AnLp−Lqversion of Hardy's theorem for spherical Fourier transform on semisimple Lie groups

Abstract

We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We prove anLp−Lqversion of Hardy's theorem for the spherical Fourier transform onG. More precisely, leta,bbe positive real numbers,1≤p,q≤∞, andfaK-bi-invariant measurable function onGsuch thatha−1f∈Lp(G)andeb‖λ‖2ℱ(f)∈Lq(𝔞+*)(hais the heat kernel onG). We establish that ifab≥1/4andporqis finite, thenf=0almost everywhere. Ifab<1/4, we prove that for allp,q, there are infinitely many nonzero functionsfand ifab=1/4withp=q=∞, we havef=const ha.