Hypercontractivity: A Bibliographic Review

Hypercontractivity:
A Bibliographic Review
E.8rian Dilvies, iI Leonard Gross b and Barry Simone
a.
Departmen t of Mathematics, King's College, The Strand, London \-VC2R 2 LS, England
b.
Department of Mathematics, \Vhite Hall , Cornell University, Ithaca, NY 14853
c. Division of Physics , Mathematics, and Astronomy, California Institute of Technology, 25337, Pasadena, CA 91125; Resea rch partially f unded under NSF grant number DMS·8S01 918.
370
Hypercontractivi ty
371
§l. Overview and Self-Adjojntness
For the past twen t y years, a set of ideas known as hypercontractivi ty has played a
continu ing role in analysis with ramification s in quantum field theory, self-adjointn ess of
Schrodinge r operators, best constants in classical inequalities and bounds on se migroup kernels.
A signi ficant role was played by a paper of Raphael Hoegh-Krohn an d one of us 198] w hich
codified previous work and coined the term "'hypercontractive."
Our goal here is t o gi\"e a
brief historical review and a rathe r complete bibliography.
In looking at th e history , one must bear in mind that for several papers there \\-ere
len gthy delays between submission and publication, roughly two years for Simon and HoeghKrohn [981 and Gross [481.
The theory of hypercontractive semigroup s was int roduced in a funda mental paper of
Nelson [i4J, who also discove red the simplest and most basic example.
Definition Let (O,/Jo) be a probability measure space. Ho ~ 0 is a self-adjoint operator
on L2 ( O,J.! ). \\Ie say that e(a)
e-
tHo
tHo
(t~O) is a hvpercont ractjve semigroyp if and only if:
is a contraction on L"( O.J.!o) fo r all t>O.
General p rinciples
(interpolation and duality) imply that e-
tHo is then a contraction
from any L P to itself and bounded from any L P to a"ny L Q(l <p,q< oo) if t is sufficiently large
(depending on p ,q ).
Example CNelson (74D
dx).
Let 0 0
Let H
~ , the harmonic oscillator on L 2(R ,
(1!yl/4 exp ( _ ~x2 ) be the unique positive unit vector with HO o
a and
E. Davies et al
372
consider
Ho
n o- 1 Hn 0 -- 2
l( - ..A..
d) . Then edx 1 + ?x
- dx
tHo
is h)'percontractive on L2 (R. n 0 2 dx) .
Nelson applied this result to deduce semiboundedness of certain cutoff quantum field
Hamiltonians.
Glimm
[461
hypercontractive. If HoI
U (m, 00), then e
-THo
=
made an
important
observati on :
Suppose
0, zero is a simple eigenvalue and for some m
is actually a contraction from L
2
to L
4
>
that e-
tHo
IS
0: u(Ho) c {OJ
for T sufficiently la rge.
This
allows one to extend Nelson's example to t he second quantization of any strictly positive
operator (see [98J).
Nelson implemented his semiboundedness proof by making extensive use of path
integrals.
An alternative prool of semiboundedness was given by P. Feder bush (44 } based on
differentiating the already established hypercontractive inequality lIe-tHfl1p(t) ~
respect to t at t = O.
IIfll2
with
This yielded the first log Sobolev inequalities, provid ing a precurso r to
the work we will discuss in
S2.
Segal (90,91,92] studied an abstract version of the theory, and in particular showed
that essential self-adjointness followed f rom the same L P properties that Nelson used. Simon Hoegh -Krohn [98] codified and extended this work, and. in particular show that :
Theorem 1; Suppose that e
- tHo
2
is a hypercontractive semigroup on L (O,jJ).
a function on (} (and also the associated multiplication operator). Suppose that e- v E
and VE L P for some p>2. Then Ho
+
Let V be
n
L
p<ox-
P
V is bounded from below and essentially self-adjoint on
O ( RO) n D(Y) . Th e same result is true ifY 2: 0 and V E L2.
At the time of this work, self-adjointness of t he spatially cut off quantum field
Hamiltonian was important in the construction of infinite volume asymptotics of the dynamics.
