CC BY
13Вестник Сыктывкарского университета. Сер Л. Вып.И.2010
УДК 512.8
THE GROWTH OF I/-NORMS IN PRESENCE OF LOGARITHMIC SOBOLEV INEQUALITIES1 2
Sergey G. Bobkov
The growth of Lp-norms is considered under various hypotheses, including LSq (logarithmic Sobolev) inequalities.
1. The graph setting
Let G = (V, £) be a connected, non-oriented, finite graph. It will be equipped with the uniform probability measure assigning the mass 1/card(V) to each point of V, where card(V) is the number of vertices.
If (x,y) E we will say that the points x and у are neighbours and write у ~ x. With any real-valued function / on V, associate the modulus of the gradient |V/| > 0, defined on V by
ivf(X)i2 = ]Tim-mi2-
Various relations between the distributions of / and |V/| under the measure ц are the subject of the theory of Sobolev inequalities on finite graphs, which may be included as part in the framework of finite Markov chains, cf. e.g. [SC], [L2], [B-T], [G-M-T], [М-Т]. Of the most interest are Poincare-type and logarithmic Sobolev inequalities
AiVar(/)< J | Vjfd/i, (1.1)
pEnt(/2) < 2 J |V/|2 d/i, (1.2)
^^Key words: Poincare-type inequalities, logarithmic Sobolev inequalities, finite graphs
2 Supported in part by NSF grant
© Sergey G. Bobkov, 2010.
which are supposed to hold for all / : V —> R. Here, Var(/) = J f2 dp — (f f dp)2 stands for the variance of /, and
for the entropy of /2 under p. An optimal constant Ai > 0 is called the spectral gap, while an optimal value p > 0 is referred to as the logarithmic Sobolev constant of G. It is well-known that p < Ai, which immediately follows from the relation
One of the interesting problems is how to relate the Lp-norms \\f\\p — (f \f\P dp)1^ of the functions on V to the Lp-norms of their moduli of the gradients
Usually, one considers a more specific problem about the probability
of large deviations of / from the mean E/ = J f dp in the class of Lipschitz functions. See e.g. [A-M], [Ll-2], and references therein. A closely related question is the one about the rate of the growth of Z/p-norms ([R2]). Note that in the graph setting the Lipschitz property may be understood in many senses. As a natural choice, the value ||/||LiP = max^y |V/(rr)| is called the Lipschitz constant of /, and one says that / is Lipschitz, if ||/||LiP < 1-
However, often the Lipschitz constant does not reflect many properties of the distributions of functions, and it is desirable to have more sensitive bounds for large deviations. For example, in presence of the logarithmic Sobolev inequality (1.2), one always has
for any / on V with p-mean zero ([B-G]). This can also be formulated in terms of the Orlicz norms || • generated by the Young functions ipait) — e'4'" — 1 with a — 1 and a — 2. Let us recall that
lim Ent ((/ + C)2) = 2 Var(/).
p{\f-Ef\>r}, r> 0,
(1.3)
Put also ||V/||^ = || |V/| Then, by the very definition, (1.3) yields the relation
||/lk<^||V/|U. (1.4)
It would be interesting to study whether the converse implication is true, as well. One may also wonder how to refine the inequality (1.4) in terms of ZAnorms. Indeed, as is known and easy to see (see Appendix), up to some absolute constants c1,c2 > 0,
Ci sup liili^ < ll/ll^ < c2 supMI^ (1.5)
P> i P P> I P
while
Cl sup№< 11/11 <C2SupMk. (L6)
P> 1 y/P p> 1 y/P
Hence, as (1.4) suggests, one may expect that ||/||p may be bounded by || V/||p with factors growing like y/p. For the first time, an observation of this kind was apparently made in the 1994 work by S. Aida and D. Stroock [A-S] for an abstract scheme of Markov transition kernels. They also considered defective logarithmic Sobolev inequalities
PEnt(/2) < 2 (E/, /) + /5 J |V/|2 d/i,
where (E/, /) is a given Dirichlet form, and where an additional term /3 f | V/|2 d/ji is referred to as defect.
