TY - JOUR

T1 - On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

AU - Karlovich, Alexei Yu

AU - Spitkovsky, Ilya M.

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/12/15

Y1 - 2011/12/15

N2 - Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at -∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(.)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I-S)/2 is Fredholm on the variable Lebesgue space LNp(.)(R), then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces LNql(R) and LNqr(R) with some exponents ql and qr lying in the segments between the lower and the upper limits of p at -∞ and +∞, respectively.

AB - Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at -∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(.)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I-S)/2 is Fredholm on the variable Lebesgue space LNp(.)(R), then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces LNql(R) and LNqr(R) with some exponents ql and qr lying in the segments between the lower and the upper limits of p at -∞ and +∞, respectively.

KW - Almost-periodic function

KW - Fredholmness

KW - Invertibility

KW - Semi-almost periodic function

KW - Singular integral operator

KW - Slowly oscillating function

KW - Variable Lebesgue space

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U2 - 10.1016/j.jmaa.2011.06.066

DO - 10.1016/j.jmaa.2011.06.066

M3 - Article

AN - SCOPUS:79961020530

VL - 384

SP - 706

EP - 725

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -