Extremal function for Moser-Trudinger type inequality with logarithmic weight
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
Authors: Roy, Prosenjit
Abstract
On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If beta is an element of [0,1) and w(0)(x) = vertical bar log vertical bar x vertical bar vertical bar(beta) then integral(B)vertical bar del(u)vertical bar(2)w(0)<= 1,u is an element of II0,rad1(w(0),B)integral(B)e(alpha u2/1-beta) dx < infinity, if and only if alpha <= alpha(3) = 2 [2 pi(1-beta)]1/1-beta. We prove the existence of an extremal function for the above inequality for the critical case when alpha = a(beta) thereby generalizing the result of Carleson Chang who proved the case when beta = 0. (C) 2016 Elsevier Ltd. All rights reserved.
Detection of Arabidopsis thaliana AtRAD1 cDNA variants and assessment of function by expression in a yeast rad1 mutant
GENE
Authors: Vonarx, EJ; Howlett, NG; Schiestl, RH; Kunz, BA
Abstract
The Saccharomyces cerevisiae RAD1 and human XPF genes encode a subunit of a nucleotide excision repair endonuclease that also is implicated in some forms of homologous recombination. An Arabidopsis thaliana gene (AtRAD1) encoding the orthologous plant protein has been identified recently. Here we report the isolation of three structurally distinct AtRAD1 cDNAs from A. thaliana leaf tissue RNA. One of the isolates (AtRAD1-1) corresponds to the cDNA previously shown to encode the full-length AtRad1 protein, whereas the other two (AtRAD1-2, AtRAD1-3) differ slightly in size due to variations at the 5' end of exon 6 or the 3' end of exon 7, respectively. The sequence differences argue that these cDNAs were probably templated by mRNAs generated via alternative splicing. Diagnostic polymerase chain reaction pointed to the presence of the AtRAD1-1 and AtRAD1-2 but not AtRAD1-3 transcripts in bud and root tissue, and to a fourth transcript (AtRAD1-4), having both alterations identified in AtRAD1-2 and AtRAD1-3, in root tissue. However, the low frequency of detection of AtRAD1-3 and AtRAD1-4 makes the significance of these tissue-specific patterns unclear. The predicted AtRad1-2, AtRad1-3 and AtRad1-4 proteins lack part of the region likely required for endonuclease complex formation. Expression of AtRAD1-2 and AtRAD1-3 in a yeast rad1 mutant did not complement the sensitivity to ultraviolet radiation or the recombination defect associated with the rad1 mutation. These results suggest that alternative splicing may modulate the levels of functional AtRad1 protein. (C) 2002 Elsevier Science B.V. All rights reserved.