Volume 243, Nº 6 (2019)

We show that for n = 2 and α > 1/2, Khabibullin’s conjecture is not true.

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing at the origin together with first derivatives is a symmetric operator with deficiency indices (3). Krein’s theory allows one to derive an expression for the resolvent kernel of a self-adjoint extension of the operator in question as a sum of the Green’s function of the vector Laplace operator and some additional kernel of finite rank.

It is proved that for every ∈ > 0 there exists a measurable set E ⊂ [0, 1] with measure |E| > 1 − ∈ such that for every function f(x) ∈ L[0, 1] one can find a function g(x) ∈ L[0, 1] coinciding with f(x) on E such that its Fourier–Haar series absolutely converges in the metric of L^p(0, 1), 0 < p < 1.

We characterize those holomorphic symbols g for which the extended Cesàro operator V_g maps the Hardy space H^p(B) into the weighted Bergman space\( {A}_{\beta}^q(B) \), 0 < p < q < ∞, β > −1, on the unit ball B of ℂ^d.

The kernel of a Toeplitz operator on the Hardy class H² in the unit disk is a nearly invariantsubspace of the backward shift operator, and, by D. Hitt’s result, it has the form g · K_ω where ω is an inner function, K_ω = H² ⊝ ωH², and g is an isometric multiplier on K_ω. We describe the functions ω and g for the kernel of the Toeplitz operator with symbol .\( \overline{\theta}\varDelta \) where θ is an inner function and Δ is a finite Blaschke product.

This is an ℝⁿ-counterpart of certain considerations on a similar subject for compact Abelian groups exposed by P. Ivanishvili and the author in 2010. The main difference with that paper is that certain notions and results of measure theory should be invoked in the case of ℝⁿ.

We extend the notion of the variation V_f([a; b]) of a function f : [a; b] → ℝ to the variation V_f(A) of a continuous map f : G → ℝⁿ, where G is an open subset of ℝⁿ, over a set A ⊂ G of the form A = ∪_i ∈ IK_i where I is countable and all K_i are compact.

Let f : G → ℝ^m where G ⊂ ℝⁿ with n ≤ m, and let f₁, . . . , f_m be the coordinate functions of f. For α = {i₁, . . . , i_n} where 1 ≤ i₁ < i₂ < ⋯ < i_n ≤ m, let f_α be the map with coordinate functions \( {f}_{i_1},\dots, {f}_{i_n} \). The main result of the paper states that if f is a continuous injective map, G is an open subset of ℝⁿ, and a subset A ⊂ G has the form A = ∪_i ∈ IK_i where I is countable and all K_i are compact, then \( {V}_{f_{\alpha }}(A)\le {H}_n\left(f(A)\right) \) where \( {V}_{f_{\alpha }}(A) \) is the variation of f_α over A and H_n is the n-dimensional Hausdorff measure in ℝ^m.

We study the boundary behavior of closed open discrete mappings from the Sobolev and Orlicz–Sobolev classes in ℝⁿ, n ≥ 3. It is proved that such a mapping f can be extended by continuity to a boundary point x₀ ∈ ∂ D of a domain D ⊂ ℝⁿ whenever its inner dilatation of order α > n− 1 has a majorant from the finite mean oscillation class at the point in question. Another sufficient condition for the existence of a continuous extension is the divergence of some integral. We also prove some results on the continuous extension of such a mapping to an isolated boundary point.

Using duality, we show that there exist near-minimizers for the distance functionals for the couple (L^∞, L^p), 1 < p < ∞, that are stable under the action of singular integral operators.

Let \( E=\underset{k=1}{\overset{n}{\cup }}\left[{a}_k,{b}_k\right]\subset \mathbb{R} \); if n > 1, then we assume that the segments [a_k, b_k] are pairwise disjoint. Assume that the following property holds: E ∩ (E + 2πν) = ∅, ν ∈ ℤ, ν ≠ 0. Denote by H^ω + r(E) the space of functions f defined on E such that |f^(r)(x₂) − f^(r)(x₁)| ≤ c_fω(|x₂ − x₁|), x₁, x₂ ∈ E, f⁽⁰⁾ ≡ f. Assume that a modulus of continuity ω satisfies the condition

\( \underset{0}{\overset{x}{\int }}\frac{\omega (t)}{t} dt+x\underset{x}{\overset{\infty }{\int }}\frac{\omega (t)}{t^2} dt\le c\omega (x). \)

We find a constructive description of the space H^ω + r(E) in terms of the rate of nonuniform approximation of a function f ∈ H^ω + r(E) by trigonometric polynomials if E and ω satisfy the above conditions.