Miroslav Pavlović

Inequalities for the gradient of eigenfunctions of the invariant laplacian in the unit ball

Abstract The classical theorem of Hardy and Littlewood on differentiation in mixed normed spaces is discussed in the context of the class Xλ={f: f= λf}. The main result is that there is a difference between the cases λ = 0 and λ ≠ 0. In particular, if λ ≠ 0, then an Hp-norm is proportional to the corresponding Bloch type norm.

On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

Let QC(K, g) be a family of K-quasiconformal mappings of the open unit disk onto itself satisfying the PDE Δw = g, g ∈ C(U), w(0) = 0. It is proved that QC(K,g) is a uniformly Lipschitz family. Moreover, if |g| ∞ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as K → 1 and |g| ∞ → 0, so w ∈ QC(K, g) behaves almost like a rotation for sufficiently small K and |g| ∞ .

Lipschitz conditions on the modulus of a harmonic function

It is proved that if u is a real valued function harmonic in the open unit ball BN ⊂ R and continuous on the closed ball, then the following conditions are equivalent, for 0 < α < 1: • |u(x)− u(w)| ≤ C|x− w|, x, w ∈ BN ; • | |u(y)| − |u(ζ)| | ≤ C|y − ζ|, y, ζ ∈ ∂BN ; • | |u(y)| − |u(ry)| | ≤ C(1− r), y ∈ ∂BN , 0 < r < 1. The Lipschitz condition on |u| is also considered.

A Schwarz lemma for the modulus of a vector-valued analytic function

It is proved that |∇|f|(z)|≤1-|f(z)| 2 /1-|z| 2 , z∈D, where f: D ↦ B k is an analytic function from the unit disk D into the unit ball B k C ℂ k . Applications to the Lipschitz condition of the modulus of a ℂ k -valued function are given.

Mean values of harmonic conjugates in the unit disc

It is proved that ifƒ = u +iv is analytic in the unit disc Δ and 0 < ρ <1, then This inequality is applied to generalize some results of Hardy and Littlewood on ρ-th mean values of harmonic conjugates u and v

A short proof of an inequality of Littlewood and Paley

A very short proof is given of the inequality where p > 2, and u is the Poisson integral of f ∈ L p (∂D), D = {z: |z| < 1}.

Analytic functions with decreasing coefficients and Hardy and Bloch spaces

The following rather surprising result is noted. (1) A function f ( z ) = ∑ a n z n such that a n ↓ 0 ( n → ∞) belongs to H 1 if and only if ∑( a n /( n + 1)) A more subtle analysis is needed to prove that assertion (2) remains true if H 1 is replaced by the predual, 1 (⊂ H 1 ), of the Bloch space. Assertion (1) extends the Hardy–Littlewood theorem, which says the following. (2) f belongs to H p (1 p n + 1) p −2 a n p A new proof of (2) is given and applications of (1) and (2) to the Libera transform of functions with positive coefficients are presented. The fact that the Libera operator does not map H 1 to H 1 is improved by proving that it does not map 1 into H 1 .

COMPLEX CONVEXITY AND VECTOR-VALUED LITTLEWOOD¿PALEY INEQUALITIES

Let 2 p 0s uch thatfHp(X) (� f(0)� p + λ (1 −| z| 2 ) p−1 � f � (z)� p dA(z)) 1/p ,f or all f ∈ H p (X). Applications to embeddings between vector-valued BMOA spaces defined via Poisson integral or Carleson measures are provided.

ON THE HOLLAND–WALSH CHARACTERIZATION OF BLOCH FUNCTIONS

It is proved that the Bloch norm of an arbitrary C1-function defined on the unit ball Bn ⊂ Rn is equal to sup x,y∈Bn, x =y (1− |x|2)1/2(1− |y|2)1/2 |f(x)− f(y)| |x− y| .

On the Hahn-Banach Extension Property in Hardy and mixed norm spaces on the unit ball

AbstractFor a nonempty setE of nonnegative integers letHEp, q, a andHEp be the closed linear span of