\mathrm{\parallel }}_p\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$. The latter is $L^p$ because $p>2$, proving $\mathrm{\parallel }\mathcal{C}f{\mathrm{\parallel }}_p\le C_{p,R}\mathrm{\parallel }f{\mathrm{\parallel }}_p$ and the stated decay. Choose a fixed $\chi \in C_c^{\mathrm{\infty }}(D_{R+1})$ equal to $1$ on ${\bar{D}}_R$. By the global density in [F2], take $g_n\in C_c^{\mathrm{\infty }}(\mathbb{C})$ with $g_n\to f$ in $L^p$, and set $f_n=\chi g_n$. Since $\chi f=f$ almost everywhere, $\mathrm{\parallel }{f}_n-f{\mathrm{\parallel }}_p\le \mathrm{\parallel }\chi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{g}_n-f{\mathrm{\parallel }}_p\to 0$, and each $f_n$ is smooth and supported in $D_{R+1}$. Their Cauchy transforms converge in $L^p$, and their two weak derivatives converge to $f$ and $\mathcal{B}f$ by step 2.1. Passing the derivative identities against tests proves the two weak identities and the $W^{1,p}$ bound.

If $\kappa =0$, the final operator is the identity. Otherwise choose $2<p_{\kappa }\le 4$ so close to $2$ that $\kappa A^{2-4\mathrm{/}{p}_{\kappa }}\le (1+\kappa )\mathrm{/}2<1$; this is possible by continuity and the value $1$ of the bound at $p=2$. For $2\le p\le p_{\kappa }$, the operator $Tf=\mu \mathcal{B}f+\nu \overline{\mathcal{B}f}$ has pointwise modulus at most $(\mathrm{\mid }\mu \mathrm{\mid }+\mathrm{\mid }\nu \mathrm{\mid })\mathrm{\mid }\mathcal{B}f\mathrm{\mid }$, and therefore norm at most $(1+\kappa )\mathrm{/}2$. Regard $L^p(\mathbb{C};\mathbb{C})$ as a real Banach space. The norm-convergent series ${\sum }_{j=0}^{\mathrm{\infty }}{T}^j$ is a two-sided inverse to $I-T$, since the finite geometric-series remainder has norm tending to zero. Its norm is at most $1\mathrm{/}(1-(1+\kappa )\mathrm{/}2)=2\mathrm{/}(1-\kappa )$, proving the assertion.