Weak type (p,p) bounds for Schrödinger groups via generalized Gaussian estimates

Let L be a non-negative self-adjoint operator acting on 2 ( X ) , where is space of homogeneous type with dimension n . Suppose that the heat e − t satisfies generalized Gaussian p 0 ′ -estimates order m for some 1 ≤ < It known I + s i bounded ≥ | / and ∈ (see example, [5] [7] [9] [10] [13] [26] ). In this paper we study endpoint case = show weak is, there constant C > independent f so μ { x : α } ‖ R when ∞ Our results can applied to Schrödinger operators rough potentials higher elliptic measurable coefficients although in general, their semigroups fail satisfy upper bounds.