*Ali Taheri*

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733133
- eISBN:
- 9780191797712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733133.003.0006
- Subject:
- Mathematics, Analysis

This chapter introduces the two main interpolations theorems: Marcinkiewicz and Riesz-Thorin. It also presents a number of important applications. The chapter includes a discussion of weak-Lebesgue ...
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This chapter introduces the two main interpolations theorems: Marcinkiewicz and Riesz-Thorin. It also presents a number of important applications. The chapter includes a discussion of weak-Lebesgue spaces, Lorentz spaces and their interpolation properties.Less

This chapter introduces the two main interpolations theorems: Marcinkiewicz and Riesz-Thorin. It also presents a number of important applications. The chapter includes a discussion of weak-Lebesgue spaces, Lorentz spaces and their interpolation properties.

*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/9780691170541.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's ...
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This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's interpolation theorem for analytic families of operators. It constructs analytic families of complex measure μsubscript Greek small letter zeta, for ζ in the complex strip Σ given by 0 ≤ Reζ ≤ 1, by introducing complex coefficients in the sums defining the measures νsubscript Greek small letter delta,jsuperscript V and νsubscript Greek small letter delta,jsuperscript V I, respectively. These coefficients are chosen as exponentials of suitable affine-linear expression in ζ in such a way that, in particular, μsubscript Greek small letter theta subscript c = νsubscript Greek small letter delta,jsuperscript V I, respectively, μsubscript Greek small letter theta subscript c = νsubscript Greek small letter delta,jsuperscript V I. As it turns out, the main problem consists in establishing suitable uniform bounds for the measure μsubscript Greek small letter zeta when ζ lies on the right boundary line of Σ.Less

This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's interpolation theorem for analytic families of operators. It constructs analytic families of complex measure μsubscript Greek small letter zeta, for ζ in the complex strip Σ given by 0 ≤ Reζ ≤ 1, by introducing complex coefficients in the sums defining the measures νsubscript Greek small letter delta,*j*superscript *V* and νsubscript Greek small letter delta,*j*superscript *V I*, respectively. These coefficients are chosen as exponentials of suitable affine-linear expression in ζ in such a way that, in particular, μsubscript Greek small letter theta subscript *c* = νsubscript Greek small letter delta,*j*superscript *V I*, respectively, μsubscript Greek small letter theta subscript *c* = νsubscript Greek small letter delta,*j*superscript *V I*. As it turns out, the main problem consists in establishing suitable uniform bounds for the measure μsubscript Greek small letter zeta when ζ lies on the right boundary line of Σ.