Inequalities for second-order Riesz transforms associated with Bessel expansions

• Autorzy: Osękowski Adam

Identyfikatory

DOI

10.4064/ba200302-5-10

• Języki publikacji: EN

Abstrakty

EN

The paper contains the proofs of L^p, logarithmic and weak-type estimates for the second-order Riesz transforms arising in the context of multidimensional Bessel expansions. Using a novel probabilistic approach, which rests on martingale methods and the representation of Riesz transforms via associated Bessel-heat processes, we show that these estimates hold with constants independent of the dimension.

Słowa kluczowe

EN

Riesz transforms; Hankel transform; Bessel expansion; martingale; differential subordination

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2020
• Tom: Vol. 68, no. 1
• Strony: 75--88
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Osękowski Adam

• Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, reprint of the 1972 edition, Dover Publ., New York, 1992.
• [2] R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations, Duke Math. J. 80 (1995), 575-600.
• [3] J. J. Betancor, J. C. Fariña, D. Buraczewski, T. Martínez and J. L. Torrea, Riesz transforms related to Bessel operators, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 701-725.
• [4] J. J. Betancor, J. C. Fariña, T. Martinez and L. Rodríguez-Mesa, Higher order Riesz transforms associated with Bessel operators, Ark. Mat. 46 (2008), 219-250.
• [5] J. J. Betancor and K. Stempak, Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform, Tohoku Math. J. (2) 53 (2001), 109-129.
• [6] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702.
• [7] D. L. Burkholder, A sharp and strict Lp-inequality for stochastic integrals, Ann. Probab. 15 (1987), 268-273.
• [8] C. Dellacherie and P.-A. Meyer, Probabilities and Potential B: Theory of Martingales, North-Holland, Amsterdam, 1982.
• [9] A. Osękowski, Sharp Martingale and Semimartingale Inequalities, IMPAN Monografie Mat. 72, Birkhäuser, 2012.
• [10] A. Osękowski, Logarithmic inequalities for Fourier multipliers, Math. Z. 274 (2013), 515-530.
• [11] A. Osękowski, Weak type inequalities for Fourier multipliers with applications to Beurling-Ahlfors transform, J. Math. Soc. Japan 66 (2014), 745-764.
• [12] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer, 1999.
• [13] G. Wang, Differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522-551.
• [14] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1966.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).