The Hausdorff–Young inequality on Lie groups

Mathematische Annalen - Tập 375 Số 1 - Trang 93-131 - 2019
Cowling, Michael G.1, Martini, Alessio2, Müller, Detlef3, Parcet, Javier4
1School of Mathematics and Statistics, University of New South Wales, Sydney, Australia
2School of Mathematics, University of Birmingham, Birmingham, UK
3Mathematisches Seminar, Christian-Albrechts-UniversitÄt zu Kiel, Kiel, Germany
4Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain

Tóm tắt

We prove several results about the best constants in the Hausdorff–Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for general Lie groups.

Tài liệu tham khảo

Andersson, M.E.: The Hausdorff–Young inequality and Fourier type. Ph.D. thesis, Uppsala (1993)

citation_journal_title=Acta Math. Sin. (Engl. Ser.); citation_title=Estimate of the

-Fourier transform norm on strong

-regular exponential solvable Lie groups; citation_author=A Baklouti, J Ludwig, L Scuto, K Smaoui; citation_volume=23; citation_publication_date=2007; citation_pages=1173-1188; citation_doi=10.1007/s10114-005-0845-5; citation_id=CR4

Bardet, I., Rouzé, C.: Hypercontractivity and logarithmic Sobolev inequality for nonprimitive quantum Markov semigroups and estimation of decoherence rates. Preprint (2018).

arXiv:1803.05379

citation_journal_title=Ann. Math. (2); citation_title=Inequalities in Fourier analysis; citation_author=W Beckner; citation_volume=102; citation_publication_date=1975; citation_pages=159-182; citation_doi=10.2307/1970980; citation_id=CR6

Bennett, J., Bez, N., Buschenhenke, S., Cowling, M.G., Flock, T.C.: On the nonlinear Brascamp–Lieb inequality. Preprint (2018).

arXiv:1811.11052v1

citation_title=Interpolation Spaces. An Introduction; citation_publication_date=1976; citation_id=CR8; citation_author=J Bergh; citation_author=J Löfström; citation_publisher=Springer

citation_journal_title=Adv. Math.; citation_title=Best constants in Young’s inequality, its converse, and its generalization to more than three functions; citation_author=HJ Brascamp, EH Lieb; citation_volume=20; citation_publication_date=1976; citation_pages=151-173; citation_doi=10.1016/0001-8708(76)90184-5; citation_id=CR9

citation_title=Functional Analysis, Sobolev Spaces and Partial Differential Equations; citation_publication_date=2011; citation_id=CR10; citation_author=H Brezis; citation_publisher=Universitext. Springer

citation_title=Representations of Compact Lie Groups; citation_publication_date=1985; citation_id=CR11; citation_author=T Bröcker; citation_author=T tom Dieck; citation_publisher=Springer

citation_journal_title=Boll. Un. Mat. Ital. B; citation_title=A commutativity condition for algebras of invariant functions; citation_author=G Carcano; citation_volume=7; citation_issue=1; citation_publication_date=1987; citation_pages=1091-1105; citation_id=CR12

citation_journal_title=J. Oper. Theory; citation_title=The

-Fourier transform on locally compact quantum groups; citation_author=M Caspers; citation_volume=69; citation_publication_date=2013; citation_pages=161-193; citation_doi=10.7900/jot.2010aug22.1949; citation_id=CR13

citation_journal_title=J. Funct. Anal.; citation_title=On the spatial theory of von Neumann algebras; citation_author=A Connes; citation_volume=35; citation_publication_date=1980; citation_pages=153-164; citation_doi=10.1016/0022-1236(80)90002-6; citation_id=CR14

citation_journal_title=Int. J. Math.; citation_title=A Hausdorff-Young inequality for locally compact quantum groups; citation_author=T Cooney; citation_volume=21; citation_publication_date=2010; citation_pages=1619-1632; citation_doi=10.1142/S0129167X10006677; citation_id=CR15

citation_journal_title=Can. J. Math.; citation_title=Representing multipliers of the Fourier algebra on non-commutative

spaces; citation_author=M Daws; citation_volume=63; citation_publication_date=2011; citation_pages=798-825; citation_doi=10.4153/CJM-2011-020-2; citation_id=CR17

