

Recent Issues 
Volume 12, 4 issues
Volume 12
Issue 4, 867–1148
Issue 3, 605–866
Issue 2, 259–604
Issue 1, 1–258
Volume 11, 8 issues
Volume 11
Issue 8, 1841–2148
Issue 7, 1587–1839
Issue 6, 1343–1586
Issue 5, 1083–1342
Issue 4, 813–1081
Issue 3, 555–812
Issue 2, 263–553
Issue 1, 1–261
Volume 10, 8 issues
Volume 10
Issue 8, 1793–2041
Issue 7, 1539–1791
Issue 6, 1285–1538
Issue 5, 1017–1284
Issue 4, 757–1015
Issue 3, 513–756
Issue 2, 253–512
Issue 1, 1–252
Volume 9, 8 issues
Volume 9
Issue 8, 1772–2050
Issue 7, 1523–1772
Issue 6, 1285–1522
Issue 5, 1019–1283
Issue 4, 773–1018
Issue 3, 515–772
Issue 2, 259–514
Issue 1, 1–257
Volume 8, 8 issues
Volume 8
Issue 8, 1807–2055
Issue 7, 1541–1805
Issue 6, 1289–1539
Issue 5, 1025–1288
Issue 4, 765–1023
Issue 3, 513–764
Issue 2, 257–511
Issue 1, 1–255
Volume 7, 8 issues
Volume 7
Issue 8, 1713–2027
Issue 7, 1464–1712
Issue 6, 1237–1464
Issue 5, 1027–1236
Issue 4, 771–1026
Issue 3, 529–770
Issue 2, 267–527
Issue 1, 1–266
Volume 6, 8 issues
Volume 6
Issue 8, 1793–2048
Issue 7, 1535–1791
Issue 6, 1243–1533
Issue 5, 1001–1242
Issue 4, 751–1000
Issue 3, 515–750
Issue 2, 257–514
Issue 1, 1–256
Volume 5, 5 issues
Volume 5
Issue 5, 887–1173
Issue 4, 705–885
Issue 3, 423–703
Issue 2, 219–422
Issue 1, 1–218
Volume 4, 5 issues
Volume 4
Issue 5, 639–795
Issue 4, 499–638
Issue 3, 369–497
Issue 2, 191–367
Issue 1, 1–190
Volume 3, 4 issues
Volume 3
Issue 4, 359–489
Issue 3, 227–358
Issue 2, 109–225
Issue 1, 1–108
Volume 2, 3 issues
Volume 2
Issue 3, 261–366
Issue 2, 119–259
Issue 1, 1–81
Volume 1, 3 issues
Volume 1
Issue 3, 267–379
Issue 2, 127–266
Issue 1, 1–126





Abstract

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
$BHT\otimes BHT$.

Keywords
bilinear and multiparameter Hilbert transforms, $L^p$
estimates, Hölder estimates, polydiscs, multiparameter
paraproducts, wave packets

Mathematical Subject Classification 2010
Primary: 42B15, 42B20

Milestones
Received: 3 May 2014
Accepted: 22 January 2015
Published: 3 June 2015

