L p -improving properties of measures supported on curves on the Heisenberg group

Silvia Secco

Studia Mathematica (1999)

  • Volume: 132, Issue: 2, page 179-201
  • ISSN: 0039-3223

Abstract

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L p - L q boundedness properties are obtained for operators defined by convolution with measures supported on certain curves on the Heisenberg group. We find the curvature condition for which the type set of these operators can be the full optimal trapezoid with vertices A=(0,0), B=(1,1), C=(2/3,1/2), D=(1/2,1/3). We also give notions of right curvature and left curvature which are not mutually equivalent.

How to cite

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Secco, Silvia. "$L^p$-improving properties of measures supported on curves on the Heisenberg group." Studia Mathematica 132.2 (1999): 179-201. <http://eudml.org/doc/216594>.

@article{Secco1999,
abstract = {$L^p$-$L^q$ boundedness properties are obtained for operators defined by convolution with measures supported on certain curves on the Heisenberg group. We find the curvature condition for which the type set of these operators can be the full optimal trapezoid with vertices A=(0,0), B=(1,1), C=(2/3,1/2), D=(1/2,1/3). We also give notions of right curvature and left curvature which are not mutually equivalent.},
author = {Secco, Silvia},
journal = {Studia Mathematica},
keywords = {Heisenberg group; measure; convolution operator; boundedness properties},
language = {eng},
number = {2},
pages = {179-201},
title = {$L^p$-improving properties of measures supported on curves on the Heisenberg group},
url = {http://eudml.org/doc/216594},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Secco, Silvia
TI - $L^p$-improving properties of measures supported on curves on the Heisenberg group
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 2
SP - 179
EP - 201
AB - $L^p$-$L^q$ boundedness properties are obtained for operators defined by convolution with measures supported on certain curves on the Heisenberg group. We find the curvature condition for which the type set of these operators can be the full optimal trapezoid with vertices A=(0,0), B=(1,1), C=(2/3,1/2), D=(1/2,1/3). We also give notions of right curvature and left curvature which are not mutually equivalent.
LA - eng
KW - Heisenberg group; measure; convolution operator; boundedness properties
UR - http://eudml.org/doc/216594
ER -

References

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  1. [1] W. Drury, Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc. 108 (1990), 89-96. Zbl0708.42011
  2. [2] A. M. Mantero, Asymmetry of convolution operators on the Heisenberg group, Boll. Un. Mat. Ital. A (6) 4 (1985), 19-27. Zbl0561.43004
  3. [3] D. Oberlin, Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), 56-60. Zbl0613.43002
  4. [4] Y. Pan, A remark on convolution with measures supported on curves, Canad. Math. Bull. 36 (1993), 245-250. Zbl0820.43002
  5. [5] Y. Pan, Convolution estimates for some degenerate curves, Math. Proc. Cambridge Philos. Soc. 116 (1994), 143-146. Zbl0812.42006
  6. [6] Y. Pan, L p -improving properties for some measures supported on curves, preprint. 
  7. [7] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds, J. Funct. Anal. 86 (1989), 360-389. Zbl0684.22006
  8. [8] S. Secco, Fractional integration along homogeneous curves in ℝ 3 , preprint. 

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