Skip to main content Accesibility Help
×
×
Home

ANALYSIS IN THE MULTI-DIMENSIONAL BALL

  • Peter Sjögren (a1) and Tomasz Z. Szarek (a2) (a3)
Abstract

We study the heat semigroup maximal operator associated with a well-known orthonormal system in the $d$ -dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted $L^{p}$ estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.

Copyright
Footnotes
Hide All

The second author was partially supported by the National Science Centre of Poland, project no. 2015/19/D/ST1/01178, and by the Foundation for Polish Science START Scholarship.

Footnotes
References
Hide All
1. Ciaurri, Ó., The Poisson operator for orthogonal polynomials in the multidimensional ball. J. Fourier Anal. Appl. 19 2013, 10201028.
2. Coulhon, T., Kerkyacharian, G. and Petrushev, P., Heat kernel generated frames in the setting of Dirichlet spaces. J. Fourier Anal. Appl. 18 2012, 9951066.
3. Dai, F. and Xu, Y., Approximation Theory and Harmonic Analysis on Spheres and Balls (Springer Monographs in Mathematics), Springer (New York, NY, 2013).
4. Dunkl, C. F. and Xu, Y., Orthogonal Polynomials of Several Variables, Cambridge University Press (Cambridge, 2001).
5. Duoandikoetxea, J., Extrapolation of weights revisited: new proofs and sharp bounds. J. Funct. Anal. 260 2011, 18861901.
6. Fukushima, M., Oshima, Y. and Takeda, M., Dirichlet Forms and Symmetric Markov Processes (De Gruyter Studies in Mathematics 19 ), De Gruyter (Berlin, 2011).
7. Gyrya, P. and Saloff-Coste, L., Neumann and Dirichlet Heat Kernels in Inner Uniform Domains (Astérisque 336 ), Société Mathématique de France (2011).
8. Hebisch, W. and Saloff-Coste, L., On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51 2001, 14371481.
9. Kerkyacharian, G., Petrushev, P. and Xu, Y., Gaussian bounds for the weighted heat kernels on the interval, ball and simplex. Preprint, 2018, arXiv:1801.07325.
10. Kerkyacharian, G., Petrushev, P. and Xu, Y., Gaussian bounds for the heat kernels on the ball and simplex: Classical approach. Preprint, 2018, arXiv:1801.07326.
11. Saloff-Coste, L., Aspects of Sobolev-type Inequalities (London Mathematical Society Lecture Note Series 289 ), Cambridge University Press (Cambridge, 2002).
12. Szegö, G., Orthogonal Polynomials, 4th edn (Colloquium Publications 23 ), American Mathematical Society (Providence, RI, 1975).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed