## Riesz transform and perturbation.(English)Zbl 1122.58014

Suppose that $$A\colon\mathbb{R}^d\to\mathbb{R}^{d\times d}$$ is measurable and satisfies the ellipticity and boundedness estimate $C^{-1}| v| ^2\leq v^TA(x)v\leq C| v| ^2$ with some $$C>0$$. Let $$H$$ denote the form closure of the divergence form operator $$-\text{div}A\nabla$$ on $$L^p(\mathbb{R}^d)$$. The Riesz transform of $$H$$ is the operator $$\nabla H^{-1/2}$$ on $$L^p$$.
It is known that the Riesz transform of $$H$$ is bounded for $$p\in(1,2+\varepsilon)$$, where $$\varepsilon>0$$ depends on $$d$$ and $$C$$. The present article gives a perturbation type criterion to extend this result to higher values of $$p$$ and to closed noncompact Riemannian manifolds. The first result pertains to the Euclidean case. Suppose one is given two divergence form operators $$H_0$$ and $$H$$, induced by the corresponding matrix functions $$A_0$$ and $$A$$ as above. Consider $$H$$ as a perturbation of $$H_0$$, assuming that $$A_0-A\in L^q$$ for some $$q\in[1,\infty)$$. If the Riesz transform of $$H_0$$ is bounded in $$L^{p_0}$$ for some $$p_0>2$$, and if $$\nabla(I+H)^{-1/2}$$ (the local Riesz transform of $$H$$) is bounded in $$L^p$$ for all $$p\in(2,p_0)$$, then the Riesz transform of $$H$$ is bounded in $$L^p$$ for all $$p\in(2,p_0)$$. A generalization of this result to weighted $$L^p$$ spaces, where the weights are positive, bounded, and bounded away from $$0$$, leads to a theorem for a noncompact, connected smooth manifold $$M$$ of dimension $$d$$: Suppose that $$G_0$$ and $$G$$ are two Riemannian metrics on $$M$$ with uniformly equivalent associated norms on the tangent spaces. Assume that $$G$$ is a $$L^q$$-perturbation of $$G_0$$ in a specified sense, for some $$q\in[1,\infty)$$. Denote by $$H_0$$ and $$H$$ the positive Laplace operators associated with $$G_0$$ and $$G$$. Moreover, assume the norm estimate $\| e^{-tH_0}\| _{{\mathcal L}(L^1,L^\infty)}\leq C_1t^{-C_2},\qquad t\geq 1,$ with positive constants $$C_1,C_2$$. Then there holds a theorem similar to the result in $$\mathbb{R}^d$$, only differing in the additional assumption that the Riesz transform of $$H$$ is bounded in $$L^{p'}$$ for $$p'\in (p_0',2)$$, where $$p_0'$$ is the conjugate exponent of $$p_0$$.

### MSC:

 58J37 Perturbations of PDEs on manifolds; asymptotics 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47F05 General theory of partial differential operators 47B44 Linear accretive operators, dissipative operators, etc.
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### References:

  Auscher, P. On necessary and sufficient conditions forL p -estimates of Riesz transforms associated to elliptic operators on $$\mathbb{R}$$ n : A survey, to appear inMem. A.M.S. 186, 871, (2007).  Auscher, P. and Coulhon, T. Riesz transform on manifolds and Poincaré inequalities,Ann. Sc. Norm. Sup. Pisa 5(3), IV, 531–555, (2005). · Zbl 1116.58023  Auscher, P., Coulhon, T., Duong, X. T., and Hofmann, S. Riesz transforms on manifolds and heat kernel regularity,Ann. Scient. E.N.S. Paris 37(6), 911–950, (2004). · Zbl 1086.58013  Auscher, P. and Shen, Z. Private communication.  Auscher, P. and Tchamitchian, P. Square root problem for divergence operators and related topics,Astérisque 249 (1998). · Zbl 0909.35001  Bakry, D. Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée inSéminaire de Probabilités XXI, Springer L.N.1247, 137–172, 1987.  Barbatis, G. Stability of weighted Laplace-Beltrami operators underL p -perturbation of the Riemannian metric,J. Anal. Math,68, 253–276 (1996). · Zbl 0868.35025  Coulhon, T. and Duong, X. T. Riesz transforms for 1 ,Trans. A.M.S. 351, 1151–1169 (1999). · Zbl 0973.58018  Coulhon, T. and Duong, X. T. Riesz transforms forp>2,C.R.A.S. Paris 332(11), série I, 975–980, (2001). · Zbl 0987.43001  Coulhon, T. and Duong, X. T. Riedz transform and related inequalities on noncompact Riemannian manifolds,Comm. Pure Appl. Math. 56(12), 1728–1751, (2003). · Zbl 1037.58017  Coulhon, T. and Zhang, Q. Large time behavior of heat kernels on forms, to appear inJ. Differential Geom. · Zbl 1137.58013  Dungey, H. Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds,Math. Z. 247(4), 765–794, (2004). · Zbl 1080.58022  Grigor’yan, A. Heat kernel on a manifold with a local Harnack inequality,Comm. Anal. Geom. 2, 111–138, (1994). · Zbl 0845.58056  Hofmann, S. and Martell, J. M.L p bounds for Riesz transforms and square roots associated to second order elliptic operators,Publicacions Matematiques 47, 497–515, (2003). · Zbl 1074.35031  Ishiwata, S. Gradient estimate of the heat kernel on modified graphs, preprint. · Zbl 1196.58012  Li, X. D. Riesz transforms and Schrödinger operators on complete Riemannian manifolds with negative Ricci curvature,Rev. Mat Iberoamericana 22, 591–648, (2006). · Zbl 1119.53022  Shen, Z. Bounds of Riesz transforms onL p spaces for second order elliptic operators,Ann. Inst. Fourier 55, 173–197, (2005). · Zbl 1068.47058
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