ArticleOriginal scientific text
Title
Some remarks on Bochner-Riesz means
Authors 1
Affiliations
- Stat-Math Division, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore-560 059, India
Abstract
We study norm convergence of Bochner-Riesz means associated with certain non-negative differential operators. When the kernel satisfies a weak estimate for large values of m we prove norm convergence of for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.
Keywords
unitary representations, Schrödinger operators, Bochner-Riesz means, nilpotent groups, Rockland operators, Heisenberg group, summability
Bibliography
- A. Bonami et J. L. Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223-263.
- L. Carleson and P. Sjölin, Oscillatory integrals and multiplier problem for the disc, Studia Math. 44 (1972), 287-299.
- J. Dziubański, W. Hebisch and J. Zienkiewicz, Note on semigroups generated by positive Rockland operators on graded homogeneous groups, ibid. 110 (1994), 115-126.
- C. Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1972), 44-52.
- G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ., Princeton, 1982.
- W. Hebisch, Almost everywhere summability of eigenfunction expansions associated to elliptic operators, Studia Math. 96 (1990), 263-275.
- B. Helffer et J. Nourrigat, Caractérisation des opérateurs hypoelliptiques homogènes à gauche sur un groupe nilpotent gradué, Comm. Partial Differential Equations 4 (1979), 899-958.
- L. Hörmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, in: Some Recent Advances in the Basic Sciences, Vol. 2, Yeshiva Univ., New York, 1969, 155-202.
- A. Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math. 78 (1984), 253-266.
- A. Hulanicki and J. Jenkins, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc. 278 (1983), 703-715.
- A. Hulanicki and J. Jenkins, Nilpotent Lie groups and summability of eigenfunction expansions of Schrödinger operators, Studia Math. 80 (1984), 235-244.
- G. Karadzhov, Riesz summability of multiple Hermite series in
spaces, C. R. Acad. Bulgare Sci. 47 (1994), 5-8. - C. E. Kenig, R. Stanton and P. Tomas, Divergence of eigenfunction expansions, J. Funct. Anal. 46 (1982), 28-44.
- G. Mauceri, Riesz means for the eigenfunction expansions for a class of hypoelliptic differential operators, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 4, 115-140.
- G. Mauceri and S. Meda, Vector valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (1990), 141-154.
- B. S. Mitjagin [B. S. Mityagin], Divergenz von Spektralentwicklungen in
-Räumen, in: Linear Operators and Approximation II, Internat. Ser. Numer. Math. 25, Birkhäuser, Basel, 1974, 521-530. - D. Müller and E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. 73 (1994), 413-440.
- J. Peetre, Remarks on eigenfunction expansions for elliptic differential operators with constant coefficients, Math. Scand. 15 (1964), 83-97.
- J. Peetre, Some estimates for spectral functions, Math. Z. 92 (1966), 146-153.
- C. D. Sogge, Concerning the
norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-134. - C. D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math. 126 (1987), 439-447.
- E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1971.
- K. Stempak and J. Zienkiewicz, Twisted convolution and Riesz means, J. Anal. Math. 76 (1998), 93-107.
- S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton Univ. Press, Princeton, 1993.
- S. Thangavelu, Hermite and special Hermite expansions revisited, Duke Math. J. 94 (1998), 257-278.
- S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progr. Math. 159, Birkhäuser, Boston, 1998