On the L^p boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Let H=−Δ+V be a Schrödinger operator on L²(R²) with real-valued potential V, and let H₀=−Δ. If V has sufficient pointwise decay, the wave operators W_±=s−lim_t→±∞⁡e^itHe^−itH₀ are known to be bounded on L^p(R²) for all 1<p<∞ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on L^p(R²) for 1<p<∞. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents p.