citation_journal_title=Bull. Soc. Math. Fr.; citation_title=Formes linéaires sur un anneau d’opérateurs; citation_author=J Dixmier; citation_volume=81; citation_publication_date=1953; citation_pages=9-39; citation_doi=10.24033/bsmf.1436; citation_id=CR18

citation_journal_title=J. Funct. Anal.; citation_title=On the regular representation of a nonunimodular locally compact group; citation_author=M Duflo, CC Moore; citation_volume=21; citation_publication_date=1976; citation_pages=209-243; citation_doi=10.1016/0022-1236(76)90079-3; citation_id=CR19

citation_journal_title=Bull. Soc . Math. Fr.; citation_title=L’algèbre de Fourier d’un groupe localement compact; citation_author=P Eymard; citation_volume=92; citation_publication_date=1964; citation_pages=181-236; citation_doi=10.24033/bsmf.1607; citation_id=CR20

Eymard, P., Terp, M.: La transformation de Fourier et son inverse sur le groupe des

$$ax+b$$

d’un corps local. In: Eymard, P., Faraut, J., Schiffmann, G., Takahashi, R. (eds.) Analyse harmonique sur les groupes de Lie (Sém. Nancy-Strasbourg 1976–1978), II. Lecture Notes in Mathematics, vol. 739, pp. 207–248. Springer, Berlin (1979)

citation_title=Harmonic Analysis in Phase Space; citation_publication_date=1989; citation_id=CR22; citation_author=GB Folland; citation_publisher=Princeton University Press

citation_title=A Course in Abstract Harmonic Analysis; citation_publication_date=1995; citation_id=CR23; citation_author=GB Folland; citation_publisher=CRC Press

citation_journal_title=Proc. Lond. Math. Soc.; citation_title=Projectivity of modules over Fourier algebras; citation_author=BE Forrest, HH Lee, E Samei; citation_volume=3; citation_issue=102; citation_publication_date=2011; citation_pages=697-730; citation_doi=10.1112/plms/pdq030; citation_id=CR24

citation_journal_title=J. Math.; citation_title=Sharpness in Young’s inequality for convolution Pacific; citation_author=JJF Fournier; citation_volume=72; citation_publication_date=1977; citation_pages=383-397; citation_id=CR25

citation_title=Abstract Harmonic Analysis of Continuous Wavelet Transforms; citation_publication_date=2005; citation_id=CR26; citation_author=H Führ; citation_publisher=Springer

citation_journal_title=Can. Math. Bull.; citation_title=Hausdorff–Young inequalities for group extensions; citation_author=H Führ; citation_volume=49; citation_publication_date=2006; citation_pages=549-559; citation_doi=10.4153/CMB-2006-052-9; citation_id=CR27

citation_journal_title=Trans. Am. Math. Soc.; citation_title=Sharp Fourier type and cotype with respect to compact semisimple Lie groups; citation_author=J García-Cuerva, JM Marco, J Parcet; citation_volume=355; citation_publication_date=2003; citation_pages=3591-3609; citation_doi=10.1090/S0002-9947-03-03139-8; citation_id=CR28

citation_journal_title=Proc. Lond. Math. Soc. (3); citation_title=Vector-valued Hausdorff–Young inequality on compact groups; citation_author=J García-Cuerva, J Parcet; citation_volume=88; citation_publication_date=2004; citation_pages=796-816; citation_doi=10.1112/S0024611503014527; citation_id=CR29

citation_journal_title=J. Funct. Anal.; citation_title=Les espaces

d’une algèbre de von Neumann définies par la derivée spatiale; citation_author=M Hilsum; citation_volume=40; citation_publication_date=1981; citation_pages=151-169; citation_doi=10.1016/0022-1236(81)90065-3; citation_id=CR30

citation_journal_title=Pac. J. Math.; citation_title=

-Fourier transforms on nilpotent Lie groups and solvable Lie groups acting on Siegel domains; citation_author=J Inoue; citation_volume=155; citation_publication_date=1992; citation_pages=295-318; citation_doi=10.2140/pjm.1992.155.295; citation_id=CR32

citation_journal_title=Int. J. Math.; citation_title=Constructions of non-commutative