The aim of this note is to adapt the approach of [A-S] to the graph setting as above and to the setting of abstract metric spaces with local moduli of the gradients. We will also see that more general hypotheses in comparison with (1.2), such as, the so-called LS^-inequalities may be involved in a similar analysis.
2. The growth of Lp-norms on finite graphs
Keeping the same notations as in Section 1, the following theorem holds.
Theorem 2.1. For any function f on V and any p > 2,
11/11? - 11/111 <" /V/||?df, (2.1)
P J 2
where p is the logarithmic Sobolev constant. In particular,
l-\\f\\l (2.2)
Essentially, this statement is a variation of Theorem 3.4 in [A-S] (in the case where there is no defect).
Note that, dividing the inequality (2.2) by p — 2 and letting p —>> 2, one arrives at the logarithmic Sobolev inequality (1.2) with an additional factor 2 on the right-hand side. In this sense, (2.2) and (1.2) are almost equivalent.
Under an additional mean assumption E/ = 0, the term ||/||2 in (2.2) may further be estimated by using the Poincare-type inequality (1.1). More precisely, it gives
A\ p
Hence, we also have:
Corollary 2.2. For any function f on V with ¡i-mean zero,
11/11, < J^f^ IIV/II,. (2.3)
Let us remind the argument of [A-S], which goes back to the seminal work of L. Gross [G]. It is based on the general formula
|ll/llp = ^ll/rPEnt(|/n, p>0, (2.4)
holding for any measurable function / on an arbitrary probability space, such that 0 < ||/||p+£ < +oo, for some e > 0. In the graph setting the latter just means that / is not identically zero. Therefore, for all p > 0,
|ll/lli = ^ll/rpEnt(|/n. (2.5)
Let p > 2 and ||/||p > 0. According to the hypothesis (1.2), applied to the function we may write
pEnt(|/n<2 J\V\f\p/2\2 dp. (2.6)
Here the right-hand side is
2 [Y,(\f(x)\P/2-\f(y)\P/2)2 =
4 EE(i/wr/2-i/wr/2)21<
X yr^X
"tzttttv LL(l/Wr-l/(i/)rj 1{|/(X)|
^ ' T' Hr^jnf
Since ar — br < rar 1 (a — 6), for all a > b > 0 and r > 1, the expression inside the double sum is estimated by
j (I/O*)I - \f(v)\)21mr2 i{\m\>\M\} < j (f(x) - f(y))21f(x)r2.
Thus,
2J\v\f\p,2\2¿a < ^yEEi№)-/W)2l/Wr2
= P2 J\Vf(x)\2\f(x)r2dfi(x),
and by (2.6),
Ent(|/n < ¿J |V/|2|/r2^.
The last integral may further be bounded by virtue of Holder's inequality with the exponents | and (§)* = and we get
Ent(|/n < ^||V/||J||/||r2.
Hence from (2.5) we obtain a differential inequality ^ ||/||p < ~p ||V/||^, which can easily be integrated to yield (2.1).
3. LS^-inequalities
In the literature one can find a variety of more general or modified forms of the logarithmic Sobolev inequality (1.2), which have been introduced and analysed for different aims; cf. e.g. [L2-3], [B-T]. Here we consider one natural generalization, which in the graph setting as above takes the form
Entd/I9) < Cqq j \Vf(x)\ldn(x), (3.1)
where 1 < q < 2 is a fixed parameter, Cq is a positive constant (independent of the function / on V), and where
/ \1/9 |v/(®)|, = ( £l/(s)-/(y)N ' (3-2)
y^x '
Definition. The inequality (3.1) will be called a logarithmic Sobolev inequality with the power parameter g, or for short, an LS^-inequality (with constant Cq).
