Hypercontractivity: A Bibliographic Review - Mathematics ...

Hypercontractivity:
A Bibliographic Review
E.8rian Dilvies, iI Leonard Gross b and Barry Simone
a. Department of Mathematics, King's College, The Strand, London \-VC2R 2LS, England
b. Department of Mathematics, \Vhite Hall, Cornell University, Ithaca, NY 14853
c. Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 253-
37, Pasadena, CA 91125; Resea rch partially funded under NSF grant number DMS·8S01918.
370

374 E. Davies et al
§2 Logarithmic Sobolev Inequalities and hypercontactivity
Some aspects of these concepts are most easily understood in finite dimensions. Let
-n/2 1"/ J n
dll(x) = (2'1r) exp[ -lix [- 2 dx denote Gauss measure on R . The inequality
is the prototype o f logarithmic Sobolev inequalities. If one defines an operator N on L2([Rn, p)
by (Nf,g) ') = JRll vf(x) . vg(x)dp(x) then (2.1) reads
L - (p)
n If(xJI2'nlf(xJldp(x) JR :s «Nf.f)" + Ilf ll2? 'n ll fll 2 .
L-(pJ V(PJ L (pJ
In P. Federbush's proof [44J of semibouncleclness of HO+V, he Drst differentiated the
-tH
hypercontractivity inequality for e 0 at t = O to obtain the inequality (2.2) for HO (with R n
replaced by an infinite dimensional space and N replaced by HO) ' He then showed that the
inequality (2.2) was itself sufficient to prove semiboundedness. L. Gross [48] later sho, ... ·ed
that, conversely, one could recover hypercont.ractivity of e· tN
(2.2)
from (2.1) so tllat
hypercontractivity and logarithmic Sobolev inequalities were actually equivalent for Dirichlet
form operators (such as N). The techniques in Gross' a rgument for the direction "log.
-tHo
Sobolev for HO implies bounds on e .. has yielded tremendous advances recently in the
understanding of heat kernels. This will be discussed in the next section. A direct proof of
(2.1) with the best constant c = 1 was also given by G ross [48} using a central limit theorem
argument applied to a logarithmic Soholev-like inequality for an operator on L2 of a two point
measure space. The "two point inequality" was an outgrowt h of his earlier work [49] on
hypercontractivity fo r Fermions but was also anticipated by Bonami [14J in his work on
harmonic analysis on finite groups.

Hypercontractivity 375
There are now many proofs of these two types of inequalities fot Gauss measure. Some
prove hypercont ractivity directly [9, 10, 19, 46 , 74, 75, 76, 77, 91] while others prove the
logarithmic Soholev inequality directly [2, 5, 13. 43, 48, 84]. The most elementary direct proof
of hypercontractivity with bes t constants is that of E. Nelson [76] wbile the most elementary
and simplest direct proof of the logarithmic Soholev inequality with best constants (c = 1) is
is that of O. Rothalls [84].
Both kinds of inequalities were developed in variollS directions in the 1970·s. Inequality
(2.2) can be interpreted as saying that (N + 1)-1/2 is a bounded operator from L2(1l) to the
Orlicz space L2 ln L. G. Feissner showed more generally that (N + 1)-k/2 is a bounded
operator from LP(JJ) to LP ink L. See aha [8}. Furthermore one may ask whether, given a
measure 1/ on jR" with a reasonable density, its Dirichlet form operator satisfies a logarithmic
Soholev inequality. There is a procedure by which Dirichlet form operators arise naturally in
quantum mechanics and quantum field theory; if V is a suitable real valued function on an
then the operator H := -.6. + V is a self-adjoint operator with a unique lowest eigenvector '" of
unit norm whi ch may be taken st.rictly positive. If dll(x) = 1,b(x)2dx and >. = inf spectrum H,
then the unitary operator U : f - !II/; from L2(Rn, dx) to L2(]Rn J dv) converts (H - J.) into
an operator H := UCR - J.)U - l on L2(Jln, dv) which turns out to be a Dirichlet operator
for 1/ [40]. Hence, by Gross' theorem, hypercontractivity and the logarithmic Sobolev
inequality are equivalent for iI. Conditions on V which assure that both hold were obtained
by J . P. Eckmann [40}, R. Carmona [25], J. Rosen [82], J. Hooton [58]. Moreover J. Rosen [82]
showed that if the density of 1/ decreases very quickly at 00, then for 1 < P < 00,
lIe-tHUL2 _ LP < 00 for all t > O. This is strictly stronger than hypercontractivity and was
called supercont ractivity. For a review of studies of H, see B. Simon [94]. An even stronger
notion, ultracontractivity, will be discussed in §3.
An application of hypercontractive ideas in yet another direction was made hy W.

376 E. Davies et al
Beckner [10] who used extensions of tbe above-mentioned t.wo point inequality to get the exact
bounds in the Hausdorff-Young inequality. The study of e- zN , for complex z , from the point of
view of hypercontracti"'ity has recently been completed in a definitive manner by J. Epperson [41].
One may ask whether the Laplace-Beltrami operator on a manifold other than R" generates
a logarithmic Soholev inequality. This has been addressed in a number of works [32, 33, 34, 35, 87, 89]
..... ith startlingly complete resuiLs for SO in [1061. Finally we mention that applications of
logarithmic SoboleY inequalities in infinite dimensions to statistiea] mechanics were made by Holley
and Stroock [55, 56, 57J. The bibliography contains many more works which touch on hypercontractivity
or logarit.hrruc Soboley inequalities in one way or another and not described here or below.

Hypercontractivity 379
[30, 33, 35J. See also [31, 42, 21 , 34], and [33] for an analogous result for certain second order
hypoelliptic operators. By comparison with earlier literature (3, 81] the advantage of (3 .4) is
that one has obtained the sharp constant 4 in the exponential.
The possibility of obtaining sharp cor.stants is a recurring feature of the use of
logarithmic Sobolev inequalities, and demonstrates their deep significance. Many developments
of the above applications are being currently investigated, and one particularly looks forward
to the proof of lower bounds on heat kernels which are of comparable accuracy to the upper
bounds mentioned above.

380 E. Davies et al
References
[I] R . A. Adams, General logarithmic Sobolev inequalities and Orlicz imbeddings. J. Funct.
[2)
An al. II (1979 ), 292-303.
_______ and F. H. Clarke, Gross's logarithmic Sobolev in equality: a simple proof,
Amer. J. Math. ill (1979 ), 1265-1270.
[3] D. G. A ronson, Non-negative solutions of linear parabolic equations, Ann. Sci. ;\orm. Sup.
Pi,. (3) 22 (1968), 607-694 .
[4] Joel Avrim, Perturba tion of Logarithmic Sobolev generators by electric and m agnetil:
potent ials, Berkeley Thesis 1982.
[5) D. Bakry, Une remarque ,ur les dirr"'on' hyperoo",a",,,e,, P repr,nt Nov. 1987,3 pp.
[6J D. Bakry and M. Emery, Hypercontractivite de semi-groupes de diffus ion, Compte-Re ndus
[7)
299 (1984), 775-778.
_ _ _ _____ • Diffusions hypercontractives, pp. 177-207 in Sem . de P robabilites X IX.
Lecture Notes in Mathematics #1123, eds. J . Azema a nd M. Yor, Springe r' Verlag,
New York, 1985.
[8] D. Bakry and P. A. Meyer, Sur les inequalities de Sobolev logarithmique I &. II Preprints,
Vniv. de Strasbourg 1980/81.
(9] W . Beckner. Inequalities in Fourier Analysis on R n , P roc. Nat. Acad. Sci. U.S .A . 72
(1975),638-641.
[10] _______ _ , Inequalities in Fourier A nalysis, Ann. of Math . 102 ( 1975), 159·1$2.
Ill) ________ , S. Janson and D. Jerison, Convolution inequalities on the circle,
Conference on ha r monic analysis in honor of Anton i Zygm u nd , \Vadsworth,
Belmont 1983. vol. 1, 32·43.
[12J Iwo Bialynicki.Birula a nd J erzy Mycielski, Uncertainty relations for information ent·rapy in

384 E. Davies et al
(1972),52-109.
[50] _______ , Hypercontractivity and logarith mic Sobolev inequalities fo r the Clifford-
Dirichlet form. Duke Math. J. 42 ( 1975). 383-396.
[51] O. Gue nnoun, Inegalites logarithmiques de Gross-Sobolev et hypercontractivite • T hese
3eme cycle, Universite Lyon 1 ( 1980 ). 90 pages.
[52) F. Guerra, L. Rosen and B. Simon, The P (4i)2 Euclidean quantum field theory as classica l
statistical mechanics. Ann. Math . .!Ql (1975), 111·259.
[53] ______ _ , Boundary condit-ions for the P (¢)2 Euclidean quantu m field tlleory ,
Ann. Inst. H. Poincare, 2..Q.A (1976), 231-334.
[54] R. Hoegh-Krohn, A general class of quantum field s without cutoffs in t wo space-time
dimensions, Commun. Math. Phys. 21 (1971). 224-255.
[55J R. Holley and O. Stroock. Diffusions on an infinite dimensional torus. J. Funct. Anal. 42
[56J
(1981),29·63.
________ , Logarithmic Sobolev inequalit ies and stochastic Ising model.:;;, J. of
Stat istical Physics 1§. (1987).1159-1194.
[57J _ _______ , Uniform and L2 convergence in one dimensional stochastic Ising models,
M.LT. preprint, Sept. 1988,8 p p .
[58J J. G. Hooton, Dirichlet (o rms associated with hypercontractive semigroups, Trans. A.;\-\'S.
ill (1979), 237-256.
[591 ____ ____ , Dirichlet semigroups on bounded domains, Rocky Mountain J . r.,·fath.
12 (1982), 283-297.
[60] _ ______ , Compact Sobolev imbeddings on finite m easure spaces, J. Math. Anal.
and Applications a.;l (1981), 570-581.
[6 1) S. Janson, On hypercont ractivity for multipliers on orthogonal poly nomials [Best constant
for Laplacian on S2 J Ark. Mat. 21 ( 1983),97-110.

Hypercontractivity 389
[1091 A. Klei n, Self-adjointness of the locally coned generators of Lorentz transformations
for P (lPh, in Mathematics of Contemporary Physics, ed R. F. Streater,
Academic Press, London (1972), 221-236.
[110] _ _ ___ Quadratic expressions of a free Boson field , 'llans. Amer. t.'lat.h. Soc. ill (1973) ,
439-456.
[Ill] A. Klein and L. J. Landau, Construction of a unique self·adjoint generator for a symmetric
local semigroup, J. F1mct. Anal. .11 (1981),121-137.
[112] _ __________ , Singular perturbations of positivity preserving semigroups via path
space techniques, J. Funet. Anal. .2..0. (1975), 44-82.
[113) A. Klei n, L. J. Landau and D. Shucker, Decoupling inequalities for stationary Gaussian
processes, Annals of Probability l.Q (1982), 702- 708.