-spaces with a complex parameter arising from modular actions; citation_author=H Izumi; citation_volume=8; citation_publication_date=1997; citation_pages=1029-1066; citation_doi=10.1142/S0129167X97000494; citation_id=CR33

citation_journal_title=Int. J. Math.; citation_title=Natural bilinear forms, natural sesquilinear forms and the associated duality on non-commutative

-spaces; citation_author=H Izumi; citation_volume=9; citation_publication_date=1998; citation_pages=975-1039; citation_doi=10.1142/S0129167X98000439; citation_id=CR34

citation_title=A new local variant of the Hausdorff–Young inequality; citation_inbook_title=Complex Analysis and Related Topics; citation_publication_date=2000; citation_pages=107-130; citation_id=CR35; citation_author=A Kamaly; citation_publisher=Basel

citation_journal_title=J. Funct. Anal.; citation_title=Divergence of eigenfunction expansions; citation_author=CE Kenig, RJ Stanton, PA Tomas; citation_volume=46; citation_publication_date=1982; citation_pages=28-44; citation_doi=10.1016/0022-1236(82)90042-8; citation_id=CR36

citation_journal_title=Bull . (N. S.) Am. Math. Soc.; citation_title=Merits and demerits of the orbit method; citation_author=AA Kirillov; citation_volume=36; citation_publication_date=1999; citation_pages=433-488; citation_doi=10.1090/S0273-0979-99-00849-6; citation_id=CR37

citation_journal_title=Math. Ann.; citation_title=Sharp inequalities for Weyl operators and Heisenberg groups; citation_author=A Klein, B Russo; citation_volume=235; citation_publication_date=1978; citation_pages=175-194; citation_doi=10.1007/BF01405012; citation_id=CR38

citation_journal_title=Ann. Sci. École Norm. Sup. (4); citation_title=The Plancherel formula for group extensions. I; citation_author=A Kleppner, RL Lipsman; citation_volume=I; citation_issue=5; citation_publication_date=1972; citation_pages=459-516; citation_doi=10.24033/asens.1235; citation_id=CR39

citation_journal_title=Ann . Sci. École Norm. Sup. (4); citation_title=The Plancherel formula for group extensions. II; citation_author=A Kleppner, RL Lipsman; citation_volume=6; citation_publication_date=1973; citation_pages=103-132; citation_doi=10.24033/asens.1246; citation_id=CR40

citation_title=Lie Groups Beyond an Introduction; citation_publication_date=2002; citation_id=CR41; citation_author=AW Knapp; citation_publisher=Birkäuser

citation_journal_title=J. Funct. Anal.; citation_title=Applications of the complex interpolation method to a von Neumann algebra: noncommutative

-spaces; citation_author=H Kosaki; citation_volume=56; citation_publication_date=1984; citation_pages=29-78; citation_doi=10.1016/0022-1236(84)90025-9; citation_id=CR42

citation_journal_title=Trans. Am. Math. Soc.; citation_title=

Fourier transforms on locally compact unimodular groups; citation_author=RA Kunze; citation_volume=89; citation_publication_date=1958; citation_pages=519-540; citation_id=CR43

Lebedev, N.N.: Special Functions and Their Applications. Revised edition. Translated from the Russian and edited by R.A. Silverman. Dover Publications, New York (1972)

citation_journal_title=C. R. Acad . Sci. Paris Sér. A; citation_title=Sur l’algèbre de Fourier d’un groupe localement compact; citation_author=H Leptin; citation_volume=266; citation_publication_date=1968; citation_pages=1180-1182; citation_id=CR45

citation_journal_title=Invent. Math.; citation_title=Gaussian kernels have only Gaussian maximizers; citation_author=EH Lieb; citation_volume=102; citation_publication_date=1990; citation_pages=179-208; citation_doi=10.1007/BF01233426; citation_id=CR46

citation_journal_title=Bull . Sci. Math. (2); citation_title=Non-abelian Fourier analysis.; citation_author=RL Lipsman; citation_volume=98; citation_publication_date=1974; citation_pages=209-233; citation_id=CR47

citation_journal_title=Anal. Math.; citation_title=Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter; citation_author=C Markett; citation_volume=8; citation_publication_date=1982; citation_pages=19-37; citation_doi=10.1007/BF02073769; citation_id=CR48

citation_journal_title=II. Ann. Math. (2); citation_title=Unitary representations of locally compact groups; citation_author=F Mautner; citation_volume=52; citation_publication_date=1950; citation_pages=528-556; citation_doi=10.2307/1969431; citation_id=CR50