Under the same name a similar family of analytic inequalities for local gradients was treated in [B-Z] to study super-Gaussian tails for distributions of Lipschitz functions. As it turns out, (3.1) may also be used to obtain improved rates for the growth of ZAnorms in comparison with the particular case q — 2 of Theorem 2.1.
Indeed, given p > q, and starting with the identity (2.4), we get
^ 11/11? = jHI/lir^a/n* (3-3)
where one should assume that ||/||p > 0. On the other hand, by (3.1), applied to the function \f\p^q, we have
Ent(|/n < CI J \V\flP/q\qqdp
y~x
2 Cq
° EEd/^l^-l/^W^I/WW/to)!)-
card(y)
V 7 X y~x
The expression inside the double sum can be bounded by
(\f(x)\-\f(y)\)q\f(x)\p~qimx)\>\fm < (^)'lf(x)-f(v)\q\f(x)r<. Hence,
2 Cq in\q
Ent(l/n ^ ^dmU) EEl/^-^N/^r'
2vq Cq r
= ^ \vm\;\mr<M*)-
By Holder's inequality with the exponents | and (|)* = the above integral is estimated by
/\ Q/P / r \ (p-Q)/P
\vf(x)rqdp(x)j (J \f(x^dp(x)j = iiiv/ui'ii/iij;-«,
and we get
Ent(|/r) < ^
q
2vq Cq
Entdm < JL-±\\\vf\q\\i\\f\\p-q.
Combining this with (3.3), we obtain a differential inequality
I; 11/112 < 2^111 v/IJ,3.4)
If q > 1, integrate (3.4) between q and p with respect to the variable p, and then we arrive at
2Cq ip 2Cq rV?-1 — a
11/112 - m ^ / tQ~2 II IWI, II? dt < P q II |V/|, \\l
H J q H H
Thus, we have obtained the following:
Theorem 3.1. Whenever p > q > 1, under an LSq-inequality with constant Cq, we have, for any function f on V,
\\m - wm < 2q ii iv/i, in. (3.5)
4. LSi-inequality and isoperimetry
The limit case q — 1 in Theorem 3.1 is special, since then (3.4) turns into the differential inequality
< ^Iliv/Ul,
Clearly, it yields
ll/llp - \\f\\i < 2Ci J* -t II |V/U lit dt < 2C\ logp II |V/|i ||p,
which may also be obtained by letting q 1 in (3.5).
Hence, if an LSi-inequality is satisfied for the graph G — (V,£) with constant Ci, that is, if for any function / on V,
Ent(|/|) < Ci I/(®) - /Ml (4-1)
then, for any p > 1, we have
i < 2Ci logp ( J (g №) " ^^l)" ^i®)) (42)
Thus, while in case q > 1 the coefficient in front of |||V/|g||p grows polynomially fast as in Theorem 3.1, it grows logarithmically in case q — 1.
Now, it might be useful to realize that the hypothesis (4.1) has a simple geometric description of an isoperimetric nature. The main feature of this inequality is that it holds for all /, if and only if it holds in the class of the indicator functions / = A C V.
To see this, first note that, by homogeneity and due to the triangle inequality | \a\ — \b\ | < \a — b\ (a, 6 G R), without loss of generality one may restrict (4.1) to the functions with values 0 < / < 1. Let us assume this. Introduce a measure Q on V x V, assigning the mass l/card(y) to each couple (x, y) of neighbours in V. Put At — {x^V\ f(x) > i}, so that
f(x)= [ lAt(x)dt (4.3)
Jo
and
[ E №) - /(f) I Mx) = 2 [ Q(At X At) dt,
J y~x
where At denotes the complement of At in V. Then, (4.1) takes the form
Ent(/)<2Ci [ Q{At x At) dt. (4.4)
Jo
But the entropy / —>> Ent(/) is a convex functional, so, by (4.3),
Ent(/) < [ Ent(lAt)dt. Jo
Hence, (4.4) would follow from
Ent(lAt) < 2C1! Q(At x At), 0 < t < 1.
