BCAM Course | Bilinear Spherical Maximal Functions

The mini-course will consist on four in-person lectures of 2 hours each (six-hour course) given by Saurabh Shrivastava (IISER Bhopal, India).

Spherical maximal functions have been extensively studied in the literature for their importance in harmonic analysis and PDEs. The bilinear (or multilinear) analogues of spherical maximal functions attracted a lot of interest in the recent past due to their analogy to the classical linear counterpart and recent developments in the multilinear theory. The study of Lp-estimates for bilinear spherical means and various maximal functions associated with them is an active research area in the multilinear theory. Many new ideas have emerged to understand the issue of Lp-boundedness of these objects. We plan to discuss the following aspects of bilinear spherical means.

• Lp-estimates for the full bilinear spherical maximal function 
• Lp-estimates for the lacunary bilinear spherical maximal function: Dimension one case requires trilinear smoothing estimates
• Lp-improving properties of bilinear spherical means
• Variants of bilinear spherical maximal functions: There are many interesting varinats of the classical bilinear spherical maximal functions. We shall discuss two of the important variants in these lectures.

The lectures will be based on the following references.

(1) Greenleaf, A.; Iosevich, A.; Krause, B.; Liu, A., Lp-estimates for bilinear generalized Radon transforms in the plane. Combinatorial and additive number theory V, 179-198, Springer Proc. Math. Stat., 395.

(2) Bhojak, Ankit; Choudhary, Surjeet Singh; Shrivastava, Saurabh; Under preparation article.

(3) Bhojak, Ankit; Choudhary, Surjeet Singh; Shrivastava, Saurabh; Shuin, Kalachand, Sharp endpoint Lp-estimates for Bilinear spherical maximal functions. arXiv:2310.00425

(4) Christ, Michael; Zhou, Zirui, A class of singular bilinear maximal functions. J.Funct. Anal. 287 (2024), no. 8, Paper No. 110572, 37 pp.

(5) Dosidis, Georgios, Multilinear spherical maximal function. Proc. Amer. Math. Soc.149 (2021), no. 4, 1471{1480.

(6) Jeong, E., and Lee, S. Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators. J. Funct. Anal. 279 (2020), no. 7, 108629, 29 pp.

(7) Roncal, Luz; Shrivastava, Saurabh; Shuin, Kalachand, Bilinear spherical maximal functions of product type. J. Fourier Anal. Appl. 27 (2021), no. 4, Paper No. 73,

42 pp.

(8) Shrivastava, Saurabh; Shuin, Kalachand, Lp-estimates for multilinear convolution operators de ned with spherical measure. Bull. Lond. Math. Soc. 53 (2021), no.

4, 1045-1060.