论文标题
迭代的奇异积分均质化，并应用于随机准文化图
Homogenization of iterated singular integrals with applications to random quasiconformal maps
论文作者
论文摘要
我们使用随机的Beltrami系数研究了Beltrami微分方程的迭代随机奇异积分和同构溶液的同质化。更确切地说，让我们 $（f_j）_ {j \ geq 1} $是平面beltrami方程的标准化同质形态解决方案$ \ overline {\ partial} f_j（z）=μ_j（z，z，ω，ω）\ partial f_j（z）\ pottial f_j（z），$ lanagial notial notial unaties $ | $μ_j（z，ω）类型的示例= ϕ（z）\ sum_ {n \ in \ mathbf {z}^2} g（2^j z-z-n，x_-n，x_ {n}（ω）），$ g（z，ω）$ in $ g（z，ω）$在$ z $中迅速衰减，$ z $，$ x_ $ x_ n. c^\ infty_0 $。我们将MAPS $ f_j $的$ j \ to \ infty $ f_j $的$ j \ to \ f_j $ $ j \ to $ j \ to $ j \ to $ jj $建立到确定性的准符号限制$ f_ \ infty $。 该结果是作为我们的主要定理的应用，该应用涉及迭代随机奇异积分的均质化。作为我们定理的特殊情况，令$ t_1，\ ldots，t_ {m} $为翻译和扩张不变的单数积分，上面是$ {\ bf r}^d，$，并考虑$ d $ -D $ -Dimensional版本的$μ_j$，例如，如上所述，如上所述或更常规的设置。然后，我们证明存在确定性函数$ f $，因此几乎可以肯定地像$ j \ to \ infty $，$$μ_jt_ {m}μ_j\ ldotst_1μ_Jj\ to f \ quad \ quad \ quad \ quad \ textrm {弱{in} l^p，p，quad 1 <quad 1 <p <p <p < $$
We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=μ_j(z,ω) \partial F_j(z),$ where the random dilatation satisfies $|μ_j|\leq k<1$ and has locally periodic statistics, for example of the type $$μ_j (z,ω)=ϕ(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(ω)), $$ where $g(z,ω)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d., and $ϕ\in C^\infty_0$. We establish the almost sure and local uniform convergence as $j\to\infty$ of the maps $F_j$ to a deterministic quasiconformal limit $F_\infty$. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let $T_1,\ldots , T_{m}$ be translation and dilation invariant singular integrals on ${\bf R}^d, $ and consider a $d$-dimensional version of $μ_j$, e.g., as defined above or within a more general setting. We then prove that there is a deterministic function $f$ such that almost surely as $j\to\infty$, $$ μ_j T_{m}μ_j\ldots T_1μ_j\to f \quad \textrm{weakly in } L^p,\quad 1 < p < \infty\ . $$
