Muscalu, Pipher, Tao and Thiele proved that the standard
bilinear and biparameter Hilbert transform does not satisfy any
${L}^{p}$
estimates. They also raised a question asking if a bilinear and biparameter multiplier
operator defined by
$${T}_{m}\left({f}_{1},{f}_{2}\right)\left(x\right):={\int}_{{\mathbb{R}}^{4}}m\left(\xi ,\eta \right){\widehat{f}}_{1}\left({\xi}_{1},{\eta}_{1}\right){\widehat{f}}_{2}\left({\xi}_{2},{\eta}_{2}\right){e}^{2\pi ix\cdot \left(\left({\xi}_{1},{\eta}_{1}\right)+\left({\xi}_{2},{\eta}_{2}\right)\right)}\phantom{\rule{0.3em}{0ex}}d\xi \phantom{\rule{0.3em}{0ex}}d\eta $$
satisfies any
${L}^{p}$ estimates,
where the symbol
$m$
satisfies
$$\left{\partial}_{\xi}^{\alpha}{\partial}_{\eta}^{\beta}m\left(\xi ,\eta \right)\right\lesssim \frac{1}{dist{\left(\xi ,{\Gamma}_{1}\right)}^{\left\alpha \right}}\cdot \frac{1}{dist{\left(\eta ,{\Gamma}_{2}\right)}^{\left\beta \right}}$$
for sufficiently many multiindices
$\alpha =\left({\alpha}_{1},{\alpha}_{2}\right)$
and
$\beta =\left({\beta}_{1},{\beta}_{2}\right)$,
${\Gamma}_{i}$
($i=1,2$) are
subspaces in
${\mathbb{R}}^{2}$
and
$dim{\Gamma}_{1}=0$,
$dim{\Gamma}_{2}=1$.
Silva partially answered this question and proved that
${T}_{m}$ maps
${L}^{{p}_{1}}\times {L}^{{p}_{2}}\to {L}^{p}$ boundedly
when
$\frac{1}{{p}_{1}}+\frac{1}{{p}_{2}}=\frac{1}{p}$
with
${p}_{1}$,
${p}_{2}>1$,
$\frac{1}{{p}_{1}}+\frac{2}{{p}_{2}}<2$ and
$\frac{1}{{p}_{2}}+\frac{2}{{p}_{1}}<2$.
One notes that the admissible range here for these tuples
$\left({p}_{1},{p}_{2},p\right)$ is a
proper subset of the admissible range of the bilinear Hilbert transform (BHT) derived
by Lacey and Thiele.
We establish the same
${L}^{p}$
estimates as BHT in the full range for the bilinear and
$d$parameter
($d\ge 2$) Hilbert
transforms with arbitrary symbols satisfying appropriate decay assumptions and having
singularity sets
${\Gamma}_{1},\dots ,{\Gamma}_{d}$
with
$dim{\Gamma}_{i}=0$ for
$i=1,\dots ,d1$ and
$dim{\Gamma}_{d}=1$. Moreover, we establish
the same
${L}^{p}$ estimates
as BHT for bilinear and biparameter Fourier multipliers of symbols with
$dim{\Gamma}_{1}=dim{\Gamma}_{2}=1$ and
satisfying some appropriate decay estimates. In particular, our results include the
${L}^{p}$
estimates as BHT in the full range for certain modified bilinear
and biparameter Hilbert transforms of tensorproduct type with
$dim{\Gamma}_{1}=dim{\Gamma}_{2}=1$ but with
a slightly better logarithmic decay than that of the bilinear and biparameter Hilbert
transform
$BHT\otimes BHT$.
