Power decay for oscillatory complex-linear Beltrami phases

Write $K=\mathrm{\mid }v\mathrm{\mid }$, $\nu =c_v{e}^{iv\cdot z}\mu$, and $h_n=(\nu \mathcal{B}{)}^n\nu$. The Neumann series $h_v={\sum }_{n\ge 0}{h}_n$ converges in $L^r$ by [F1]. For $g_N={\sum }_{n=0}^N{h}_n$ and $b_N=h_v-g_N$, one has $\mathrm{\parallel }{b}_N{\mathrm{\parallel }}_r\le C{c}^{N+1}$ and $\mathrm{\parallel }{g}_N{\mathrm{\parallel }}_r\le C$. Also $\mathrm{\parallel }{g}_N{\mathrm{\parallel }}_2\le C$, because $\mathrm{\parallel }\nu \mathcal{B}{\mathrm{\parallel }}_{2\to 2}\le \kappa <1$. All these functions have support in ${\bar{D}}_R$. Define $f_n=c_v^{-(n+1)}{e}^{-i(n+1)v\cdot z}{h}_n$. Then $f_0=\mu$ and $f_n=\mu T_n{f}_{n-1}$, where $T_{n}u=e^{-inv\cdot z}\mathcal{B}(e^{inv\cdot z}u)$. This is a translation-invariant Fourier multiplier, and $\mathrm{\parallel }{T}_n{\mathrm{\parallel }}_{p\to p}=\mathrm{\parallel }\mathcal{B}{\mathrm{\parallel }}_{p\to p}$, since the two modulation factors have modulus one.

Put $\mathrm{\ell }=2q\mathrm{/}(q-2)$, so $1\mathrm{/}2=1\mathrm{/}q+1\mathrm{/}\mathrm{\ell }$, and $M=2+\mathrm{\parallel }\mathcal{B}{\mathrm{\parallel }}_{\mathrm{\ell }\to \mathrm{\ell }}$. Iteration of the product norm gives $\mathrm{\parallel }{f}_n{\mathrm{\parallel }}_{\mathrm{\ell }}\le C_R{\kappa }^{n+1}\mathrm{\parallel }\mathcal{B}{\mathrm{\parallel }}_{\mathrm{\ell }\to \mathrm{\ell }}^n$. The translation product identity and Hölder give ${\omega }_2(f_n,t)\le {\omega }_q(\mu ,t)\mathrm{\parallel }{T}_n{f}_{n-1}{\mathrm{\parallel }}_{\mathrm{\ell }}+\kappa {\omega }_2(T_n{f}_{n-1},t)$. By translation commutation and the $L^2$ norm one, the second term is at most $\kappa {\omega }_2(f_{n-1},t)$. For $n=0$, bounded support gives ${\omega }_2(f_0,t)\le C_R{\omega }_q(\mu ,t)$. Induction therefore yields ${\omega }_2(f_n,t)\le C_{R}A{\kappa }^n{M}^n{t}^{\theta }$ for $0<t\le 1$: the two induction contributions are bounded by $M^{n-1}$ and $\mathrm{\parallel }\mathcal{B}{\mathrm{\parallel }}_{\mathrm{\ell }\to \mathrm{\ell }}^n$, whose sum is at most $M^n$ after increasing the fixed constant.

For any $L^2$ function $f$ and $T>0$, averaging $\mathrm{\parallel }f(\cdot -y)-f{\mathrm{\parallel }}_2^2$ over $[-2\mathrm{/}T,2\mathrm{/}T{]}^2$ and using Plancherel gives the multiplier $2[1-\mathrm{sinc}(2{\xi }_1\mathrm{/}T)\mathrm{sinc}(2{\xi }_2\mathrm{/}T)]$. On $\mathrm{\mid }\xi \mathrm{\mid }\ge T$, one sinc factor has absolute value at most $1\mathrm{/}\sqrt{2}$; the multiplier is bounded below by a positive absolute constant. Hence $\mathrm{\parallel }{\mathbf{1}}_{\{\mathrm{\mid }\xi \mathrm{\mid }\ge T\}}\widehat{f}{\mathrm{\parallel }}_2\le C{\omega }_2(f,3\mathrm{/}T)$. Since ${\widehat{h}}_n$ is the shifted transform of $f_n$ by $(n+1)v$, the disk $\mathrm{\mid }\xi \mathrm{\mid }\le K\mathrm{/}2$ samples ${\widehat{f}}_n$ only at frequencies at least $K\mathrm{/}2$. Step 2.1 therefore gives, when $K\ge 6$, $\mathrm{\parallel }{\widehat{g}}_N{\mathrm{\parallel }}_{L^2(D_{K\mathrm{/}2})}\le CA{M}^{N+1}{K}^{-\theta }$, by summing the geometric upper bounds and discarding the beneficial factors ${\kappa }^n$.

