Functional Analysis and Applications Seminar 

27/05/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Universidade Nova de Lisboa e CEAF, IST

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

Let $a$ be a semi-almost periodic matrix function with the almost periodic representatives $b$ and $c$ at $-\mathrm{\infty }$ and $+\mathrm{\infty }$, respectively. Suppose $p(\cdot )$ is a slowly oscillating exponent such that the Cauchy singular integral operator $S$ is bounded on the variable Lebesgue space $L_p(\cdot )$. 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 $L_p(\cdot )$, then the operators $bP+Q$ and $cP+Q$ are invertible on standard Lebesgue spaces $L_q$ and $L_r$ with some exponents $q$ and $r$ lying in the segments between the lower and the upper limits of $p(\cdot )$ at $-\mathrm{\infty }$ and $+\mathrm{\infty }$, respectively. This is a joint work with Ilya Spitkovsky.