# Improved Estimates by Means of Airy-Type Analysis

# Improved Estimates by Means of Airy-Type Analysis

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.

*Keywords:*
improved estimates, Airy-type analysis, interpolation theorem, endpoint result, uniform bounds, Airy-type decompositions, complex interpolation

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