Research activity Shaposhnikova is the author of more than 70 research papers and of four books: her research mainly belongs to the following fields.
Function spaces From 1979 on, the
theory of multipliers in various
spaces of
differentiable functions has been the main research theme of her work. She found conditions for the
boundedness of
singular integrals and
pseudodifferential operators acting between pairs of
Sobolev spaces in 1995. In 1989 she showed that multipliers in
Bessel potential spaces are
traces of multipliers belonging to a certain class of differentiable functions with a
weighted mixed norm. A large part of her joint work with
Vladimir Maz'ya on the theory of multipliers involves their analytic characterization,
trace inequalities and relations between traces and
extension of multipliers, relations of
Sobolev multipliers and other
function spaces, maximal
subalgebras of multiplier spaces, estimates of their
essential norm and
compactness of multipliers.
Linear and non-linear PDEs Based on her researches on the
theory of multipliers, T. Shaposhnikova gave various applications of this theory to the study of
solutions to second order
linear and
quasilinear elliptic partial differential equations and
systems of such equations: this was a consequence of the fact that, in several cases, such solutions can be considered as multipliers in certain
spaces of
differentiable functions on a given
domain (1986, 1987). She described the structure of
composition operators in spaces of multipliers between
Sobolev spaces and gave applications of those results to
semilinear elliptic systems of equations (1987). She also showed that
multipliers can be naturally suited to deal with the
Lp coercivity of the
Neumann problem (1989). Various other applications of multipliers, for example to the problem of
higher regularity in
single and
double layer potential theory for
Lipschitz domains, to the problem of
regularity at the boundary in the -theory of elliptic
boundary value problems and to
singular integral operators in
Sobolev spaces are summarized in the book .
History of mathematics Her prize winning book on
Jacques Hadamard, coauthored with
V. Maz'ya, Her recent activity in this field includes the paper telling three stories of scientists who were forced to answer a mathematical question under rather trying circumstances.
Translation and editing activity Shaposhnikova has
translated and
edited several
mathematical monographs: it is worth to note the works by and by , the book on
Sobolev spaces by , and the books by and by . However, her work is not restricted only to the translation of monographs: for example she translated into
Russian a
play by
Lars Gårding, titled "
Mathematics, Life and Death", published the
mathematical journal Algebra i Analiz (Алгебра и анализ). Shaposhnikova began translating fiction while still living in
Russia. In the 1970s she translated into
Russian "
The Voyage of the Dawn Treader", "
The Silver Chair" and the "
Screwtape Letters" by
C. S. Lewis. These
translations were impossible to publish due to ideological reasons and were distributed as
samizdat: In 2005 she began translating
Swedish children's books into
Russian. "
Mechanical Santa Claus" by
Sven Nordqvist and two books of the "
Loranga" series by
Barbro Lindgren. ==Selected publications==