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Multipliers in Spaces of Bessel Potentials:
The Case of Indices of Nonnegative Smoothness
A. A. Belyaev,
A. A. Shkalikov Lomonosov Moscow State University
Abstract:
The aim of the paper
is to study spaces of multipliers acting from
the Bessel potential space
$H^s_p(\mathbb{R}^n)$
to the other Bessel potential space
$H^t_q(\mathbb{R}^n)$.
We obtain conditions
ensuring the equivalence of uniform
and
standard multiplier norms
on the space of multipliers
$$
M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]\qquad
\text{for}\quad s,t \in \mathbb{R},\quad p,q > 1.
$$
In the case
$$
p,q > 1,\qquad
p \le q,\qquad s > \frac np,\qquad
t \ge 0,\qquad s-\frac np \ge t-\frac nq,
$$
the space
$M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]$
can be described explicitly.
Namely,
we prove in this paper that
the latter space coincides
with the space
$H^t_{q,\mathrm{unif}}(\mathbb{R}^n)$
of uniformly localized Bessel potentials introduced by Strichartz.
It is also proved that
if both smoothness indices
$s$
and
$t$
are nonnegative,
then
such a description
is possible
only
for
the given values of the indices.
Keywords:
Bessel potential space, multiplier,
Strichartz theorem, uniform localization principle.
UDC:
517.518.23 Received: 06.09.2017
DOI:
10.4213/mzm11795
Fulltext:
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