## 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.
Full Text:

### References:

 [1] 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). [2] 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 [3] 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 [4] Auscher, P. and Shen, Z. Private communication. [5] Auscher, P. and Tchamitchian, P. Square root problem for divergence operators and related topics,Astérisque 249 (1998). · Zbl 0909.35001 [6] 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. [7] 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 [8] Coulhon, T. and Duong, X. T. Riesz transforms for 1 ,Trans. A.M.S. 351, 1151–1169 (1999). · Zbl 0973.58018 [9] 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 [10] 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 [11] Coulhon, T. and Zhang, Q. Large time behavior of heat kernels on forms, to appear inJ. Differential Geom. · Zbl 1137.58013 [12] Dungey, H. Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds,Math. Z. 247(4), 765–794, (2004). · Zbl 1080.58022 [13] Grigor’yan, A. Heat kernel on a manifold with a local Harnack inequality,Comm. Anal. Geom. 2, 111–138, (1994). · Zbl 0845.58056 [14] 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 [15] Ishiwata, S. Gradient estimate of the heat kernel on modified graphs, preprint. · Zbl 1196.58012 [16] 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 [17] Shen, Z. Bounds of Riesz transforms onL p spaces for second order elliptic operators,Ann. Inst. Fourier 55, 173–197, (2005). · Zbl 1068.47058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.