arXiv subjects

Alexei A. Belyaev

Publications and source records attributed to Alexei A. Belyaev.

3 recordsLinked to original sources

arXiv2022

Multipliers in the scale of periodic Bessel potential spaces with smoothness indices of different signs

We prove a general type description result for the multipliers acting between two periodic Bessel potential spaces, defined on the $n$--dimensional torus, in a case when their smoothness indices are of different signs. This is done through the detailed examination of a periodic analogue of the linear operator $J_s$, which is employed in the definition of the scale of the Bessel potential space defined on the whole space $\mathbb{R}^n$. Our method of defining this periodic analogue of $J_s$ uses the results about an asymptotic behaviour of the generalized Fourier coefficients and existence of a natural homeomorphism between the spaces $\mathcal{D}'(\mathbb{T}^n)$ and $S'_{2 \cdot π}(\mathbb{R}^n)$, where the latter consists of all $2 \cdot π$--periodic distributions from the dual Schwartz space $\mathcal{S}'(\mathbb{R}^n)$.

math.FA↗

arXiv2022

Multipliers in the Bessel potential spaces with positive smoothness indices: bilateral continuous embeddings and their exact character

We investigate the problem of establishing bilateral continuous embeddings of the uniformly localized Bessel potential spaces $H^γ_{r, \: unif}(\mathbb{R}^n)$ into the multiplier spaces between Bessel potential spaces with positive smoothness indices. This problem is considered in the model situation when the natural norms of both of these Bessel potential spaces are generated by some inner product yet the description theorems for the corresponding multiplier space in terms of the spaces $H^γ_{r, \: unif}(\mathbb{R}^n)$ can not be established. The optimal character of the indices figuring in these embeddings is also examined.

math.FA↗

arXiv2019

Sharp embeddings of uniformly localized Bessel potential spaces into multiplier spaces

For $p > 1, γ\in \mathbb{R}$, denote by $H^γ_p(\mathbb{R}^n)$ the Bessel potential space, by $H^γ_{p, unif}(\mathbb{R}^n)$ the corresponding uniformly localized Bessel potential space and by $M[s, -t]$ the space of multipliers from $H^s_2(\mathbb{R}^n)$ into $H^{-t}_2(\mathbb{R}^n)$. Assume that $s, t \geqslant 0, n/2 > \max(s, t) > 0, r: = \min(s, t), p_1: = n/max(s, t)$. Then the following embeddings hold $$ H^{-r}_{p_1, unif}(\mathbb{R}^n) \subset M[s, -t] \subset H^{-r}_{2, unif}(\mathbb{R}^n). $$ The main result of the paper claims the sharpness of the left embedding in the following sense: it does not hold if the lower index $p_1$ is replaced by $p_1 -\varepsilon$ with any sufficiently small $\varepsilon > 0$.

math.FA↗