It remains to note that the inequality of the form
Ent(lA) < 2C1Q(A x A) (4.5)
is exactly (4.1) for / = 1A.
Now, with every set A in V we associate the edge boundary function
^a(^) = the number of neighbours of x outside A, x e A,
putting Ka{x) — 0, if x ^ A. Then clearly Q(A x A) = f dji.
In addition, Ent(l^) = p(A) log^y. Replacing C = 2C\ in (4.2) and (4.5), we arrive at:
Theorem 4.1. Assume that, for any set A C V,
p(A) log —^ <C J hA(x) dp(x) (4.6)
with some constant C. Then, for any function f on V, and any p > 1,
||/||p-||/||i < Clogp (| tV■
The inequality (4.6) is of an isoperimetric type: the integral on the right-hand side represents the size of the edge boundary of A.
5. Connection between LS^ and Poincare-type inequalities
In order to judge about the rate of the growth of the functions / on V with mean E/ = J f dp = 0 in Theorem 3.1, so that to obtain a corresponding analogue of Corollary 2.2, we need to relate LS^-inequalities
Ent(|/n < q / E №) - f(y)\q (5.1)
J y~x
to the inequalities of Poincare-type
11/ - mi < [ E №) - f(v)\9 Mx). (5.2)
Here we assume that Cq and Aq are some positive constants independent of /•
Note that (5.2) makes sense for any q > 1, while (5.1) requires that q < 2. Indeed, otherwise, if / is not identically zero, Ent(|/ + C\q) +oo, as C +oc, while the right-hand side of (5.1) does not change. So, (5.1) may not hold for q > 2.
As was already mentioned, in the basic case q — 2, (5.1) easily implies (5.2) with A\ — \ Cf, by applying the logarithmic Sobolev inequality to the functions of the form f+C with growing C. This argument is no longer valid in case 1 < q < 2, since then Ent(|/ + C\q) 0, as C +oo. Therefore, we need to choose a different route.
Lemma 5.1. For any function f > 0 on V,
Ent(/) > — log/i{f > 0} E/. (5.3)
This observation is elementary, and we refer to [B-Z], Lemma 2.2. Now, to derive (5.2) from (5.1), take an arbitrary function f on V with ¿/-median at zero, that is, such that ¡i{f > 0} < \ and ¡i{f < 0} < Apply (5.3) to the functions (f+)g and (f )q, where
f+ — max{/, 0}, /" = max{-/,0}.
Then we get
Ent((/+)') > log 2 E (/+)', Ent((/")') > log 2 E (/")', and after summing
E|/|9 < (Ent((/+)') + Ent((/")')). (5-4)
On the other hand, \ f+(x) — f+(y)\ < \f(x) — f(y)\, and similarly for /". Hence, combining (5.1) with (5.4), we obtain that
2 Cg ('
E\f\q ^ w| / E i/(*) - f(y)\q (5-5)
Finally, ||/ - E/H, < ||/||9 + E |/| < 2)1/11,, so E |/ - E/|* < 2"E \.f\q, and by (5.5),
21+i cq f
E |/ - E/|' < -r^—f / E I№) ~ M\q M*)- (5-6)
y~x
The both sides of this inequality are invariant under adding any constant to /, so the assumption about the median of / may be removed at this step. Hence, (5.6) provides a Poincare-type inequality such as (5.2).
Theorem 5.2. Under the LSq-inequality (5.1) with constant Cq, the Poincare-type inequality (5.2) holds true with
2i+g d AQ = _1
q log 2 "
We can now return to Theorem 3.1. Under the mean assumption E/ = 0, using Theorem 5.2, we get that
1 - 1 q-1
2q (S)«"1 " 1
i < wm+wy9' , ii iv/|g iii
To simplify the constant, assume p > 2. Clearly, < p9-1, and
pq-i _ i
> logp > log 2 > C:
q — 1 log 2
with Hence, II/HJ < 2(1 + 1)^^ || |V/|, |||.
