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Sharp logarithmic inequalities for Riesz transforms. (English) Zbl 1247.42016

The author gives a detailed discussion about sharp logarithmic inequalities for Riesz transforms. The author establishes the following inequality and gives the optimal constant \(L\) \[ \int_A |R_jf(x)|dx\leq K\int_{\mathbb{R}^d}\Psi(|f(x)|)dx+|A|\cdot L, \] where \(R_j(j=1,2\dots,n)\) denotes Riesz transform, \(A\) denotes any Borel subset in \(\mathbb{R}^d\), \(K>\frac{2}{\pi}\) and \(\Psi(t)=(t+1)\log(t+1)-t, (t\geq 0)\). The proof is based on probabilistic techniques and the existence of certain special harmonic functions. At last he also gives related sharp estimates for the so-called re-expansion operator as an application.
The reviewer considers this paper to be interesting and deep.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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[1] Bañuelos, R.; Wang, G., Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations, Duke Math. J., 80, 575-600 (1995) · Zbl 0853.60040
[2] Bañuelos, R.; Wang, G., Sharp inequalities for martingales under orthogonality and differential subordination, Illinois J. Math., 40, 678-691 (1996) · Zbl 0883.60042
[3] Birman, M. S., Re-expansion operators as objects of spectral analysis, (Linear and Complex Analysis Problem Book. Linear and Complex Analysis Problem Book, Lecture Notes in Math., vol. 1043 (1984), Springer), 130-134
[4] Burkholder, D. L., Explorations in martingale theory and its applications, (École dʼEte de Probabilités de Saint-Flour XIX—1989. École dʼEte de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464 (1991), Springer: Springer Berlin), 1-66 · Zbl 0771.60033
[5] Davis, B., On the weak type \((1, 1)\) inequality for conjugate functions, Proc. Amer. Math. Soc., 44, 307-311 (1974) · Zbl 0259.42016
[6] Dellacherie, C.; Meyer, P.-A., Probabilities and Potential B: Theory of Martingales (1982), North-Holland: North-Holland Amsterdam
[7] Gokhberg, I.; Krupnik, N., Norm of the Hilbert transformation in the \(L_p\) space, Funct. Anal. Pril.. Funct. Anal. Pril., Funct. Anal. Appl., 2, 180-181 (1968), (in Russian); English transl.:
[8] Gokhberg, I.; Krupnik, N., Introduction to the Theory of One-Dimensional Singular Integral Operators (1973), Izdat. “Štiinca”: Izdat. “Štiinca” Kishinev, (in Russian)
[9] Gundy, R. F.; Silverstein, M., On a Probabilistic Interpretation for Riesz Transforms, Lecture Notes in Math., vol. 923 (1982), Springer: Springer Berlin/New York · Zbl 0495.60053
[10] Gundy, R. F.; Varopoulos, N. Th., Les transformations de Riesz et les intégrales stochastiques, C. R. Acad. Sci. Paris Sér. A-B, 289, A13-A16 (1979) · Zbl 0413.60003
[11] Hollenbeck, B.; Kalton, N. J.; Verbitsky, I. E., Best constants for some operators associated with the Fourier and Hilbert transforms, Studia Math., 157, 237-278 (2003) · Zbl 1016.42006
[12] Ilʼin, E. M., Scattering characteristics of a problem of diffraction by a wedge and by a screen, Investigations on Linear Operators and the Theory of Functions, X. Investigations on Linear Operators and the Theory of Functions, X, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 107, 193-197 (1982), (in Russian) · Zbl 0504.47012
[13] Ilʼin, E. M., The scattering matrix for a problem of diffraction by a wedge, (Operator Theory and Function Theory, vol. 1 (1983), Leningrad University), 87-100, (in Russian) · Zbl 0571.47009
[14] Iwaniec, T.; Martin, G., The Beurling-Ahlfors transform in \(R^n\) and related singular integrals, J. Reine Angew. Math., 473, 25-57 (1996)
[15] Janakiraman, P., Best weak-type \((p, p)\) constants, \(1 \leqslant p \leqslant 2\) for orthogonal harmonic functions and martingales, Illinois J. Math., 48, 3, 909-921 (2004) · Zbl 1063.31002
[16] Kolmogorov, A. N., Sur les fonctions harmoniques conjugées et les séries de Fourier, Fund. Math., 7, 24-29 (1925) · JFM 51.0216.04
[17] A. Osȩkowski, On the best constants in the weak type inequalities for re-expansion operator and Hilbert transform, Trans. Amer. Math. Soc., http://dx.doi.org/10.1090/S0002-9947-2012-05640-6; A. Osȩkowski, On the best constants in the weak type inequalities for re-expansion operator and Hilbert transform, Trans. Amer. Math. Soc., http://dx.doi.org/10.1090/S0002-9947-2012-05640-6 · Zbl 1319.42018
[18] Pichorides, S. K., On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math., 44, 165-179 (1972) · Zbl 0238.42007
[19] Riesz, M., Sur les fonctions conjugées, Math. Z., 27, 218-244 (1927) · JFM 53.0259.02
[20] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0207.13501
[21] Wang, G., Differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities, Ann. Probab., 23, 522-551 (1995) · Zbl 0832.60055
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