The reverse Riesz inequality ‖Δ^{1/2}f‖_p ≤ C‖|∇f|‖_p holds for all 1 < p < ∞ on connected sums of Euclidean-ended manifolds, contrasting with the Riesz transform's sharp range p < min n_i.
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Reverse Riesz Inequality on Manifolds with Ends
The reverse Riesz inequality ‖Δ^{1/2}f‖_p ≤ C‖|∇f|‖_p holds for all 1 < p < ∞ on connected sums of Euclidean-ended manifolds, contrasting with the Riesz transform's sharp range p < min n_i.