It was first proven for (¢4h field theories by Glimm-Jaffe [47] using different methods and
373
Hyperconrractivity
then by Rosen (83] and Segal [91] for general P(¢h theories. Segal used variants of the above
theo rem. Hoegh-Kroh n [54} applied it to the: exp ( aQ ): interaction.
A second example of some historical significance is in the Simon-Hoegh-Krohn paper
[98[.
Let
V2.0
argument, -A
,
be in L2 ( Rd , e- x dx).
+ x 2 + Vex)
Then, by th e above theorem and a small additional
is essent ially self-adjoint on C?,( Rd). This is of interest because
prio r to this, all such t heorems had required V to be iocally L P with
Simon [96] showed how to get rid of the x 2 and prove -~
if V
2:
ax2
0 in L 2(R d , e- dx)'some a>O.
+
p 2: dj2
if d 2: 4.
V essen tially self adjoint on
cf'
Motivated by this, in a cel ebrated work , Kata {62J,
using differen t methods, showed that V E L 2 10,(R d ) suffices if V?:D.
It is interesting that the se lf-adjoint ness result is now mainly of histori cal interest.
Kato's work has replaced any application to Schrodinger operators.
And, because of the
Euclidean re\'olution in quantum field theory, self-adjointness became irrelevant, although the
L P estimates associated to hypercontractivity are still significant; see Guerra et al. [52,53}.
374
E. Davies et al
§2 Logarithmic Sobolev Inequalities and hypercontactivity
Some aspects of these concepts are most easily understood in finite dimensions .
dll(x)
= (2'1r) -n/2 exp[
Let
-lix 1"/
[- 2 Jdx denote Gauss measure on R n . The inequality
is th e 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)
JR n If(xJI2'nlf(xJldp(x) :s «Nf.f)"L-(pJ + Ilf ll 2?V(PJ 'n ll fll L 2(pJ .
In P.
Federbush's proof [44J of semibouncleclness of HO+V, he Drst diffe rentiated the
-tH
hypercontractivity inequality for e
0 at t = O to obtain the inequali ty (2 .2 ) for HO (with R n
r eplaced by an infinite dimensional space and N replaced by HO) '
inequality (2 .2 ) was itself sufficient to prove semiboundedness.
that,
(2.2)
conversely,
one
could
recover
hypercont.ractivity
He then showed that the
L. Gross [48] later sho,...·ed
of e· tN
from
(2.1)
so
tllat
hypercontractivity and logarithmic Sobolev inequalities were actually equivalent for Dirichlet
form operators (such as N).
The techniques in Gross'
Sobolev for HO implies bounds on e
understanding of heat kernels.
(2.1) with the best constant c
-tHo
a rgument for the direct ion "log.
.. has yielded tremendous advances recently in the
This will be discussed in the next section.
=
A direct proof of
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 outgrow t h of his earlier work [49] on
hypercontractivity fo r Fermio ns bu t was also anticipated by Bonami [14J in his work on
harmonic analysis on finite grou ps .
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.
(2.2) can be interpreted as saying that (N
Orlicz space L2 ln L.
See aha [8}.
+
1)-k/2 is a bounded
Furthermore one may ask whether, given a
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
1/
on
is a bounded operator from L2(1l) to the
G. Feissner showed more generally that (N
operator from LP(JJ) to LP ink L.
measure
+ 1)-1/2
Inequality
quantum mechanics and quantum field theory; if V is a suitable real valued fun ction on
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.
then the unitary operator U :
an operator
for
1/
[40].
H
an
f -
If dll(x) = 1,b(x)2dx and >. = inf spectrum H,
!II/; from L2(Rn,
dx) to L2(]Rn J dv) converts (H - J.) into
:= UCR - J.)U - l on L2(Jln, dv) which turns out to be a Dirichlet operator
Hence,
by Gross'
inequali ty are equivalent for
iI.
theorem,
hypercontractivity
and
Conditions on V which assure that both hold we re obtained
by J . P . Eckman n [40}, R. Carmona [25], J. Rosen [82], J. Hooton [58].
showed that if the density of
lI e-tHUL2 _ LP <
00
for all t
called supercont ractivity.
the logarithmic Sobolev
1/
>
decreases very quickly at
O.
