Analyse harmonique
Weak-type endpoint bounds for Bochner–Riesz means for the Hermite operator
Comptes Rendus. Mathématique, Tome 360 (2022) no. G2, pp. 111-126.

We obtain weak-type (p,p) endpoint bounds for Bochner–Riesz means for the Hermite operator H=-Δ+|x| 2 in n ,n2 and for other related operators, for 1p2n/(n+2), extending earlier results of Thangavelu and of Karadzhov.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.265
Classification : 42B15, 42B08, 42C10
Chen, Peng 1, 2 ; Li, Ji 3 ; Ward, Lesley 2 ; Yan, Lixin 1

1 Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P.R. China
2 School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes SA 5095, Australia
3 Department of Mathematics, Macquarie University, NSW 2109, Australia
@article{CRMATH_2022__360_G2_111_0,
     author = {Chen, Peng and Li, Ji and Ward, Lesley and Yan, Lixin},
     title = {Weak-type endpoint bounds for {Bochner{\textendash}Riesz} means for the {Hermite} operator},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {111--126},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     number = {G2},
     year = {2022},
     doi = {10.5802/crmath.265},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.265/}
}
TY  - JOUR
AU  - Chen, Peng
AU  - Li, Ji
AU  - Ward, Lesley
AU  - Yan, Lixin
TI  - Weak-type endpoint bounds for Bochner–Riesz means for the Hermite operator
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 111
EP  - 126
VL  - 360
IS  - G2
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.265/
DO  - 10.5802/crmath.265
LA  - en
ID  - CRMATH_2022__360_G2_111_0
ER  - 
%0 Journal Article
%A Chen, Peng
%A Li, Ji
%A Ward, Lesley
%A Yan, Lixin
%T Weak-type endpoint bounds for Bochner–Riesz means for the Hermite operator
%J Comptes Rendus. Mathématique
%D 2022
%P 111-126
%V 360
%N G2
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.265/
%R 10.5802/crmath.265
%G en
%F CRMATH_2022__360_G2_111_0
Chen, Peng; Li, Ji; Ward, Lesley; Yan, Lixin. Weak-type endpoint bounds for Bochner–Riesz means for the Hermite operator. Comptes Rendus. Mathématique, Tome 360 (2022) no. G2, pp. 111-126. doi : 10.5802/crmath.265. http://www.numdam.org/articles/10.5802/crmath.265/

[1] Bourgain, Jean; Guth, Larry Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal., Volume 21 (2011) no. 6, pp. 1239-1295 | DOI | MR | Zbl

[2] Carleson, Lennart; Sjölin, Per Oscillatory integrals and a multiplier problem for the disc, Stud. Math., Volume 44 (1972), pp. 287-299 | DOI | MR | Zbl

[3] Cheeger, Jeff; Gromov, Mikhail; Taylor, Michael Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom., Volume 17 (1982) no. 1, pp. 15-53 | MR | Zbl

[4] Chen, Peng; Lee, Sanghyuk; Sikora, Adam; Yan, Lixin Bounds on the maximal Bochner–Riesz means for elliptic operators (2018) (to appear in Transactions of the American Mathematical Society, https://arxiv.org/abs/1803.03369)

[5] Chen, Peng; Ouhabaz, El Maati; Sikora, Adam; Yan, Lixin Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner–Riesz means, J. Anal. Math., Volume 129 (2016), pp. 219-283 | DOI | MR | Zbl

[6] Christ, Michael Weak type endpoint bounds for Bochner–Riesz multipliers, Rev. Mat. Iberoam., Volume 3 (1987) no. 1, pp. 25-31 | DOI | MR | Zbl

[7] Christ, Michael Weak type (1,1) bounds for rough operators, Ann. Math., Volume 128 (1988) no. 1, pp. 19-42 | DOI | MR | Zbl

[8] Christ, Michael; Sogge, Christopher D. The weak type L 1 convergence of eigenfunction expansions for pseudo-differential operators, Invent. Math., Volume 94 (1988) no. 2, pp. 421-453 | DOI | Zbl

[9] Coulhon, Thierry; Sikora, Adam Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem, Proc. Lond. Math. Soc., Volume 96 (2008) no. 2, pp. 507-544 | DOI | Zbl

[10] Duong, Xuan Thinh; Ouhabaz, El Maati; Sikora, Adam Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., Volume 196 (2002) no. 2, pp. 443-485 | DOI | MR | Zbl

[11] Fefferman, Charles Inequalities for strongly singular convolution operators, Acta Math., Volume 124 (1970), pp. 9-36 | DOI | MR | Zbl

