Let G be a compact, simply connected, simple Lie group of dimension N and rank R, and let P be a Weyl-invariant convex polyhedron in t, the dual of the Lie algebra of a maximal torus in G. We form from P a summability method which, to an integrable function f, associates the partial sums S_nf(x) =∑_{λ ∈ nP} d_λ χ_λ*f(x). Stanton and Tomas proved that for p < 2 there exists f ∈ L^p such that Snf diverges almost everywhere, but that if one restricts attention to central f ∈ L^p, S_nf converges almost everywhere for p > 2N/(N +R) = p_0. The main result of this paper is that notwithstanding there exist central functions in L^{p_0}(G) with unbounded Fourier coefficients, S_nf converges almost everywhere if f belongs to the Lorentz space L^{p_0,1}(G).
Pointwise convergence of Fourier series on compact Lie groups
GIULINI, SAVERIO;
1990-01-01
Abstract
Let G be a compact, simply connected, simple Lie group of dimension N and rank R, and let P be a Weyl-invariant convex polyhedron in t, the dual of the Lie algebra of a maximal torus in G. We form from P a summability method which, to an integrable function f, associates the partial sums S_nf(x) =∑_{λ ∈ nP} d_λ χ_λ*f(x). Stanton and Tomas proved that for p < 2 there exists f ∈ L^p such that Snf diverges almost everywhere, but that if one restricts attention to central f ∈ L^p, S_nf converges almost everywhere for p > 2N/(N +R) = p_0. The main result of this paper is that notwithstanding there exist central functions in L^{p_0}(G) with unbounded Fourier coefficients, S_nf converges almost everywhere if f belongs to the Lorentz space L^{p_0,1}(G).
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