论文标题
双线性希尔伯特变换的全范围范围
The full range of uniform bounds for the bilinear Hilbert transform
论文作者
论文摘要
我们证明了均匀的均匀$ l^{p} $为双线性希尔伯特家族的界限转换为$ \ mathrm {bht}_β[f_1，f_1，f_2]（x）：= \ mathrm {p.v。} \ frac {\ mathrm {d} t} {t} $。我们表明操作员$ \ mathrm {bht}_β$ maps $ l^{p_ {1}} \ times l^{p_ {2}} $ in $ l^{p} \ frac {2} {3} $，独立于$β\ in（0,1] $。这是全部开放范围的指数范围，其中调制了不变的双线运算符，包含$ \ mathrm {bht}_β$的双线性运算符可以均匀地界定。 $ \ mathbb {r}^{3} _ {+} $在迭代的外部lebesgue空间方面。
We prove uniform uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $\mathrm{BHT}_β [f_1, f_2] (x) := \mathrm{p.v.} \int_{\mathbb{R}} f_1 (x - t) f_2 (x + βt) \frac{\mathrm{d} t}{t}$. We show that the operator $\mathrm{BHT}_β$ maps $L^{p_{1}}\times L^{p_{2}}$ into $L^{p}$ as long as $p_1 \in (1, \infty)$, $p_2 \in (1, \infty)$, and $p > \frac{2}{3}$ with a bound independent of $β\in(0,1]$. This is the full open range of exponents where the modulation invariant class of bilinear operators containing $\mathrm{BHT}_β$ can be bounded uniformly. This is done by proving boundedness of certain affine transformations of the frequency-time-scale space $\mathbb{R}^{3}_{+}$ in terms of iterated outer Lebesgue spaces. This results in new linear and bilinear wave packet embedding bounds well suited to study uniform bounds.
