Compactness for the weighted Hardy operator in variable exponent spaces

In this paper, we prove a necessary and sufficiency condition for the weighted Hardy operator

$H_{\upsilon ,\omega }f(x)=\upsilon (x)\underset{0}{\overset{x}{\int }}f(t)\omega (t)\mathrm{d}t$

to be compactly acting from $L^{p(?)}(0,\infty )$ to $L^{q(?)}(0,\infty )$.     

Analysis of some injection bounds for Sobolev spaces by wavelet decomposition

We consider the Sobolev spaces $H^s(\Omega )$ and $H_0^s(\Omega )$ and the Besov spaces $B_{2,\infty }^{1/2}(\Omega )$, where Ω is a sufficiently regular (see Lemma 2) subdomain of ${\mathbb{R}}^2$. It is well known that for the values of $s\in [0,1/2)$ the two Sobolev spaces coincide, with equivalence of the norms, and that the inclusion $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ holds. The Note is concerned with the explicit analysis of the constants appearing in the continuity bounds for the injections $H^s(\Omega )?H_0^s(\Omega )$ and $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ and of their dependence on the regularity s of the spaces. The analysis is carried out by using the wavelet characterization of the corresponding norms.     

Bilinear forms on homogeneous Sobolev spaces

In this paper we characterize the boundedness of the bilinear form defined by

$(f,g)\in {\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})\to \underset{\mathbb{R}}{\int }{(?\mathrm{\Delta })}^{s/2}(fg)(x){(?\mathrm{\Delta })}^{s/2}(b)(x)dx,$

in the product of homogeneous Sobolev spaces ${\dot{H}}^s(\mathbb{R})\times {\dot{H}}^s(\mathbb{R})$, $0<s<1/2$. We deduce a characterization of the space of pointwise multipliers from ${\dot{H}}^s(\mathbb{R})$ to its dual ${\dot{H}}^{?s}(\mathbb{R})$ in terms of trace measures.     

On mixed codes with covering radius 1 and minimum distance 2: (extended abstract)

Let R, S and T be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when r divides s, when $r=s-1$, when s is large, relative to r, when r is large, relative to s, as well as $K(3r,2r,2r;2)$. Finally, a table with bounds on $K(r,s,s;2)$ is given.     

Generalized LIL for geometrically weighted random series in Banach spaces

Let $\{X,X_n;n?0\}$ be a sequence of independent and identically distributed random variables, taking values in a separable Banach space B with topological dual $B^{?}$. Considering the geometrically weighted series $\xi (\beta )={\sum }_{n=0}^{\infty }{\beta }^n{X}_n$ for $0<\beta <1$, motivated by Einmahl and Li (2005, 2008), a general law of the iterated logarithm for $\xi (\beta )$ is established.     

Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.