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The anisotropic fractional isoperimetric problem with respect to unconditional unit balls
The boundedness of multi-linear and multi-parameter pseudo-differential operators
1. | School of Science, Xi'an University of Posts and Telecommunications, Xi'an, Shanxi 710121, China |
2. | School of Mathematical Sciences, Chongqing Normal University, Chongqing 400000, China |
In this paper, we establish the boundedness on $ L^r(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}) $ of bilinear and bi-parameter pseudo-differential operators whose symbols $ \sigma(x,\xi,\eta)\in S^{(0,0)}_{(1,1),(\delta_1,\delta_2)} $ for $ x,\xi,\eta\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2} $ and $ 0\leq\delta_1,\delta_2<1 $, which extends the result of Dai and Lu in [
References:
[1] |
Á. Bényi, D. Maldonado, V. Naibo and R. H. Torres,
On the Hörmander classes of bilinear pseudodifferential operators, Integral Equ. Oper. Theory, 67 (2010), 341-264.
doi: 10.1007/s00020-010-1782-y. |
[2] |
Á. Bényi and R. H. Torres,
Symbolic calculus and the transposes of bilinear pseudodifferential operators, Commun. Partial Differ. Equ., 28 (2003), 1161-1181.
doi: 10.1081/PDE-120021190. |
[3] |
F. Bernicot,
Local estimates and global continuities in Lebesgue spaces for bilinear operators, Anal. PDE, 1 (2008), 1-27.
doi: 10.2140/apde.2008.1.1. |
[4] |
J. Chen and G. Lu,
Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal., 101 (2014), 98-112.
doi: 10.1016/j.na.2014.01.005. |
[5] |
J. Chen and G. Lu,
Hömander type theorem on Bi-parameter Hardy spaces for Fourier multipliers with optimal smoothness, Rev. Mat. Iberoam., 34 (2018), 1541-1561.
doi: 10.4171/rmi/1035. |
[6] |
M. Christ and J. L. Journé,
Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80.
doi: 10.1007/BF02392554. |
[7] |
R. Coifman and Y. Meyer,
On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.
doi: 10.2307/1998628. |
[8] |
W. Dai and G. Lu,
$L^p$ estimates for multi-linear and multi-parameter pseudo-differential operators, Bull. Soc. Math. France., 143 (2013), 567-597.
doi: 10.24033/bsmf.2698. |
[9] |
W. Ding, G. Lu and Y. Zhu,
Multi-parameter local Hardy spaces, Nonlinear. Anal., 184 (2019), 352-380.
doi: 10.1016/j.na.2019.02.014. |
[10] |
C. Fefferman,
$L^p$ bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413-417.
doi: 10.1007/BF02764718. |
[11] |
C. Fefferman and E. M. Stein,
Some maximal inequalities, Am. J. Math., 93 (1971), 107-115.
doi: 10.2307/2373450. |
[12] |
L. Grafakos and R. H. Torres,
Multilinear Calderón-Zygmund theory, Adv. Math., 165 (2002), 124-164.
doi: 10.1006/aima.2001.2028. |
[13] |
Y. Han, G. Lu and E. Sawyer,
Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE, 7 (2014), 1465-1534.
doi: 10.2140/apde.2014.7.1465. |
[14] |
L. Hörmander,
On the $L^2$ continuity of pseudo-differential operators, Commun. Pure Appl. Math., 24 (1971), 529-535.
doi: 10.1002/cpa.3160240406. |
[15] |
C. Kenig and E. M. Stein,
Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1-15.
doi: 10.4310/MRL.1999.v6.n1.a1. |
[16] |
K. Koezuka and N. Tomita,
Bilinear pseudo-differential operators with symbols in $BS^{m}_{1,1}$ on Triebel-Lizorkin spaces, J. Fourier Anal. Appl., 24 (2018), 309-319.
doi: 10.1007/s00041-016-9518-2. |
[17] |
G. Lu and L. Zhang,
$L^p$ estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J., 66 (2017), 877-900.
doi: 10.1512/iumj.2017.66.6069. |
[18] |
A. Miyachi and N. Tomita,
Estimates for trilinear flag paraproducts on $L^{\infty}$ and Hardy spaces, Math. Z., 282 (2016), 577-613.
doi: 10.1007/s00209-015-1554-0. |
[19] |
C. Muscalu,
Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam, 23 (2007), 705-742.
doi: 10.4171/RMI/510. |
[20] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.
doi: 10.1007/BF02392566. |
[21] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Multi-parameter paraproducts, Rev. Mat. Iberoam, 22 (2006), 963-976.
doi: 10.4171/RMI/480. |
[22] |
C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis II, Cambridge Univ. Press, 2013.