Up to an absolute factor, this is a generalized form of Corollary 2.2.
Corollary 5.3. Under the LSq-inequality (5.1) with constant Cq (1 < ? < 2); for any function f on V with ¡¿-mean zero, and any p > 2,
pq~l - 1
IP ^ 2{)C«\-—r\ IMViW
1/9
6. Continuous setting
The above statements remain to hold for local gradients. To describe a general scheme, let (V, d) be a metric space, equipped with a Borel probability measure ¡i. Given a function / on V, define the generalized modulus of the gradient
\Df{x)\ = limsup xeV, (6.1)
d(x,y)^0+ d{X, y)
putting \Df(x) \ = 0, if x is an isolated point in V.
If / is continuous, then \Df\ is Borel measurable (cf. [B-H]).
A function / is called locally Lipschitz, if for any point x e V, there is r > 0 and C — C(x,r), such that, \f(x) — f(y)\ < Cd(x,y), whenever y) < r (that is, / has a finite Lipschitz constant on some neighborhood of x). For such functions, \Df\ are finite and Borel measurable.
For example, if V is the Euclidean space Rn with Euclidean metric, any locally Lipschitz function / on V is differentiate almost everywhere (with
respect to Lebesgue measure), and (6.1) leads to
df(x) "
i=1
dxi
at every point x, at which / is differentiate.
Returning to the abstract metric probability space, we say that (V, d, ¡jl) satisfies an LS^-inequality with constant Cq, where 1 < q < 2, if for any bounded locally Lipschitz function / on V,
Entd/I9) < Cqq j \Df(x)\qdp(x). (6.2)
The requirement that / is bounded may easily be removed from the definition.
In the sequel, for short, we write = ( / \Df\p djj)1^.
The corresponding analogue of Theorem 3.1 is the following.
Theorem 6.1. Whenever p > q > 1, under an LSq-inequality with constant Cqj for any locally Lipschitz function f on V with finite \\f\\p,
(py-i _ i
wm-wm < q qq_1 \\Df\\*. (6.3)
The proof is similar to the proof of Theorem 3.1 and is even simpler due to the property \Du(f)\ < \u'(f)\ • |D(/)|, where u is an arbitrary differentiate function on the real line. (As a result, the factor 2 in (3.5) can be removed).
In the limit case q — 1, the LSi-inequality (6.2) is equivalent to the isoperimetric-type inequality
ti(A)log-^<C^+(A) (6.4)
in the class of all Borel subsets A of V, where
n+(A) = Inn mf ——-—
r^ 0+ r
is the outer Minkowski content of A with respect to the measure ¡jl (Ar is the open r-neighbourhood of A for the metric d). For the entropy and related functionals, this equivalence was studied by many authors, starting with the works by V.G. Maz'ya in the early 1960s, cf. [M], [Rl], [B-H].
As a result, we obtain:
Theorem 6.2. In presence of the isoperimetric inequality (6.4), for any p > 1 and for any locally Lipschitz function f on V with finite
< Cilogp \\Df\\
V
Example. As shown in [B2], any log-concave probability measure /i, supported on a Euclidean ball in Rn of radius r > 0, satisfies the isoperimetric inequality
2r p+(A) > p(A) log -Ly + (1 - p(A)) log 1_1(Ay
Hence, (6.4) is fulfilled with C\ — 2r, so, by Theorem 6.2, ll/llp-H/lli < 2rlogp||£>/||JM p> 1.
In particular, this holds for any convex body K in Rn, contained in a Euclidean ball of radius r > 0, with respect to the normalized Lebesgue measure.
Let us return to Theorem 6.1. As shown in [B-Z], an LS^-inequality (6.2) implies a Poincare-type inequality
4 Cq
mi < i^f wm (6.5)
in the class of all locally Lipschitz functions f on V with ¡i-mean E/ = 0. Combining it with (6.3), we get, for any p > q.
