LetB be the unit ball of Cⁿ, I give necessary conditions on sequence S of points in B to be H^∞(B) interpolating in term of a Cⁿ valued holomorphic function zero on S (a substitute for the interpolating Blaschke product).

These conditions are sufficient to prove that the sequence S is interpolating for ∩_p>1 (B) and is also interpolating for H^p(B) for 1≤p<∞.

Inequalities of the form $\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $ for all f∈H¹, where {m_k} are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterize B-convexity. It is also shown that for 1< q<∞ and 0<α<∞ the space X=H(1, q,γa) consisting of analytic functions on the unit disc such that $\int_0^1 {(1 - r)^{q\alpha - 1} M_1^q (f,r) dr< \infty } $ satisfies the previous inequality for vector valued functions in H¹ (X), defined as the space of X-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {m_k}={2^k} but not for {m_k}={k^a} for any α ∈ N.

A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)^μ in Rⁿ, n≥1, is obtained, for μ∈(0,1/2n)/(1,1/2n−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev.

As in the polyharmonic case, the proof is based on a weighted positivity property of (− Δ)^μ, where the weight is a fundamental solution of this operator. It is shown that this property holds for μ as above while there is an interval [A_n, 1/2n−A_n], where A_n→1, as n→∞, with μ-values for which the property does not hold. This interval is non-empty for n≥8.

This paper presents an elementary proof of the following theorem: Given {r_j} ${}_{j}^{m}$ =1 with m=d+1, fix $fix R \geqslant \sum\nolimits_{j = 1}^m {r_j } $ and let Q=[−R, R]^d. Then any f∈ L²(Q) is completely determined by its averages over cubes of side r_j that are completely contained in Q and have edges parallel to the coordinate axes if and only if r_j/r_k is irrational for j≠k. When d=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials.

For an essentially normal operator T, it is shown that there exists a unilateral shift of multiplicity m in C^*(T) if and only if γ(T)≠0 and γ(T)/m. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic as C^*-algebras. Finally, we construct a natural C^*-algebra ε + ε_* on the Bergman space L ${}_{a}^{2}$ (B_n), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.

An asymptotic formula for the density of states of the polyharmonic periodic operator (−δ)^l+V in Rⁿ, n≥2, l>1/2 is obtained. Special consideration is given to the case of the Schrödinger equation n=3, l=1, V being a periodic potential, where the second term of the asymptotic is found.

LetF be a family of meromorphic functions on the unit disc Δ and let a and b be distinct values. If for every f∈F, f and f′ share a and b on Δ, then F is normal on Δ.

The main goal of this paper is to present an alternative, real variable proof of the T(1)-theorem for the Cauchy integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the T(1)-theorem. An example shows that the L^∞-BMO estimate for the Cauchy integral does not follow from L² boundedness when the underlying measure is not doubling.

In this paper we show that the pluripolar hull of E={(z, ω)∈C²:ω=e^−1/z, z≠0} is equal to E. This implies that E is plurithin at 0, which answers a question of Sadullaev. The result remains valid if e^−1/z is replaced by certain other holomorphic functions with an essential singularity at 0.