Vol 215, No 5 (2016)

Let \( \mathfrak{F} \)₀ and \( \mathfrak{F} \) be perfect subsets of the complex plane ℂ. Assume that \( \mathfrak{F} \)₀ ⊂ \( \mathfrak{F} \) and the set \( \Omega \overset{\mathrm{def}}{=}\mathfrak{F}\backslash {\mathfrak{F}}_0 \) is open. We say that a continuous function f : \( \mathfrak{F} \) → ℂ is an analytic continuation of a function f₀ : \( \mathfrak{F} \)₀ → ℂ if f is analytic on Ω and f|\( \mathfrak{F} \)₀ = f₀. In the paper, it is proved that if \( \mathfrak{F} \) is bounded, then the commutator Lipschitz seminorm of the analytic continuation f coincides with the commutator Lipschitz seminorm of f₀. The same is true for unbounded \( \mathfrak{F} \) if some natural restrictions concerning the behavior of f at infinity are imposed.

We develop self-adjoint extensions of the radial part of the Laplace operator for l = 1 in a special scalar product. The product arises under the passage of the standard product from ℝ³ to the set of functions parametrizing one of two components of the transverse vector field. Similar extensions are treated for the square of the inverse operator of the radial part in question. Bibliography: 8 titles.

The relationship between smoothness properties of the boundary of a domain Ω and the boundedness of the Beurling transform in the corresponding Lipschitz classes Lip(ω) for the case of a Dini-regular module of continuity ω is studied. The result is sharp. Our motivation arises from the work of Mateu, Orobitg, and Verdera.

We give an elementary proof of the sharp Bernstein type inequality

\( {\left\Vert {f}^{(s)}\right\Vert}_2\le \frac{n^s}{2^s}{\left(\frac{\kappa_{2r+1-2s}}{\kappa_{2r+1}}\right)}^{1/2}{\left\Vert {\updelta}_{\frac{\uppi}{n}}^sf\right\Vert}_2. \)

Let F be an outer function on the unit disk. It is well known that its smoothness properties can be twice worse than those of the modulus of its boundary values, but under some restrictions on log |F|, this gap becomes smaller. It is shown that the smoothness decay admits a convenient description in terms of a rearrangement invariant Banach function space containing log |F|. All the results are of pointwise nature.

We give a simple proof of Dorronsoro’s theorem and use similar ideas to establish an equivalence for embeddings of vector fields. Bibliography: 13 titles.