[12] Fefferman, Charles A note on spherical summation multipliers, Isr. J. Math., Volume 15 (1973), pp. 44-52 | DOI | MR | Zbl

[13] Grafakos, Loukas Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, 2014 | Zbl

[14] Herz, Carl S. On the mean inversion of Fourier and Hankel transforms, Proc. Natl. Acad. Sci. USA, Volume 40 (1954), pp. 996-999 | DOI | MR | Zbl

[15] Karadzhov, Georgi E. Riesz summability of multiple Hermite series in L p spaces, C. R. Acad. Bulg. Sci., Volume 47 (1994) no. 2, pp. 5-8 | MR | Zbl

[16] Kenig, Carlos E.; Stanton, Robert J.; Tomas, Peter A. Divergence of eigenfunction expansions, J. Funct. Anal., Volume 46 (1982), pp. 28-44 | DOI | MR | Zbl

[17] Koch, Herbert; Tataru, Daniel L p eigenfunction bounds for the Hermite operator, Duke Math. J., Volume 128 (2005) no. 2, pp. 369-392 | MR | Zbl

[18] Lee, Sanghyuk Improved bounds for Bochner-Riesz and maximal Bochner–Riesz operators, Duke Math. J., Volume 122 (2004) no. 1, pp. 205-232 | MR | Zbl

[19] Mitjagin, Boris S. Divergenz von Spektralentwicklungen in L P -Räumen, Linear operators and approximation, II (Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach, 1974) (International Series of Numerical Mathematics), Volume 25, Birkhäuser, 1974, pp. 521-530 | MR | Zbl

[20] Seeger, Andreas Endpoint estimates for multiplier transformations on compact manifolds, Indiana Univ. Math. J., Volume 40 (1991) no. 2, pp. 471-533 | DOI | MR | Zbl

[21] Seeger, Andreas Endpoint inequalities for Bochner–Riesz multipliers in the plane, Pac. J. Math., Volume 174 (1996) no. 2, pp. 543-553 | DOI | MR | Zbl

[22] Sikora, Adam Riesz transform, Gaussian bounds and the method of wave equation, Math. Z., Volume 247 (2004) no. 3, pp. 643-662 | MR | Zbl

[23] Sogge, Christopher D. On the convergence of Riesz means on compact manifolds, Ann. Math., Volume 126 (1987) no. 2, pp. 439-447 | DOI | MR | Zbl

[24] Sogge, Christopher D. Concerning the L p norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal., Volume 77 (1988) no. 1, pp. 123-134 | DOI | MR | Zbl

[25] Stein, Elias M. Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993 (with the assistance of Timothy S. Murphy) | Zbl

[26] Stein, Elias M.; Weiss, Guido Introduction to Fourier Analysis on Euclidean spaces, Princeton Mathematical Series, 32, Princeton University Press, 1971 | Zbl

[27] Tao, Terence Weak-type endpoint bounds for Riesz means, Proc. Am. Math. Soc., Volume 124 (1996) no. 9, pp. 2797-2805 | MR | Zbl

[28] Tao, Terence The weak-type endpoint Bochner–Riesz conjecture and related topics, Indiana Univ. Math. J., Volume 47 (1998) no. 3, pp. 1097-1124 | MR | Zbl

[29] Tao, Terence Some recent progress on the restriction conjecture, Fourier analysis and convexity (Brandolini, Luca et al., eds.) (Applied and Numerical Harmonic Analysis), Birkhäuser, 2004, pp. 217-243 | Zbl

[30] Tao, Terence; Vargas, Ana; Vega, Luis A bilinear approach to the restriction and Kakeya conjectures, J. Am. Math. Soc., Volume 11 (1998) no. 4, pp. 967-1000 | MR | Zbl

[31] Thangavelu, Sundaram Summability of Hermite expansions. I, Trans. Am. Math. Soc., Volume 314 (1989) no. 1, pp. 119-142 | DOI | MR | Zbl

[32] Thangavelu, Sundaram Summability of Hermite expansions. II, Trans. Am. Math. Soc., Volume 314 (1989) no. 1, pp. 143-170 | DOI | MR | Zbl

[33] Thangavelu, Sundaram Lectures on Hermite and Laguerre expansions, Mathematical Notes (Princeton), 42, Princeton University Press, 1993 (with a preface by Robert S. Strichartz) | DOI | Zbl

[34] Thangavelu, Sundaram Hermite and special Hermite expansions revisited, Duke Math. J., Volume 94 (1998) no. 2, pp. 257-278 | MR | Zbl

Cité par Sources :