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Restriction problems

In higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general class of square integrable functions. As such, the restriction of the Fourier transform of an L (R) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in L for 1 < p < 2. It is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature. The case when S is the unit sphere in R is of particular interest. In this case the Tomas– Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in R is a bounded operator on L provided 1 ≤ p ≤ 2 n + 2 / n + 3.

One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets E indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2 R. For a given integrable function f, consider the function f defined by: f R (x) = ∫ E R f ^ (ξ) e i 2 π x ⋅ ξ d ξ, x ∈ R n. {\displaystyle f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi)e^{i2\pi x\cdot \xi }\,d\xi,\quad x\in \mathbb {R} ^{n}.}

Suppose in addition that f ∈ L (R). For n = 1 and 1 < p < ∞, if one takes E = (− R, R), then f converges to f in L as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for n > 1. In the case that E is taken to be a cube with side length R, then convergence still holds. Another natural candidate is the Euclidean ball E = { ξ : | ξ | < R }. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in L (R). For n ≥ 2 it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless p = 2. In fact, when p ≠ 2, this shows that not only may f fail to converge to f in L, but for some functions f ∈ L (R), f is not even an element of L.