×

The boundedness of the Hilbert transformation from one rearrangement invariant Banach space into another and applications. (English) Zbl 1471.46027

Summary: In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46L51 Noncommutative measure and integration
46L52 Noncommutative function spaces
44A15 Special integral transforms (Legendre, Hilbert, etc.)
47L20 Operator ideals
47C15 Linear operators in \(C^*\)- or von Neumann algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arazy, J., Some remarks on interpolation theorems and the boundedness of the triangular projection in unitary matrix spaces, Integral Equ. Oper. Theory, 1, 4, 453-495 (1978) · Zbl 0395.47030
[2] Bennett, C.; Sharpley, R., Interpolation of Operators, Pure and Applied Mathematics, vol. 129 (1988), Academic Press · Zbl 0647.46057
[3] Birman, M.; Solomyak, M., Spectral Theory of Selfadjoint Operators in Hilbert Space, Mathematics and Its Applications (Soviet Series) (1987), D. Reidel Publishing Co.: D. Reidel Publishing Co. Dordrecht · Zbl 0744.47017
[4] Boyd, D. W., The Hilbert Transformation on rearrangement invariant Banach spaces (1966), University of Toronto, Thesis
[5] Boyd, D. W., Indices of function spaces and their relationship to interpolation, Can. J. Math., 38, 1245-1254 (1969) · Zbl 0184.34802
[6] Boyd, D., The Hilbert transform on rearrangement-invariant spaces, Can. J. Math., 19, 599-616 (1967) · Zbl 0147.11302
[7] Calderón, A. P., Spaces between \(L^1\) and \(L^\infty\) and the theorem of Marcinkiewicz, Stud. Math., 26, 273-299 (1966) · Zbl 0149.09203
[8] Carro, M.; Pick, L.; Soria, J.; Stepanov, V. D., On embeddings between classical Lorentz spaces, Math. Inequal. Appl., 4, 3, 397-428 (2001) · Zbl 0996.46013
[9] Caspers, M.; Potapov, D.; Sukochev, F.; Zanin, D., Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture, Am. J. Math., 144, 593-610 (2019) · Zbl 07069612
[10] Gohberg, I. C.; Kreı̌n, M. G., Introduction to the Theory of Linear Non-selfadjoint Operators, Transl. Math. Monogr., vol. 18 (1969), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I. · Zbl 0181.13503
[11] Gohberg, I. C.; Kreı̌n, M. G., Theory and Applications of Volterra Operators on Hilbert Spaces, Transl. Math. Monogr., vol. 24 (1970), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I. · Zbl 0194.43804
[12] Kalton, N.; Sukochev, F., Symmetric norms and spaces of operators, J. Reine Angew. Math., 621, 81-121 (2008) · Zbl 1152.47014
[13] Krein, S.; Petunin, Y.; Semenov, E., Interpolation of Linear Operators (1982), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I. · Zbl 0493.46058
[14] Lindenstrauss, J.; Tzafiri, L., Classical Banach Spaces (1979), Springer-Verlag, I, II · Zbl 0403.46022
[15] Lord, S.; Sukochev, F.; Zanin, D., Singular Traces. Theory and Applications, De Gruyter Studies in Mathematics, vol. 46 (2013), De Gruyter: De Gruyter Berlin · Zbl 1275.47002
[16] Meyer-Nieberg, P., Banach Lattices (1991), Springer-Verlag · Zbl 0743.46015
[17] de Pagter, B.; Witvliet, H.; Sukochev, F., Double operator integrals, J. Funct. Anal., 192, 52-111 (2002) · Zbl 1079.47502
[18] Potapov, D.; Sukochev, F., Lipschitz and commutator estimates in symmetric operator spaces, J. Oper. Theory, 59, 1, 211-234 (2008) · Zbl 1174.46031
[19] Sadovskaya, O.; Sukochev, F., Isomorphic classification of \(L_{p , q}\)-spaces: the case \(p = 2, 1 \leq q < 2\), Proc. Am. Math. Soc., 146, 3975-3984 (2018) · Zbl 1406.46021
[20] Soria, J.; Tradacete, P., Optimal rearrangement invariant range for Hardy-type operators, Proc. R. Soc. Edinb., 146A, 865-893 (2016) · Zbl 1354.26026
[21] Sukochev, F., Completeness of quasi-normed symmetric operator spaces, Indag. Math. (N. S.), 25, 2, 376-388 (2014) · Zbl 1298.46051
[22] Sukochev, F.; Tulenov, K.; Zanin, D., The optimal range of the Calderón operator and its applications, J. Funct. Anal., 277, 10, 3513-3559 (2019) · Zbl 1437.46036
[23] Sukochev, F.; Tulenov, K.; Zanin, D., Corrigendum to the paper “The optimal range of the Calderón operator and its applications”, J. Funct. Anal., 277, 10, 3513-3559 (2019) · Zbl 1437.46036
[24] Tulenov, K. S., The optimal symmetric quasi-Banach range of the discrete Hilbert transform, Arch. Math., 113, 6, 649-660 (2019) · Zbl 1436.46023
[25] Tulenov, K. S., Optimal rearrangement-invariant Banach function range for the Hilbert transform, Eurasian Math. J., 12, 1, 1-16 (2021), (in press)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.