Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2014 | 12 | 8 | 1198-1213
Tytuł artykułu

Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms

Treść / Zawartość
Warianty tytułu
Języki publikacji
We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures on the space of matrices of dimension 2×2, and some transference-type arguments.
Słowa kluczowe
Opis fizyczny
  • [1] Astala K., Faraco D., Székelyhidi L. Jr., Convex integration and the Lp theory of elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2008, 7(1), 1–50
  • [2] Bañuelos R., Bielaszewski A., Bogdan K., Fourier multipliers for non-symmetric Lévy processes, Banach Center Publ., 95, Polish Academy of Sciences, Warsaw, 2011, 9–25
  • [3] Bañuelos R., Bogdan K., Lévy processes and Fourier multipliers, J. Funct. Anal., 2007, 250(1), 197–213
  • [4] Bañuelos R., Osekowski A., Martingales and sharp bounds for Fourier multipliers, Ann. Acad. Sci. Fenn. Math., 2012, 37(1), 251–263
  • [5] Bañuelos R., Wang G., Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations, Duke Math. J., 1995, 80(3), 575–600
  • [6] Boros N., Székelyhidi L. Jr., Volberg A., Laminates meet Burkholder functions, J. Math. Pures Appl., 2013, 100(5), 687–700
  • [7] Burkholder D.L., An extension of a classical martingale inequality, In: Probability Theory and Harmonic Analysis, Ohio, May 10–12, 1983, Monogr. Textbooks Pure Appl. Math., 98, Marcel Dekker, New York, 1986, 21–30
  • [8] Choi K.P., A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in Lp(0, 1), Trans. Amer. Math. Soc., 1992, 330(2), 509–529
  • [9] Conti S., Faraco D., Maggi F., A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal., 2005, 175(2), 287–300
  • [10] Davis B., On the weak type (1, 1) inequality for conjugate functions, Proc. Amer. Math. Soc., 1974, 44(2), 307–311
  • [11] Dellacherie C., Meyer P.-A., Probabilities and Potential B, North-Holland Math. Stud., 72, North-Holland, Amsterdam, 1982
  • [12] Geiss S., Mongomery-Smith S., Saksman E., On singular integral and martingale transforms, Trans. Amer. Math. Soc., 2010, 362(2), 553–575
  • [13] Iwaniec T., Martin G., Riesz transforms and related singular integrals, J. Reine Angew. Math., 1996, 473, 25–57
  • [14] Janakiraman P., Best weak-type (p, p) constants, 1 < p < 2, for orthogonal harmonic functions and martingales, Illinois J. Math., 2004, 48(3), 909–921
  • [15] Kirchheim B., Rigidity and Geometry of Microstructures, Habilitation thesis, University of Leipzig, 2003, available at
  • [16] Kirchheim B., Müller S., Šverák V., Studying nonlinear pde by geometry in matrix space, In: Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2003, 347–395
  • [17] Müller S., Šverák V., Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. Math., 2003, 157(3), 715–742
  • [18] Osekowski A., Inequalities for dominated martingales, Bernoulli, 2007, 13(1), 54–79
  • [19] Osekowski A., On relaxing the assumption of differential subordination in some martingale inequalities, Electron. Commun. Probab., 2011, 16, 9–21
  • [20] Osekowski A., Sharp weak type inequalities for the Haar system and related estimates for nonsymmetric martingale transforms, Proc. Amer. Math. Soc., 2012, 140(7), 2513–2526
  • [21] Osekowski A., Sharp logarithmic inequalities for Riesz transforms, J. Funct. Anal., 2012, 263(1), 89–108
  • [22] Osekowski A., Logarithmic inequalities for Fourier multipliers, Math. Z., 2013, 274(1–2), 515–530
  • [23] Pichorides S.K., On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math., 1972, 44, 165–179
  • [24] Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., 30, Princeton University Press, Princeton, 1970
  • [25] Székelyhidi L. Jr., Counterexamples to elliptic regularity and convex integration, In: The Interaction of Analysis and Geometry, Novosibirsk, August 23–September 3, 2004, Contemp. Math., 424, American Mathematical Society, Providence, 227–245
  • [26] Wang G., Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab., 1995, 23(2), 522–551
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.