Abstract
Proceeding the study of local properties of analytic functions started in [Br] we prove new dimensionless inequalities for such functions in terms of their Chebyshev degree. As a consequence, we obtain the reverse Holder inequalities for analytic functions with absolute (i.e., independent of dimension) constants. For polynomials such inequalities were recently proved by Bobkov who sharpened and generalized the previous Bourgain result and by Sodin and Volberg.
Results & Lemmas (8)
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Theorem 1.1
Theorem 1.1 Let f ∈Or, r > 1, and I be a real interval situated in Bc(0, 1). (Hereafter we identify Cn with R2n.) There is a constant d =…
Theorem 1.1 Let f ∈Or, r > 1, and I be a real interval situated in Bc(0, 1). (Hereafter we identify Cn with R2n.) There is a constant d = d(f, r) > 0 such that for any I and any measurable subset ω ⊂I sup I |f| ≤ 4|I| |ω| !d sup ω |f| . (1.1) ∗Research supported in part by NSERC. 1991 Mathematics Subject Classification. Primary 31B05. Secondary 46E15. Key words and phrases. Analytic function, distribution, plurisubharmonic function, Chebyshev degree.
Theorem 1.2
Theorem 1.2 Let Rk ⊂Cn(∼= R2n) be a k-dimensional affine subspace, V ⊂ Bc(0, 1) ∩Rk be a k-dimensional convex body and h: Rk −→R+ be a…
Theorem 1.2 Let Rk ⊂Cn(∼= R2n) be a k-dimensional affine subspace, V ⊂ Bc(0, 1) ∩Rk be a k-dimensional convex body and h : Rk −→R+ be a log-concave function supported on V . There are absolute constants c, C > 0 (i.e. independent of dimensions k, n) such that for every r > 1 and f ∈Or (1) µh{x ∈V : |f(x)| > αfV } ≤C exp(−cα1/df (r)) and (2) µh{x ∈V : |f(x)| ≤αfV } ≤C(cα)1/df (r) (log α)1/2 , α ≤e−1 . (1.2)
Corollary 1.3
Corollary 1.3 Under the assumptions of Theorem 1.2 1 |V | Z V |f|pdµh ≤(cpdf(r))pdf (r)(fV )p ≤(cpdf(r))pdf(r) 1 |V | Z V |f|dµh !p (p > 1)…
Corollary 1.3 Under the assumptions of Theorem 1.2 1 |V | Z V |f|pdµh ≤(cpdf(r))pdf (r)(fV )p ≤(cpdf(r))pdf(r) 1 |V | Z V |f|dµh !p (p > 1) with an absoulte constant c > 0. In particular, if fV ≤1, then the Orlicz norm of f defined by the Orlicz function exp(1/df(r)) −1 on (V, dµh) is bounded by an absolute constant. 2
Corollary 1.4
Corollary 1.4 Under assumptions of Theorem 1.2 1 |V | Z V |log |f| −CV (f)| dµh ≤Cdf(r). Here C > 0 is an absoulte constant and CV (f):= 1…
Corollary 1.4 Under assumptions of Theorem 1.2 1 |V | Z V |log |f| −CV (f)| dµh ≤Cdf(r) . Here C > 0 is an absoulte constant and CV (f) := 1 |V | R V log |f|dµh. A similar result for analytic functions compairing log |f| with supV log |f| was ob- tained in [Br]. In this case the constant in the inequality is equivalent to log k (for k = dimV ). In the case of polynomials Corollary 1.3 implies the fundamental Bourgain inequality [B] (with h ≡1) and its generalizations proved by Bobkov [Bo]
Theorem 1.5
Theorem 1.5 Let f, V, r > 1 and µh be as in Theorem 1.2. Then 1 |V | Z V |f|mdµh !n 1 |V | Z V |f|−pdµh !q ≤(2e)n+q4(mn+pq)df (r)Γ(mdf(r) +…
Theorem 1.5 Let f, V , r > 1 and µh be as in Theorem 1.2. Then 1 |V | Z V |f|mdµh !n 1 |V | Z V |f|−pdµh !q ≤(2e)n+q4(mn+pq)df (r)Γ(mdf(r) + 1)n (1 −pdf(r))q provided m, n, p, q > 0 satisfy mn = pq, p <
Theorem 2.1
Theorem 2.1 [KLS] Let f1, f2, f3, f4 be four nonnegative continuous functions de- fined on Rn, and α, β > 0. Then the following are…
Theorem 2.1 [KLS] Let f1, f2, f3, f4 be four nonnegative continuous functions de- fined on Rn, and α, β > 0. Then the following are equivalent: (a) For every log-concave function F defined on Rn with compact support, Z Rn F(t)f1(t)dt α Z Rn F(t)f2(t)dt β ≤ Z Rn F(t)f3(t)dt α Z Rn F(t)f4(t)dt β .
Theorem 4.1
Theorem 4.1 Let f be as above and bf(r1) < ∞. Then the function F:= ef satisfies inequality (1.1) with cbf(r1) instead of df(r) where c ≥1…
Theorem 4.1 Let f be as above and bf(r1) < ∞. Then the function F := ef satisfies inequality (1.1) with cbf(r1) instead of df(r) where c ≥1 depends on r1 and r only.
Corollary 4.2
Corollary 4.2 All results of the presents paper are valid for ef with cbf(r) instead of df(r). Examples. (a) Let f ∈Or then we proved in…
Corollary 4.2 All results of the presents paper are valid for ef with cbf(r) instead of df(r). Examples. (a) Let f ∈Or then we proved in [Br] that blog |f|((1 + r)/2) < ∞. This motivates our definition of the Chebyshev degree. Note also that df(r) can be estimated by the general valency of f defined as maximum of valency of f restricted to each complex disk l(1+r)/2. (b) Let f1, ..., fk ∈(Cn)∗be complex linear functionals. A quasipolynomial with the spectrum f1, ..., fk is a finite sum q(z) = Pk i=