# zbMATH — the first resource for mathematics

A perturbation result for the Riesz transform. (English) Zbl 1332.58010
Let $$(M,g)$$ 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.

##### MSC:
 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
Full Text: