Endpoint maximal and smoothing estimates for Schroedinger equations

الأصل · EN

For α>1 we consider the initial value problem for the dispersive equation i∂ₜ u +(-Δ)α/² u= 0. We prove an endpoint Lᵖ inequality for the maximal function ₜ∈[₀,₁]|u(·,t)| with initial values in Lᵖ-Sobolev spaces, for p∈(2+4/(d+1),∞). This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp Lᵖ space-time estimates (local in time) for the same range of p.