One can simplify the constant like in the proof of Corollary 5.3 and derive:
Corollary 6.3. Under the LS(¡-inequality (6.2) with constant Cq (1 < Q < 2), for any locally Lipschitz function f on V with fi-mean zero, and any P>2,
fffl-1 - 1\
11/11, < lOC^^-J ||Z?/||P. (6.6)
Examples. A probability measure ¡jl on R with density ^^ = c\ exp{—c2|x|Qi} satisfies an LS^-inequality (6.2) with finite constant, if and only (1 < q < 2).
More generally, one may consider densities of the form e~u<KX\ where the function U e C2(R) is supposed to increase near +00, to decrease near —00, and to satisfy U"(x) — o(U'(x)2), as \x\ +oc. Then (cf. [B-Z]), ¡jl satisfies (6.2) with a finite Cq (1 < q < 2), if and only if
limsup v —— < +oc.
In the classical case q — 2, replacing Cf = 2/p, the LS^-inequality (6.2) takes the form
pEnt(/2) < 2 J \Df(x)\2dp(x), (6.7)
while the conclusion (6.3) of Theorem 6.1 reads as
\\f\\l-\\f\\l <P-^\\Df\\l P> 2- (6-8)
It is slightly better then the bound of Theorem 2.1 about the graph setting. Moreover, (6.7) may be obtained from (6.8) by letting p ^ 2, so these inequalities are equivalent.
As was already mentioned, (6.8) implies a Poincare-type inequality pII/II! — 11 A/112 (given that E / = 0), which is a little better than what follows from the bound (6.5) in case q — 2. Hence, one can also improve the constant in Corollary 6.4. Namely, we get
\\f-mP < y^ip/iu p>2. (6.9)
For example, the standard Gaussian measure // on Rn satisfies (6.7) with p = 1 ([G]), so, by (6.9), for any ¿/-integrable locally Lipschitz function / on Rn,
H/-E/H, < y/^T\\Df\\p, p> 2.
This inequality (with asymptotically equivalent constants) is well-known and can be obtained by different arguments (cf. e.g. [P]).
Remark 6.4. It is shown in [B-Z], cf. Theorem 6.1, that in presence of the the LS^-inequality (6.2), for any locally Lipschitz function / on V with p-mean zero,
[ e№'B*dii{x) < [ elDf^qdfi(x), (6.10)
where Bqq — q*q q Cq and g* = ^ conjugate to q e (1,2]. This is a more general form of (1.3), which corresponds to the classical case q — 2 in (6.10).
If the Orlicz norm < 1, the second integral in (6.10) does not
exceed 2, so
/«««.<,+/«-'/*<, <4.
Hence, Je^^dfi < 2, or, equivalently, H/H^ < 2Bq. Using Bq < (q — l)~l/qCq, we may conclude that
o n
\\fhi<^fyrq\\Dfhq, (6.11)
for any f on V with //-mean zero (which is a corresponding generalization of (1.4), up to a universal factor).
However, the dependence of the constant in (6.11) on <7, when it is close to 1, is asymptotically incorrect. A sharper inequality can be derived from Corollary (6.3), by using an ZAdescription of the the Orlicz norm given in Proposition 8.1 below.
Indeed, assuming that < 1 and applying a lower bound in (8.1)
with a = q, we get that ||/||p < 2px!q, for all p > 1. Hence, by (6.6), for all
11/11, <
An optimal constant in (pq — p)l^q < Ap for the range p > 2 is attained for p = 2, so A = (1 - 21~q)1/q, and
I _ l/q
\\f\\P < 20 Cq ^ j p < (20 log 2) Cqp.
For the range 1 < p < 2, one may just use ||/||p < ||/||2 < (40 log 2) Cqp.
Now, applying a lower bound in (8.1) with a = 1, we get that \\f\\ip1 < (80elog2) Cq.
This may be summarized in the following.
