*Charles Fefferman*

*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/9780691159416.003.0002
- Subject:
- Mathematics, Numerical Analysis

This chapter illustrates the continuing powerful influence of Eli Stein's ideas. It starts by recalling his ideas on Littlewood–Paley theory, as well as several major developments in pure and applied ...
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This chapter illustrates the continuing powerful influence of Eli Stein's ideas. It starts by recalling his ideas on Littlewood–Paley theory, as well as several major developments in pure and applied mathematics, to which those ideas gave rise. Before Eli, Littlewood–Paley theory was one of the deepest parts of the classical study of Fourier series in one variable. Stein, however, found the right viewpoint to develop Littlewood–Paley theory and went on to develop Littlewood–Paley theory on any compact Lie group, and then in any setting in which there is a reasonable heat kernel. Afterward, the chapter discusses the remarkable recent work of Gressman and Strain on the Boltzmann equation, and explains in particular its connection to Stein's work.Less

This chapter illustrates the continuing powerful influence of Eli Stein's ideas. It starts by recalling his ideas on Littlewood–Paley theory, as well as several major developments in pure and applied mathematics, to which those ideas gave rise. Before Eli, Littlewood–Paley theory was one of the deepest parts of the classical study of Fourier series in one variable. Stein, however, found the right viewpoint to develop Littlewood–Paley theory and went on to develop Littlewood–Paley theory on any compact Lie group, and then in any setting in which there is a reasonable heat kernel. Afterward, the chapter discusses the remarkable recent work of Gressman and Strain on the Boltzmann equation, and explains in particular its connection to Stein's work.

*Brian Street*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691162515
- eISBN:
- 9781400852758
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691162515.003.0003
- Subject:
- Mathematics, Analysis

This chapter develops the theory of multi-parameter Carnot–Carathéodory geometry, which is needed to study singular integral operators. In the case when the balls are of product type, all of the ...
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This chapter develops the theory of multi-parameter Carnot–Carathéodory geometry, which is needed to study singular integral operators. In the case when the balls are of product type, all of the results are simple variants of results in the single-parameter theory. When the balls are not of product type, these ideas become more difficult. What saves the day is the quantitative Frobenius theorem given in Chapter 2. This can be used to estimate certain integrals, as well as develop an appropriate maximal function and an appropriate Littlewood–Paley square function, all of which are essential to our study of singular integral operators.Less

This chapter develops the theory of multi-parameter Carnot–Carathéodory geometry, which is needed to study singular integral operators. In the case when the balls are of product type, all of the results are simple variants of results in the single-parameter theory. When the balls are not of product type, these ideas become more difficult. What saves the day is the quantitative Frobenius theorem given in Chapter 2. This can be used to estimate certain integrals, as well as develop an appropriate maximal function and an appropriate Littlewood–Paley square function, all of which are essential to our study of singular integral operators.

*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/9780691170541.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter shows that one may reduce the desired Fourier restriction estimate to a piece Ssubscript Greek small letter psi of the surface S lying above a small, “horn-shaped” neighborhood ...
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This chapter shows that one may reduce the desired Fourier restriction estimate to a piece Ssubscript Greek small letter psi of the surface S lying above a small, “horn-shaped” neighborhood Dsubscript Greek small letter psi of the principal root jet ψ, on which ∣x₂ − ψ(x₁)∣ ≤ εxᵐ₁. Here, ε > 0 can be chosen as small as one wishes. The proof then provides the opportunity to introduce some of the basic tools which will be applied frequently, such as dyadic domain decompositions, rescaling arguments based on the dilations associated to a given edge of the Newton polyhedron, in combination with Greenleaf's restriction and Littlewood–Paley theory, hence summing the estimates that have been obtained for the dyadic pieces.Less

This chapter shows that one may reduce the desired Fourier restriction estimate to a piece *S*subscript Greek small letter psi of the surface *S* lying above a small, “horn-shaped” neighborhood *D*subscript Greek small letter psi of the principal root jet ψ, on which ∣*x*₂ − ψ(*x*₁)∣ ≤ εxᵐ₁. Here, ε > 0 can be chosen as small as one wishes. The proof then provides the opportunity to introduce some of the basic tools which will be applied frequently, such as dyadic domain decompositions, rescaling arguments based on the dilations associated to a given edge of the Newton polyhedron, in combination with Greenleaf's restriction and Littlewood–Paley theory, hence summing the estimates that have been obtained for the dyadic pieces.