*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.

*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/9780691170541.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses the remaining cases for l = 1. With the same basic approach as in Chapter 5, the chapter again performs an additional dyadic frequency domain decomposition related to the ...
More

This chapter discusses the remaining cases for l = 1. With the same basic approach as in Chapter 5, the chapter again performs an additional dyadic frequency domain decomposition related to the distance to a certain Airy cone. This is needed in order to control the integration with respect to the variable x₁ in the Fourier integral defining the Fourier transform of the complex measures νsubscript Greek small letter delta superscript Greek small letter lamda. It first applies a suitable translation in the x₁-coordinate before performing a more refined analysis of the phase Φsuperscript Music sharp sign. The chapter then treats the case where λρ(̃δ) ≲ 1 and hereafter deals with the case where λρ(̃δ) ≲ 1 and B = 4. Finally, the chapter turns to the case where B = 3.Less

This chapter discusses the remaining cases for *l* = 1. With the same basic approach as in Chapter 5, the chapter again performs an additional dyadic frequency domain decomposition related to the distance to a certain Airy cone. This is needed in order to control the integration with respect to the variable *x*₁ in the Fourier integral defining the Fourier transform of the complex measures νsubscript Greek small letter delta superscript Greek small letter lamda. It first applies a suitable translation in the *x*₁-coordinate before performing a more refined analysis of the phase Φsuperscript Music sharp sign. The chapter then treats the case where λρ(̃δ) ≲ 1 and hereafter deals with the case where λρ(̃δ) ≲ 1 and *B* = 4. Finally, the chapter turns to the case where *B* = 3.

*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/9780691170541.003.0006
- Subject:
- Mathematics, Geometry / Topology

This chapter turns to the case where hsubscript lin(φ) ≥ 2. In a first step, the chapter performs a decomposition of the remaining piece Ssubscript Greek small letter psi of the surface S. Then, in ...
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This chapter turns to the case where hsubscript lin(φ) ≥ 2. In a first step, the chapter performs a decomposition of the remaining piece Ssubscript Greek small letter psi of the surface S. Then, in the domains Dₗ the chapter once again applies dyadic decomposition techniques in combination with rescaling arguments, making use of the dilations associated with the weight κₗ. But serious new problems arise, caused by the nonlinear change from the coordinates (x₁, x₂) to the adapted coordinates (y₁, y₂). Therefore, the chapter takes a closer look at the domain Dsubscript pr and devises a further decomposition of the domain Dsubscript pr into various subdomains of “type” Dsubscript (l) and Esubscript (l).Less

This chapter turns to the case where *h*subscript lin(φ) ≥ 2. In a first step, the chapter performs a decomposition of the remaining piece *S*subscript Greek small letter psi of the surface *S*. Then, in the domains *D*ₗ the chapter once again applies dyadic decomposition techniques in combination with rescaling arguments, making use of the dilations associated with the weight κₗ. But serious new problems arise, caused by the nonlinear change from the coordinates (*x*₁, *x*₂) to the adapted coordinates (*y*₁, *y*₂). Therefore, the chapter takes a closer look at the domain *D*subscript pr and devises a further decomposition of the domain *D*subscript pr into various subdomains of “type” *D*subscript (*l*) and *E*subscript (*l*).