Ukrainian mathematical congress - 2009
Yuriy Karlovych (Universidad Autónoma del Estado de Morelos, Morelos, Mexico)
Wiener-Hopf operators with oscillating matrix symbols on weighted Lebesgue spaces
We establish Fredholm criteria for Wiener-Hopf operators $W(a)$ with oscillating matrix symbols $a$, continuous on the real line $R$ and admitting mixed (slowly oscillating and semi-almost periodic) discontinuities at $\pm\infty$, on weighted Lebesgue spaces $L^p_N(R_+,w)$ where $1< p < \infty$, $N=1,2,\ldots$, and $w$ belongs to a subclass of Muckenhoupt weights. For $N>1$ these criteria are conditional. To obtain these results we apply the techniques of limit operators and use the theory of Wiener-Hopf operators with semi-almost periodic matrix symbols and slowly oscillating matrix symbols on weighted Lebesgue spaces elaborated by us earlier on the basis of the theory of pseudodifferential and Calder\'on-Zygmund operators, results on oscillating Fourier multipliers and factorization of almost periodic matrix functions. The talk is based on a joint work with Juan Loreto Hern\'andez.