In 1961 he defended his
candidate theses at A.Razmadze Institute of Mathematics and in 1967 at
He solved:
a)
the problem of A. Zygmund and P. L. Ulyanov concerning the almost everywhere
existence of conjugate functions and Hilbert transforms of several variables;
b)
the problem of D.E. Menshov on the convergence in measure of Cesaro and
Abel-Poisson means of multiple conjugate series.
He established:
a) most general estimates ( both in one and multi- dimension cases) for
approximations by Cesaro means of Fourier series of continuous functions as well
as of the functions which are Lebesgue integrals with the power greater or equal
to 1;
b) the connection of approximate properties of the above
mentioned means with Hardy-Littlewood Nikolskii imbedding theorems.
He proved that:
a) multidimensional Hilbert operator does not possess a weak type
inequality even on the intersection of all the Lebesgue classes with
powers greater or equal to 1;
b)
Marcinkiewich means of Fourier
series of any summable function of two
variables converge almost everywhere;
c) (multidimensional complex
analysis) if for a logarithm-scaled Lebesgue space the power of the
logarithm is less than dimension minus
1, then
for a density from this space, the multiple Schwarz integral may have no
boundary value almost everywhere.
The contributions of
L. Zhizhiashvili in educating young
scientists is considerable. Among his pupils there are 7 doctors of sciences and
more then 20 candidates.
L. Zhizhiashivli was a university prize laureate. He was a member of various
societies, he was a Honoured Scientist, was awarded the Order of Red Banner of
Labour.
L. Zhizhiashvili carried out many-sided scientific, pedagogic and organizational
activities. He was the Dean of the Faculty of Mechanics and Mathematics of
Tbilisi State University, he guided the publishing of the proceedings of the
Scientific seminar led by himself. During many years he was the chairman of a
Specialized Doctor Council.
