Weak-type endpoint bounds for Bochner–Riesz means for the Hermite operator

Chen, Peng; Li, Ji; Ward, Lesley; Yan, Lixin

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 $H=-\Delta +{\left|x\right|}^2$ in ${ℝ}^n,n\ge 2$ and for other related operators, for $1\le p\le 2n/(n+2)$, extending earlier results of Thangavelu and of Karadzhov.

Classification : 42B15, 42B08, 42C10

[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