Choose $\chi \in C_c^{\mathrm{\infty }}(\mathbb{C})$ equal to one on $\mathrm{\mid }z\mathrm{\mid }\le 2R+1$, and set $P(z)=\chi (z)\mathrm{/}(\pi z)$ for $z\mathrm{\ne }0$, with arbitrary value at zero. Thus $P$ is compactly supported and belongs to $L^1\cap L^{r\mathrm{\prime }}$ for every $1<r\mathrm{\prime }<2$, although it is singular at zero. On $D_R$, convolution $P\ast h_v$ equals $\mathcal{C}{h}_v$. The latter is holomorphic outside ${\bar{D}}_R$, continuous everywhere and tends to zero at infinity, so [F3] shows that a supremum estimate on $D_R$ controls its global supremum. Let $r^{\mathrm{\prime }}=r\mathrm{/}(r-1)\in (1,2)$, ${\theta }_0=1-2\mathrm{/}r>0$, and $P_{\delta }={\rho }_{\delta }\ast P$ for a fixed smooth nonnegative mollifier. Then $\mathrm{\parallel }P-P_{\delta }{\mathrm{\parallel }}_{r^{\mathrm{\prime }}}\le C{\delta }^{{\theta }_0}$, $\mathrm{\parallel }{P}_{\delta }{\mathrm{\parallel }}_2\le C{\delta }^{-1}$, and $\mathrm{\parallel }\mathrm{\nabla }{P}_{\delta }{\mathrm{\parallel }}_2\le C{\delta }^{-2}$. For the first assertion, split a translation difference of $P$ into disks of radius $3\mathrm{\mid }y\mathrm{\mid }$ at its two singularities and their complement. Hölder integration of $\mathrm{\mid }z{\mathrm{\mid }}^{-r^{\mathrm{\prime }}}$ on the disks gives the power $\mathrm{\mid }y{\mathrm{\mid }}^{2\mathrm{/}{r}^{\mathrm{\prime }}-1}$; off them use the derivative bound $C\mathrm{\mid }y\mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^{-2}$ and integrate its $r^{\mathrm{\prime }}$-power from $\mathrm{\mid }y\mathrm{\mid }$ to the fixed cutoff radius, giving the same power. Integrating this estimate against ${\rho }_{\delta }(y)$ proves the first bound. The other two follow from Young applied to $P\in L^1$ and the scaled $L^2$ norms of ${\rho }_{\delta }$ and its derivative.

The tail $P\ast b_N$ has supremum at most $C\mathrm{\parallel }{b}_N{\mathrm{\parallel }}_r\le C{c}^{N+1}$, and $(P-P_{\delta })\ast g_N$ has supremum at most $C{\delta }^{{\theta }_0}$, by step 4.1 and the $L^r$ bounds of step 1.1. Fourier inversion and Cauchy–Schwarz bound the remaining term by $C\mathrm{\parallel }{\widehat{P}}_{\delta }{\widehat{g}}_N{\mathrm{\parallel }}_1$. Below frequency $K\mathrm{/}2$, steps 3.1–4.1 bound it by $C{\delta }^{-1}A{M}^{N+1}{K}^{-\theta }$. Above that frequency, Plancherel gives $\mathrm{\parallel }{\widehat{P}}_{\delta }{\mathrm{\parallel }}_{L^2(\{\mathrm{\mid }\xi \mathrm{\mid }>K\mathrm{/}2\})}\le C{K}^{-1}\mathrm{\parallel }\mathrm{\nabla }{P}_{\delta }{\mathrm{\parallel }}_2\le C{K}^{-1}{\delta }^{-2}$; combine this with $\mathrm{\parallel }{g}_N{\mathrm{\parallel }}_2\le C$. We have proved $\mathrm{\parallel }{\mathrm{\Psi }}_v-z{\mathrm{\parallel }}_{\mathrm{\infty }}\le C[c^N+{\delta }^{{\theta }_0}+{\delta }^{-1}{M}^{N+1}{K}^{-\theta }+K^{-1}{\delta }^{-2}]$.

Choose $a=\theta \mathrm{/}(4\log M)$, $N=\lfloor a\log K\rfloor$ and $\delta =K^{-\theta \mathrm{/}4}$. For sufficiently large $K$, $c^N\le C{K}^{a\log c}$, $M^{N+1}\le M{K}^{\theta \mathrm{/}4}$, and the four terms in step 5.1 are bounded by $C{K}^{-b}$ with $b=\min \{-a\log c,\theta {\theta }_0\mathrm{/}4,\theta \mathrm{/}2,1\mathrm{/}2\}>0$. Indeed, the third term has exponent at most $-\theta \mathrm{/}2$ and the fourth at most $-1\mathrm{/}2$ because $\theta \le 1$. The remaining bounded interval of $1\le K$ is absorbed by increasing $C$, using the uniform principal deformation bound in [F1]. This proves the result uniformly in all stated parameters and phase directions.