A perturbation result for the Riesz transform.

*(English)*Zbl 1332.58010Let \((M,g)\) be a complete Riemannian manifold, not necessarily connected. The Riesz transform is \(\mathcal{R}:=d\Delta^{-1/2}\). The associated Riesz transformation problem is to give conditions on \(p\) and on the underlying geometry so that \(\mathcal{R}\) is a bounded operator on \(L^p\). D. Bakry [Lect. Notes Math. 1247, 137–172 (1987; Zbl 0629.58018)] showed that if the Ricci curvature is non-negative, then \(\mathcal{R}\) is bounded in \(L^p\) for any \(1<p<\infty\). The Sobolev dimension \(d_S(M)\) is the supremum of the set of \(n\) such that the Sobolev inequality of dimension \(n\) is satisfied on \(M\) (if no Sobolev inequality is satisfied on \(M\), set \(d_S=-\infty)\). The hyperbolic dimension \(d_H(M)\) is the supremum of the set of \(p\) such that \(M\) is \(p\)-hyperbolic. The author shows that only the behavior at infinity is relevant in a certain sense:

Theorem. Let \((M_0,g_0)\) and \((M_1,g_1)\) be two complete Riemannian manifolds (not necessarily connected), isometric at infinity, whose Ricci curvatures are bounded from below and which satisfy \(d_S>2\). Assume the Riesz transform action on \(M_0\) is bounded on \(L^p\) for \(p\in[p_0,p_1)\) with \(\frac{d_S}{d_S-1}<p_0\leq2\) if \(d_S>3\) and \(p_0=2\) if \(2<d_S\leq3\) and \(p_1>\frac{d_S}{d_S-2}\). Then, the Riesz transform on \(M_1\) is bounded on \(L^p\) for \(p\in[p_0,\min(d_H(M),p_1))\). Furthermore, if \(M_1\) has only one end, then the Riesz transform on \(M_1\) is bounded on \(L^p\) for \(p\in[p_0,p_1)\).

The introduction to the paper contains an elegant historical discussion of the problem to put the main theorem of the paper in proper context. The second section presents material about \(p\)-hyperbolicity. The main results are proved in the third section. The case where the manifold has several ends and the case where it has just one end are treated separately.

Theorem. Let \((M_0,g_0)\) and \((M_1,g_1)\) be two complete Riemannian manifolds (not necessarily connected), isometric at infinity, whose Ricci curvatures are bounded from below and which satisfy \(d_S>2\). Assume the Riesz transform action on \(M_0\) is bounded on \(L^p\) for \(p\in[p_0,p_1)\) with \(\frac{d_S}{d_S-1}<p_0\leq2\) if \(d_S>3\) and \(p_0=2\) if \(2<d_S\leq3\) and \(p_1>\frac{d_S}{d_S-2}\). Then, the Riesz transform on \(M_1\) is bounded on \(L^p\) for \(p\in[p_0,\min(d_H(M),p_1))\). Furthermore, if \(M_1\) has only one end, then the Riesz transform on \(M_1\) is bounded on \(L^p\) for \(p\in[p_0,p_1)\).

The introduction to the paper contains an elegant historical discussion of the problem to put the main theorem of the paper in proper context. The second section presents material about \(p\)-hyperbolicity. The main results are proved in the third section. The case where the manifold has several ends and the case where it has just one end are treated separately.

Reviewer: Peter B. Gilkey (Eugene)

##### MSC:

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |