Integral curvature estimates for solutions to Ricci flow with Lp bounded scalar curvature

Abstract In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed n -dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> sense for some $$p&gt;2,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> then the estimates imply a uniform bound on the spatial $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.