Rearranging Fourier transforms on groups
Chung Lin · Pacific Journal of Mathematics · 1975
Let G denote an infinite locally compact abelian group and X its character group.Let Θ be a suitable Haar measure on X, and 1 t}) and φ*(x) = inf {t > 0: θ φ (t) ^ x} for x > 0. φ* is called the nonincreasing rearrangement of φ.Note that even though φ is defined on X, the domain of φ* is (0, oo).A nonnegative function g defined on (0, oo) is called admissible if g is nonincreasing and lim x _, D g(x) = 0. Theorems: 1.Let G be nondiscrete with a compact open subgroup and g admissible.Then g\ N = f*\ N9 where N is the set of positive integers, for some feL p (G) if Σ?=i g(k) p k p ~2 wa.e. for some feL p (G) if \ g(x) p x p ~2dx < oo.Jo I* Introduction* As usual the Fourier transform / of a function / G L ι (G) is defined on X such that f(χ) = 1 /χcϋλ, where λ is a JGfixed but arbitrary Haar measure on G.For 1 < p < 2, feL p '(G) and p' is the conjugate exponent of p.The set of real numbers, ^-dimensional Euclidean space, the circle group, the integers, the radic integers, the countable product of the group of integers modula r and the subgroup of the circle whose elements have order a power of r are denoted by R, R n , T, Z, A r , ΠZ{r) and Z(r°°), respectively.Also p will denote any number such that 1< p < 2. Let m be l/τ/2τr7 Lebesque measure on R.Hardy and Littlewood [1], [2] characterized functions on Z such that every rearrangement is the Fourier transform of a function in L P (T), 2 < p < oo.They also characterized functions on Z such that some rearrangement is the Fourier transform of afunction in L P (T), 1< p < 2. Hewitt and Ross [4] generalized these results to arbitrary compact infinite abelian groups.We are interested in the case of LCA (locally compact abelian) groups.Here are our results.THEOREM 1.Let G be nondiscrete with a compact open subgroup, 463