## Riesz transforms for $$1\leq p\leq 2$$.(English)Zbl 0973.58018

Summary: It has been asked [see R. Strichartz, J. Funct. Anal. 52, 48-79 (1983; Zbl 0515.58037)] whether one could extend to a reasonable class of non-compact Riemannian manifolds the $$L^p$$ boundedness of the Riesz transforms that holds in $${\mathbb R}^n$$. Several partial answers have been given since.
In the present paper, we give positive results for $$1\leq p\leq 2$$ under very weak assumptions, namely the doubling volume property and an optimal on-diagonal heat kernel estimate. In particular, we do not make any hypothesis on the space derivatives of the heat kernel. We also prove that the result cannot hold for $$p>2$$ under the same assumptions. Finally, we prove a similar result for the Riesz transforms on arbitrary domains of $${\mathbb R}^n$$.

### MSC:

 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 58J35 Heat and other parabolic equation methods for PDEs on manifolds

### Keywords:

doubling volume property; heat kernel; Riesz transforms

Zbl 0515.58037
Full Text:

### References:

 [1] G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math. 44 (1992), no. 4, 691 – 727 (English, with English and French summaries). · Zbl 0792.22005 [2] Pascal Auscher and Philippe Tchamitchian, Square root of divergence operators, square functions, and singular integrals, Math. Res. Lett. 3 (1996), no. 3, 429 – 437. · Zbl 0858.47019 [3] D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. I. Étude de la dimension 1, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 130 – 144 (French). [4] Dominique Bakry, Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Séminaire de Probabilités, XXI, Lecture Notes in Math., vol. 1247, Springer, Berlin, 1987, pp. 137 – 172 (French). · Zbl 0629.58018 [5] Dominique Bakry, The Riesz transforms associated with second order differential operators, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988) Progr. Probab., vol. 17, Birkhäuser Boston, Boston, MA, 1989, pp. 1 – 43. · Zbl 0682.68058 [6] Itai Benjamini, Isaac Chavel, and Edgar A. Feldman, Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash, Proc. London Math. Soc. (3) 72 (1996), no. 1, 215 – 240. · Zbl 0853.58098 [7] Chen Jie-Cheng, Heat kernels on positively curved manifolds and applications, Ph. D. thesis, Hanghzhou University, 1987. [8] Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. · Zbl 0224.43006 [9] Thierry Coulhon, Espaces de Lipschitz et inégalités de Poincaré, J. Funct. Anal. 136 (1996), no. 1, 81 – 113 (French, with English summary). · Zbl 0859.58009 [10] Thierry Coulhon and Alexander Grigor’yan, On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J. 89 (1997), no. 1, 133 – 199. · Zbl 0920.58064 [11] Thierry Coulhon and Michel Ledoux, Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: un contre-exemple, Ark. Mat. 32 (1994), no. 1, 63 – 77 (French). · Zbl 0826.53035 [12] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. · Zbl 0699.35006 [13] E. B. Davies, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. (2) 55 (1997), no. 1, 105 – 125. · Zbl 0879.35064 [14] Duong X., McIntosh A., Singular integral operators with non-smooth kernels on irregular domains, preprint, 1995. [15] Xuan T. Duong and Derek W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), no. 1, 89 – 128. · Zbl 0932.47013 [16] Alexander Grigor$$^{\prime}$$yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana 10 (1994), no. 2, 395 – 452. · Zbl 0810.58040 [17] Alexander Grigor$$^{\prime}$$yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 2, 353 – 362. · Zbl 0812.58082 [18] Alexander Grigor$$^{\prime}$$yan, Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal. 127 (1995), no. 2, 363 – 389. · Zbl 0842.58070 [19] Grigor’yan A., Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom., 45, 33-52, 1997. CMP 97:11 [20] David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161 – 219. · Zbl 0832.35034 [21] Kenig C., unpublished notes. [22] Jiayu Li, Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications, J. Funct. Anal. 97 (1991), no. 2, 293 – 310. · Zbl 0724.58064 [23] Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153 – 201. · Zbl 0611.58045 [24] Noël Lohoué, Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemannienne à courbure non positive, J. Funct. Anal. 61 (1985), no. 2, 164 – 201 (French). · Zbl 0605.58051 [25] Noël Lohoué, Transformées de Riesz et fonctions de Littlewood-Paley sur les groupes non moyennables, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 7, 327 – 330 (French, with English summary). · Zbl 0661.43002 [26] P.-A. Meyer, Correction à: ”Démonstration probabiliste de certaines inégalités de Littlewood-Paley, IV” [Séminaire de Probabilités, X (Première partie, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), pp. 175 – 183, Lecture Notes in Math., 511, Springer, Berlin, 1976; MR 58 #18751e], Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977), Lecture Notes in Math., vol. 649, Springer, Berlin, 1978, pp. p. 741 (French). [27] Zhong Min Qian, Gradient estimates and heat kernel estimate, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 5, 975 – 990. · Zbl 0863.58064 [28] Laurent Saloff-Coste, Analyse sur les groupes de Lie à croissance polynômiale, Ark. Mat. 28 (1990), no. 2, 315 – 331 (French, with English summary). · Zbl 0715.43009 [29] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 2 (1992), 27 – 38. · Zbl 0769.58054 [30] L. Saloff-Coste, On global Sobolev inequalities, Forum Math. 6 (1994), no. 3, 271 – 286. · Zbl 0802.58055 [31] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501 [32] Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), no. 1, 48 – 79. · Zbl 0515.58037 [33] Nicholas Th. Varopoulos, Une généralisation du théorème de Hardy-Littlewood-Sobolev pour les espaces de Dirichlet, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 14, 651 – 654 (French, with English summary). · Zbl 0566.31006
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.