317,257 research outputs found
Endpoint multiplier theorems of Marcinkiewicz type
We establish sharp (H^1, L^{1,q}) and local (L \log^r L, L^{1,q}) mapping
properties for rough one-dimensional multipliers. In particular, we show that
the multipliers in the Marcinkiewicz multiplier theorem map H^1 to L^{1,\infty}
and L \log^{1/2} L to L^{1,\infty}, and that these estimates are sharp.Comment: 28 pages, no figures, submitted to Revista Mat. Ibe
Fourier multiplier theorems involving type and cotype
In this paper we develop the theory of Fourier multiplier operators
, for Banach spaces
and , and an operator-valued symbol. The case has been studied
extensively since the 1980's, but far less is known for . In the scalar
setting one can deduce results for from the case . However, in the
vector-valued setting this leads to restrictions both on the smoothness of the
multiplier and on the class of Banach spaces. For example, one often needs that
and are UMD spaces and that satisfies a smoothness condition. We
show that for other geometric conditions on and , such as the
notions of type and cotype, can be used to study Fourier multipliers. Moreover,
we obtain boundedness results for without any smoothness properties of
. Under smoothness conditions the boundedness results can be extrapolated to
other values of and as long as remains
constant.Comment: Revised version, to appear in Journal of Fourier Analysis and
Applications. 31 pages. The results on Besov spaces and the proof of the
extrapolation result have been moved to arXiv:1606.0327
On -- trace inequalities
We give necessary and sufficient conditions in order that inequalities of the
type hold for a class of integral operators with nonnegative kernels, and measures and
on , in the case where and .
An important model is provided by the dyadic integral operator with kernel
, where
is the family of all dyadic cubes in , and are
arbitrary nonnegative constants associated with .
The corresponding continuous versions are deduced from their dyadic
counterparts. In particular, we show that, for the convolution operator with positive radially decreasing kernel , the trace
inequality holds if and only if , where
. Here is a nonlinear Wolff
potential defined by and
. Analogous inequalities for
were characterized earlier by the authors using a different method
which is not applicable when
Bohl-Perron type stability theorems for linear difference equations with infinite delay
Relation between two properties of linear difference equations with infinite
delay is investigated: (i) exponential stability, (ii) \l^p-input
\l^q-state stability (sometimes is called Perron's property). The latter
means that solutions of the non-homogeneous equation with zero initial data
belong to \l^q when non-homogeneous terms are in \l^p. It is assumed that
at each moment the prehistory (the sequence of preceding states) belongs to
some weighted \l^r-space with an exponentially fading weight (the phase
space). Our main result states that (i) (ii) whenever and a certain boundedness condition on coefficients is
fulfilled. This condition is sharp and ensures that, to some extent,
exponential and \l^p-input \l^q-state stabilities does not depend on the
choice of a phase space and parameters and , respectively. \l^1-input
\l^\infty-state stability corresponds to uniform stability. We provide some
evidence that similar criteria should not be expected for non-fading memory
spaces.Comment: To be published in Journal of Difference Equations and Application
New Identities Relating Wild Goppa Codes
For a given support and a polynomial with no roots in , we prove equality
between the -ary Goppa codes where
denotes the norm of , that is In
particular, for , that is, for a quadratic extension, we get
. If has roots in
, then we do not necessarily have equality and we prove that
the difference of the dimensions of the two codes is bounded above by the
number of distinct roots of in . These identities provide
numerous code equivalences and improved designed parameters for some families
of classical Goppa codes.Comment: 14 page
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