Mitjagin, B.S.: Divergenz von Spektralentwicklungen in

$$L^p$$

-Räumen. In: Butzer, P.L., Sz.-Nagy, B. (eds.) Linear Operators and Approximation, II (Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach, 1974), pp. 521–530. Birkäuser, Basel (1974)

Parcet, J.: A local Hausdorff–Young inequality on the classical compact Lie groups and related topics. In: Ying, L.M. (ed.) Focus on Group Theory Research, pp. 1–25. Nova Sci. Publ., New York (2006)

citation_title=Introduction to Operator Space Theory; citation_publication_date=2003; citation_id=CR53; citation_author=G Pisier; citation_publisher=Cambridge University Press

Pisier, G., Xu, Q.: Non-commutative

$$L^p$$

-spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. North-Holland, Amsterdam (2003)

citation_journal_title=Houst. J. Math.; citation_title=Complex interpolation of weighted noncommutative

-spaces; citation_author=É Ricard, Q Xu; citation_volume=37; citation_publication_date=2011; citation_pages=1165-1179; citation_id=CR55

citation_journal_title=Trans. Am. Math. Soc.; citation_title=The norm of the

-Fourier transform on unimodular groups; citation_author=B Russo; citation_volume=192; citation_publication_date=1974; citation_pages=293-305; citation_id=CR56

citation_journal_title=Pac. J. Math.; citation_title=On the Hausdorff–Young theorem for integral operators; citation_author=B Russo; citation_volume=68; citation_publication_date=1977; citation_pages=241-253; citation_doi=10.2140/pjm.1977.68.241; citation_id=CR57

citation_journal_title=Ann. Math. (2); citation_title=An extension of Plancherel’s formula to separable unimodular groups; citation_author=IE Segal; citation_volume=52; citation_publication_date=1950; citation_pages=272-292; citation_doi=10.2307/1969470; citation_id=CR58

citation_journal_title=Ann. Math. (2); citation_title=A non-commutative extension of abstract integration; citation_author=IE Segal; citation_volume=57; citation_publication_date=1953; citation_pages=401-457; citation_doi=10.2307/1969729; citation_id=CR59

citation_journal_title=Math. Z.; citation_title=Contractive automorphisms on locally compact groups; citation_author=E Siebert; citation_volume=191; citation_publication_date=1986; citation_pages=73-90; citation_doi=10.1007/BF01163611; citation_id=CR60

citation_journal_title=Proc. Am. Math. Soc.; citation_title=A remark on the Hausdorff–Young inequality; citation_author=P Sjölin; citation_volume=123; citation_publication_date=1995; citation_pages=3085-3088; citation_doi=10.1090/S0002-9939-1995-1273525-5; citation_id=CR61

citation_journal_title=J. Math. Kyoto Univ.; citation_title=Plancherel formula for non-unimodular locally compact groups; citation_author=N Tatsuuma; citation_volume=12; citation_publication_date=1972; citation_pages=179-261; citation_doi=10.1215/kjm/1250523567; citation_id=CR62

citation_journal_title=Adv. Oper. Theory; citation_title=

Fourier transformation on non-unimodular locally compact groups; citation_author=M Terp; citation_volume=2; citation_publication_date=2017; citation_pages=547-583; citation_id=CR63

citation_journal_title=J. Oper. Theory; citation_title=Interpolation spaces between a von Neumann algebra and its predual; citation_author=M Terp; citation_volume=8; citation_publication_date=1982; citation_pages=327-360; citation_id=CR64

citation_title=Harmonic Analysis on the Heisenberg Group; citation_publication_date=1998; citation_id=CR65; citation_author=S Thangavelu; citation_publisher=Birkäuser

citation_title=Interpolation Theory, Function Spaces, Differential Operators; citation_publication_date=1978; citation_id=CR66; citation_author=H Triebel; citation_publisher=North-Holland Publishing Co.

citation_title=Linear Operators in Hilbert Spaces; citation_publication_date=1980; citation_id=CR67; citation_author=J Weidmann; citation_publisher=Springer