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They have proved that there exists a positive lower bound on Lp1 harmonic radius rH(M) in terms of the bounds RicM ≥ −λ0 for some λ0 ≥ 0 and inj(M) ≥ i0 for some i0 > 0. They used this result to show a C α compactness theorem for the class of compact Riemannian manifolds which have a lower bound on the Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. The main purpose of this paper is to find a positive lower bound on Lp1 harmonic radius rH(M) under weaker conditions. We have little hope to weaken the condition inj(M) ≥ i0 (see [AC]). However, there are some results based on integral norms of the Ricci curvature. For any x ∈M , let −λ(x) be the lowest nonpositive eigenvalue of the Ricci curvature at x (if RicM > 0 at x, we define λ(x) = 0). Our main result is the following theorem.

C^{\alpha} compactness theorem for Riemannian manifolds with bounds on diameter, injectivity radius, and some integral norms of Ricci curvature