TY - JOUR
T1 - Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data
AU - Karlovych, Oleksiy
AU - Karlovich, Yuri I.
AU - Lebre, Amarino B.
N1 - This work was partially supported by the Portuguese Foundation for Science and Technology, project Nos. UID/MAT/00297/2013) (Centro de Matematica e Aplicagoes) and UID/MAT/04721/2013 (Centro de Analise Funcional, Estruturas Lineares e Aplicagoes). The second author was also supported by the SEP-CONACYT, project Nos. 168104 and 169496 (Mexico).
PY - 2017
Y1 - 2017
N2 - We extend the main result of [12] to the case of more general weighted singular integral operators with two shifts of the form (aI - bUα)Pγ + + (cI - dUβ)Pγ -, acting on the space Lp(ℝ+), 1 < p < ∞, where Pγ ± = (I ± Sγ)/2 are operators associated with the weighted Cauchy singular integral operator Sγ, given by with γ εℂ satisfying 0 < 1/p + γ < 1, and Uα, Uβ are the isometric shift operators given by Uαf = (α')1/p(f ο α), Uβf = (β')1/p(f ο β), generated by diffeomorphisms α β of ℝ+ onto itself having only two βxed points at the endpoints 0 and ∞, under the assumptions that the coefficients a; b; c; d and the derivatives α', β' of the shifts are bounded and continuous on ℝ+ and admit discontinuities of slowly oscillating type at 0 and ∞.
AB - We extend the main result of [12] to the case of more general weighted singular integral operators with two shifts of the form (aI - bUα)Pγ + + (cI - dUβ)Pγ -, acting on the space Lp(ℝ+), 1 < p < ∞, where Pγ ± = (I ± Sγ)/2 are operators associated with the weighted Cauchy singular integral operator Sγ, given by with γ εℂ satisfying 0 < 1/p + γ < 1, and Uα, Uβ are the isometric shift operators given by Uαf = (α')1/p(f ο α), Uβf = (β')1/p(f ο β), generated by diffeomorphisms α β of ℝ+ onto itself having only two βxed points at the endpoints 0 and ∞, under the assumptions that the coefficients a; b; c; d and the derivatives α', β' of the shifts are bounded and continuous on ℝ+ and admit discontinuities of slowly oscillating type at 0 and ∞.
KW - Fredholmness
KW - Orientation-preserving shift
KW - Slowly oscillating function
KW - Weighted Cauchy singular integral operator
UR - http://www.scopus.com/inward/record.url?scp=85011568264&partnerID=8YFLogxK
U2 - 10.1216/JIE-2017-29-3-365
DO - 10.1216/JIE-2017-29-3-365
M3 - Article
AN - SCOPUS:85011568264
SN - 0897-3962
VL - 29
SP - 365
EP - 399
JO - Journal of Integral Equations and Applications
JF - Journal of Integral Equations and Applications
IS - 3
ER -