00,
Moreover J. Rosen [82]
then for 1
< P <
00,
This is strictly stronger than hypercontractivity and was
For a review of studies of H, see B. Simon [94].
An even stronger
notion , ultracontractivity, will be discussed in §3.
An application of hyp ercontractive ideas in yet another direction was made hy W.
E. Davies et al
376
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- z N , 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
377
§3 Ultracontractivity
?
It was shown in [28] that if H = -
<
b
<
00,
d -,)
dx-
+
Von L2(a, b) where V E Ll and -
then the semigroup e- tA is bounded from L:!(dv ) to Loo(dv) for all
t
>
= <
O.
a
T hi s
. .
. eqUlva
. 1en t t 0 e -tf! h
property .IS ca11 e d u 1tracontractlvlty
an d IS
aVLng·
a ·
pOlntv.iise bou nde d
integral kernel K(t, x, y) for all t
>
O.
A more general investigation of such inequalities from the point of view of logarit hmi c
Sobolev inequalities was initiated by Davies and Simon in [38].
If ( 0. p) is a measure space
then there is a very dose relationship between bounds of the form
(3 .1 )
for aU t
>
0, and
+8(,)
for all f EDam ( H
1/2
) and all
f
>
II f l l~
+
II r I I ~
log
II r ll 2
( 3 .2)
O. In particular one has an equivalence between (3.1) and
(3.2) in the case wh ere
e(t) = c t
-N/4
( 3.3)
and
Although theorems of this type are not applicable to the harmonic oscillator, th ey are
important for some other Schrodinger operators H =
if A
>
0 and
-
fl.
+
V on L 2 (R n ),
For example
E. Davies et aI
378
then e - tH is ultracontractive if and only if).
much mo re rapidly than (3.3).
>
I, with constants c( t) which diverge as
0
t -
The harmonic oscillator therefore stands at the borderline
between ultracontractive behavior and the absence of any L2 -
L P smoothing properties.
It was subsequently shown by Varopoulos [102] and Fabes and Stroock [42] that less
singu la r p roblems of this type, for which ( 3.3) is valid, could be handled by the use of ordinary
Sobolev inequalities or what were called Nash ineq ualities. \'\ 'e re mark that the paper of Nash
[73] involves "entrop), inequalities" which antedate the in troduction of logarithmic Soholev
inequalities by a decade.
The paper {3S] has s pawned a substantial literat u re concerning pointwise bounds
Oil
the
heat kernels of second order elliptic ope rato rs in d ive rgence form; see[34] for a comprehensive
survey.
One can easily handle uniformly elliptic operators with measurable coefficients on R n
and on regions in Rn subject to Dirichlet or Neumann boundary conditions.
The use of
logarithmic Sobolev inequalities also allows one to obtain pointwise bounds on the heat kernels
of many singular elliptic operators of second order (32,78].
\Vhen one turns to the study of the Laplace- Beltrami operator o n a Riemannian
manifold, many different techniques can be used to obtain pointwise bounds on the heat kernel
(34] . Some [26, 71, 36] make little or no use of logarithmic Soholev inequalities , while others
(32, 33, 35] depend essen tially upon such a use.
Suppose one has a uniform hound on the heat kernel of the form (3.1), or equivalently
o
~
K(t, x, y)
~
a Ct)
<
00.
By proving a logarithmic Soholev inequ ality for the non-self-adjoint operator", e- Ht ", - 1,
wh ere IfJ is a suitable weight, it is often possible to show that
0::; K(t, x, y) ::; ',it) exp (- d(x, y)2/(4
for any 6
>
+ 0) tl
(3.4 )
0, where d (x, y) is a Riemannian metric constructed di rectly from the operator
379
Hypercontractivity
[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 th e above applications are being currently investigat ed, 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
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