## Restriction and isoperimetric inequalities in harmonic analysis

##### Abstract

We study two related inequalities that arise in Harmonic Analysis: restriction
type inequalities and isoperimetric inequalities.
The (Lp, Lq) Restriction type inequalities have been the subject of much interest
since they were first conceived in the 1960s. The classical restriction type
inequality involving surfaces of non-vanishing curvature is only fully resolved in
two dimensions and there have been a lot of recent developments to establish the
conjectured (p,q) range in higher dimensions. However, it also interesting to consider
what can be said for curves where the curvature does vanish. In particular
we build upon a restriction result for homogeneous polynomial surfaces, using
what is considered the natural weight - the one induced by the affine curvature of
the surface. This is known to hold with a non-universal constant which depends
in some way on the coefficients of the polynomial. In this dissertation we shall
quantify that relationship.
Restriction estimates (for curves or surfaces) using the affine curvature weight
can be shown to lead to an affine isoperimetric inequality for such curves or
surfaces. We first prove, directly, this inequality for polynomial curves, where
the constant depends only on the degree of the underlying polynomials. We then
adapt this method, to prove an isoperimetric inequality for a wide class of curves,
which includes curves for which a restriction estimate is not yet known.
Next we state and prove an analogous result of the relative affine isoperimetric
inequality, which applies to unbounded convex sets. Lastly we demonstrate that
this relative affine isoperimetric inequality for unbounded sets is in fact equivalent
to the classical affine isoperimetric inequality.