*Isroil A. Ikromov and Detlef Müller*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691170541
- eISBN:
- 9781400881246
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691170541.003.0007
- Subject:
- Mathematics, Geometry / Topology

This chapter mostly considers the domains of type Dsubscript (l), which are in some sense “closest” to the principal root jet, since it turns out that the other domains Dsubscript (l) with l ≥ 2 are ...
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This chapter mostly considers the domains of type Dsubscript (l), which are in some sense “closest” to the principal root jet, since it turns out that the other domains Dsubscript (l) with l ≥ 2 are easier to handle. In a first step, by means of some lower bounds on the r-height, this chapter establishes favorable restriction estimates in most situations, with the exception of certain cases where m = 2 and B = 3 or B = 4. In some cases the chapter applies interpolation arguments in order to capture the endpoint estimates for p = psubscript c. Sometimes this can be achieved by means of a variant of the Fourier restriction theorem. However, in most of these cases the chapter applies complex interpolation in a similar way as has been done in Chapter 5.Less

This chapter mostly considers the domains of type *D*subscript (*l*), which are in some sense “closest” to the principal root jet, since it turns out that the other domains *D*subscript (*l*) with *l* ≥ 2 are easier to handle. In a first step, by means of some lower bounds on the r-height, this chapter establishes favorable restriction estimates in most situations, with the exception of certain cases where *m* = 2 and *B* = 3 or *B* = 4. In some cases the chapter applies interpolation arguments in order to capture the endpoint estimates for *p* = *p*subscript *c*. Sometimes this can be achieved by means of a variant of the Fourier restriction theorem. However, in most of these cases the chapter applies complex interpolation in a similar way as has been done in Chapter 5.

*Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0012
- Subject:
- Mathematics, Numerical Analysis

This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and ...
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This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and associated maximal functions. It focuses on the Littlewood–Paley bounds for two square functions introduced by Stein, who had stressed their importance in harmonic analysis and many important variants and generalizations in various monographs. The chapter proves new endpoint estimates for these square functions, for the maximal Bochner–Riesz operator, and for more general classes of radial Fourier multipliers. The majority of the chapter is devoted to these proofs, such as for convolutions with spherical measures.Less

This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and associated maximal functions. It focuses on the Littlewood–Paley bounds for two square functions introduced by Stein, who had stressed their importance in harmonic analysis and many important variants and generalizations in various monographs. The chapter proves new endpoint estimates for these square functions, for the maximal Bochner–Riesz operator, and for more general classes of radial Fourier multipliers. The majority of the chapter is devoted to these proofs, such as for convolutions with spherical measures.