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We prove uniform uniform $L^{p}$ bounds for the family of bilinear Hilberttransforms $\mathrm{BHT}_{\beta} [f_1, f_2] (x) := \mathrm{p.v.}\int_{\mathbb{R}} f_1 (x - t) f_2 (x + \beta t) \frac{\mathrm{d} t}{t}$. Weshow that the operator $\mathrm{BHT}_{\beta}$ 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 $\beta\in(0,1]$. This is the full openrange of exponents where the modulation invariant class of bilinear operatorscontaining $\mathrm{BHT}_{\beta}$ can be bounded uniformly. This is done byproving boundedness of certain affine transformations of thefrequency-time-scale space $\mathbb{R}^{3}_{+}$ in terms of iterated outerLebesgue spaces. This results in new linear and bilinear wave packet embeddingbounds well suited to study uniform bounds.

The full range of uniform bounds for the bilinear Hilbert transform