Лебедев Владимир Владимирович

The Sobolev space W_2^{1/2}: Simultaneous improvement of functions by a homeomorphism of the circle

2026 · ARTICLE · en

It is known that for every continuous real-valued function $f$ on the circle $\mathbb T=\mathbb R/2\pi\mathbb Z$ there exists a change of variable, i.e., a self-homeomorphism $h$ of $\mathbb T$, such that the superposition $f\circ h$ is in the Sobolev space $W_2^{1/2}(\mathbb T)$. We obtain new results on simultaneous improvement of functions by a single change of variable in relation to the space $W_2^{1/2}(\mathbb T)$. The main result is as follows: there does not exist a self-homeomorphism $h$ of $\mathbb T$ such that $f\circ h\in W_2^{1/2}(\mathbb T)$ for every $f\in \mathrm{Lip}_{1/2}(\mathbb T)$. Here $\mathrm{Lip}_{1/2}(\mathbb T)$ is the class of all functions on $\mathbb T$ satisfying the Lipschitz condition of order $1/2$.

The Sobolev space W_2^{1/2} : Simultaneous improvement of functions by a homeomorphism of the circle

2025 · PREPRINT · en

On extension to Fourier transforms

2023 · PREPRINT · en

On extension to Fourier transforms

2023 · ARTICLE · en

For n-point sets K in an LCA group Г we obtain an estimate for the norm of ``the best'' extension operator from the space of bounded functions on K to the space A(Г) of Fourier transforms. As a simple consequence our estimate implies that if K is an infinite closed subset of Г then there does not exist a bounded linear extension operator from the space C_0(K) of continuous functions on K vanishing at infinity to A(Г). The latter result generalizes a result by Graham who considered the case of compact subsets K.

Tame semicascades and cascades generated by affine self-mappings of the d -torus

2021 · ARTICLE · en

We give a complete characterization of tame semicascades and cascades generated by affine self-mappings of the d-torus.

Homeomorphic Changes of Variable and Fourier Multipliers

2020 · ARTICLE · en

We consider the algebras M_p of Fourier multipliers and show that for every bounded continuous function f on R^d there exists a self-homeomorphism h of R^d such that the superposition foh$ is in M_p(R^d) for all p, 1 under certain assumptions on a family K of continuous functions, one h will suffice for all f\in K. A similar result holds for functions on the torus T^d. This may be contrasted with the known solution of Luzin's problem related to the Wiener algebra.

Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form

2019 · ARTICLE · en

We consider the Wiener algebra A(T^d) of absolutely convergent Fourier series on the d-torus. For phase functions \phi of a certain special form we obtain lower bounds for the A -norms of e^{i\lambda\varphi} as \lambda tends to \infty.

Homeomorphic Changes of Variable and Fourier Multipliers

2019 · PREPRINT · en

We consider the algebras M_p of Fourier multipliers and show that for every bounded continuous function f on R^d there exists a self-homeomorphism h of R^d such that the superposition f oh is in M_p(R^d) for all p, 1 < p < \infty. Moreover, under certain assumptions on a family K of continuous functions, one h will sffice for all f in K. A similar result holds for functions on the torus T^d. This may be contrasted with the known solution of Luzin's problem related to the Wiener algebra.

Sets with distinct sums of pairs, long arithmetic progressions, and continuous mappings

2018 · PREPRINT · en

We show that if ϕ: R → R is a continuous mapping and the set of nonlinearity of ϕ has nonzero Lebesgue measure, then ϕ maps bijectively a certain set that contains arbitrarily long arithmetic progressions onto a certain set with distinct sums of pairs.

Uniformly convergent Fourier series and multiplication of functions

2018 · PREPRINT · en

Let U(T) be the space of all continuous functions on the circle T whose Fourier series converges uniformly. Salem's well-known example shows that a product of two functions in U(T) does not always belongs to U(T) even if one of the factors belongs to the Wiener algebra A(T). In this paper we consider pointwise multipliers of the space U(T), i.e., the functions m such that mf is in U(T) whenever f is in U(T). We present certain sufficient conditions for a function to be a multiplier and also obtain some results of Salem type.