Corollary 6.5. Under the LSq-inequality (6.2) with constant Cq, for any locally Lipschitz function f on V with ¡¿-mean zero,
ll/IU <mcq\\Df\\i,q. (6.12)
7. Discrete cube
As an example of a graph, consider the discrete cube V = {0, l}n with the uniform probability measure p, assigning the mass 2~n to each point. Each point x — (xi,...,xn) in V has exactly n-neighbours ^(x), i = 1,..., n, with coordinates
= Xj (j ^ i), M^)); = 1 -x{.
Hence, any function f on V has the associated modulus of the discrete gradient, given by
n i = 1
According to the Bonami-Gross theorem [Bon], [G], in this case Ai = p — 4, that is, for any / on the discrete cube, we have
Ent (f2)<U |V/|2 d\x. (7.1)
Therefore, applying Theorem 2.1 and Corollary 2.2, we obtain the following: Corollary 7.1. For any function f : {0, l}n —>> R; and any p > 2,
11/11? - 11/111 <^HV/||2. (7.2)
In particular, if f has ¡¿-mean zero,
11/11, < l|V/||P. (7.3)
It is interesting that a similar statement continues to hold for the usual modulus of the gradient \Df\ like in the previous section within a more narrow class of functions. Namely, assume / is defined and non-negative on the cube [0, l]n, is smooth and coordinatewise convex (that is, convex with respect to each coordinate). Then, whenever f(x) > f(si(x)), we have f{x) - f(Si(x)) < Hence, for any p > 2,
f(Xy/2 - f(8i(x)y'2 < | (f(x) - f(8i(x))) fix
< f
and so,
E\Vf(x)p/2\2 = | E E №)P/2 - f^P/2)2 ^»/(v»
= ^~Ef(x)p~2\Df(x)\2.
p.
Therefore, by (7.1) and Holder's inequality,
Ent(/*) < ^Efp~2\Df\2 < j\\f\\pp-2\\Df\\2p.
We are in a similar situation as in Section 2. By (2.5), we obtain < | 11D/which can be integrated to yield:
Corollary 7.2. For any smooth, coordinatewise convex f : [0, l]n —>> [0, +oc); and any p > 2,
This inequality may be used to bound probabilities of large deviations of Lipschitz coordinatewise convex functions above the mean (similarly to the approaches of [Bl] and [LI]).
8. Appendix
Here we indicate how to derive the inequalities (1.5)-(1.6) and their generalizations for the class of the Orlicz norms || • generated by the Young functions i/ja(t) = — 1 with an arbitrary parameter a > 1.
Given a measurable function / on a probability space (V, //), first assume that || f\\p < p1/^, for all p > 1. By the Taylor expansion and using the bound nn < en n!, we obtain that, for any t e [0,
\\f\\2P-\\f\\l <
W 4-U ^ 4-U
E = 1 + V - E | f\an <l + yl- (an)11 < ^ n\ 11 ~ n\
OO
oo
1
1 — aet
71 = 1
For the value t — the ratio on the right-hand side does not exceed 2, which means that \\f\\^a < 2ae. Hence, \\f\\^a < (2ae)1/°i < 2e, since the function a —>> (2ae)1/Qi is decreasing in a > 1.
Conversely, assume ||/||^a < 1, that is, Ee^l" < 2. Using an elementary inequality xp < CexOL (x > 0) with an optimal constant C = we get
that E \ f\p < 2C, so, ||/||p < 2l'p < 2pl'<*.
One can now combine the two bounds, using the homogeneity of the inequalities like (1.5)-(1.6) with respect to /.
Proposition 8.1. Given a>l, for any measurable function f on V,
\ sup Mt < II/H < 2e sup №. (8.1)
2 p>i pl'a p>i p1/^
Thus, the equality ||/||(Q) = supp>1 \p~l/a ||/||p] provides a family of the norms that are equivalent to H/H^. These norms grow with a, and limc>i—)>oo II/11 (a) = 11 /11 oo'
Acknowledgement. I would like to thank Prasad Tetali for reading the manuscript and useful comments.
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University of Minnesota
Ilocmynujia 09.02.2010
