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Riesz transform on manifolds and Poincaré inequalities. (English) Zbl 1116.58023
The authors study the validity of the \(L^p\) inequality for the Riesz transform when \(p>2\) and of its reverse inequality when \(1<p<2\) on complete Riemannian manifolds under the doubling property and some Poincaré inequality.
Let \(M\) be a non-compact complete Riemannian manifold. Consider the \(L^p\) inequality \[ ||\;|\nabla \Delta^{-1/2}f|\;||_p \leq C_p ||f||_p, \tag{1} \] and the “reverse inequality” \[ ||\Delta^{1/2}f||_p \leq C_p ||\;|\nabla f|\;||_p. \tag{2} \] A manifold is said to satisfy the volume doubling property if there exists a constant \(C\) such that for all \(x\in M\) and \( r>0\), \[ V(x,2r)\leq C V(x,r). \] It satisfies the scaled Poincaré inequalities \(P_p\) if there exists \(C>0\) such that for every ball \(B(x,r)\), and every \(f\) with \(f\), \(\nabla f\) locally \(p\)-integrable, then \[ \int_B |f-f_B|^p d\mu \leq Cr^p \int_B |\nabla f|^p\, d\mu, \tag{\(P_p\)} \] where \(f_B\) is the mean of \(f\) on \(B\).
The main theorem is: Let \(M\) be a complete non-compact Riemannian manifold satisfying the volume doubling property, and \((P_2)\). Then there exists \(\epsilon>0\) such that (1) holds for \(2<p<2+\epsilon\).
An important step in the proof is the following theorem: Let \(M\) be a complete non-compact Riemannian manifold satisfying the volume doubling property and \(P_q\) for some \(q\in [1,2]\). Then (2) holds for \(q<p<2\). If \(q=1\), there is a weak-type \((1,1)\) estimate.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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