![]() ![]() |
show all references
References:
[1] |
Á. Bényi, D. Maldonado, V. Naibo and R. H. Torres,
On the Hörmander classes of bilinear pseudodifferential operators, Integral Equ. Oper. Theory, 67 (2010), 341-264.
doi: 10.1007/s00020-010-1782-y. |
[2] |
Á. Bényi and R. H. Torres,
Symbolic calculus and the transposes of bilinear pseudodifferential operators, Commun. Partial Differ. Equ., 28 (2003), 1161-1181.
doi: 10.1081/PDE-120021190. |
[3] |
F. Bernicot,
Local estimates and global continuities in Lebesgue spaces for bilinear operators, Anal. PDE, 1 (2008), 1-27.
doi: 10.2140/apde.2008.1.1. |
[4] |
J. Chen and G. Lu,
Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal., 101 (2014), 98-112.
doi: 10.1016/j.na.2014.01.005. |
[5] |
J. Chen and G. Lu,
Hömander type theorem on Bi-parameter Hardy spaces for Fourier multipliers with optimal smoothness, Rev. Mat. Iberoam., 34 (2018), 1541-1561.
doi: 10.4171/rmi/1035. |
[6] |
M. Christ and J. L. Journé,
Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80.
doi: 10.1007/BF02392554. |
[7] |
R. Coifman and Y. Meyer,
On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.
doi: 10.2307/1998628. |
[8] |
W. Dai and G. Lu,
$L^p$ estimates for multi-linear and multi-parameter pseudo-differential operators, Bull. Soc. Math. France., 143 (2013), 567-597.
doi: 10.24033/bsmf.2698. |
[9] |
W. Ding, G. Lu and Y. Zhu,
Multi-parameter local Hardy spaces, Nonlinear. Anal., 184 (2019), 352-380.
doi: 10.1016/j.na.2019.02.014. |
[10] |
C. Fefferman,
$L^p$ bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413-417.
doi: 10.1007/BF02764718. |
[11] |
C. Fefferman and E. M. Stein,
Some maximal inequalities, Am. J. Math., 93 (1971), 107-115.
doi: 10.2307/2373450. |
[12] |
L. Grafakos and R. H. Torres,
Multilinear Calderón-Zygmund theory, Adv. Math., 165 (2002), 124-164.
doi: 10.1006/aima.2001.2028. |
[13] |
Y. Han, G. Lu and E. Sawyer,
Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE, 7 (2014), 1465-1534.
doi: 10.2140/apde.2014.7.1465. |
[14] |
L. Hörmander,
On the $L^2$ continuity of pseudo-differential operators, Commun. Pure Appl. Math., 24 (1971), 529-535.
doi: 10.1002/cpa.3160240406. |
[15] |
C. Kenig and E. M. Stein,
Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1-15.
doi: 10.4310/MRL.1999.v6.n1.a1. |
[16] |
K. Koezuka and N. Tomita,
Bilinear pseudo-differential operators with symbols in $BS^{m}_{1,1}$ on Triebel-Lizorkin spaces, J. Fourier Anal. Appl., 24 (2018), 309-319.
doi: 10.1007/s00041-016-9518-2. |
[17] |
G. Lu and L. Zhang,
$L^p$ estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J., 66 (2017), 877-900.
doi: 10.1512/iumj.2017.66.6069. |
[18] |
A. Miyachi and N. Tomita,
Estimates for trilinear flag paraproducts on $L^{\infty}$ and Hardy spaces, Math. Z., 282 (2016), 577-613.
doi: 10.1007/s00209-015-1554-0. |
[19] |
C. Muscalu,
Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam, 23 (2007), 705-742.
doi: 10.4171/RMI/510. |
[20] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.
doi: 10.1007/BF02392566. |
[21] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele,
Multi-parameter paraproducts, Rev. Mat. Iberoam, 22 (2006), 963-976.
doi: 10.4171/RMI/480. |
[22] |
C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis II, Cambridge Univ. Press, 2013.
![]() ![]() |
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