Let (X, d, μ) be a space of homogeneous type and Ω an open subset of X. Given a bounded operator T: L^p(Ω) → L^q(Ω) for some 1 ≤ p ≤ q < ∞, we give a criterion for T to be of weak type (p_0, a) for p_0 and a such that 1/p_0 - 1/a = 1/p-1/q. These results are illustrated by several applications including estimates of weak type (p_0, a) for Riesz potentials L^-2 or for Riesz transform type operators ∇ Δ^-2 as well as L^p-L^q boundedness of spectral multipliers F(L) when the heat kernel of L satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space H^1_L associated with L into L^a(X). By duality this gives boundedness from L^a'(X) into BMO_L.
For every unitary irreducible representation of a nilpotent Lie group we prove that the representation Hilbert space is the only nonzero invariant Hilbert space of distribution vectors.
We obtain sharp upper bounds for three-term segments of a bounded power series. Along the way we show that the Taylor polynomials of a certain algebraic function do not vanish in the unit disk.
