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Let \(p(\cdot )\) be a variable exponent in the class \(LH^*(\mathbb {R})\) and \(\varrho \) be a Khvedelidze weight. We prove that if \(a\in S_{1,0}^0(\mathbb {R}\times \mathbb {R})\) slowly oscillates at infinity in the first variable, then the condition \(\displaystyle \lim _{R\to \infty }\inf _{|x|+|\xi |\ge R}|a(x,\xi )|>0 \) is sufficient for the Fredholmness of the pseudodifferential operator \(\operatorname {Op}(a)\) on the weighted variable Lebesgue space \(L^{p(\cdot )}(\mathbb {R},\varrho )\) .

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On Pseudodifferential Operators with Slowly Oscillating Symbols on Variable Lebesgue Spaces with Khvedelidze Weights

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  • 发表日期：

    2025年

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  • Cláudio Fernandes,
  • Oleksiy Karlovych

摘要

Let \(p(\cdot )\) be a variable exponent in the class \(LH^*(\mathbb {R})\) and \(\varrho \) be a Khvedelidze weight. We prove that if \(a\in S_{1,0}^0(\mathbb {R}\times \mathbb {R})\) slowly oscillates at infinity in the first variable, then the condition \(\displaystyle \lim _{R\to \infty }\inf _{|x|+|\xi |\ge R}|a(x,\xi )|>0 \) is sufficient for the Fredholmness of the pseudodifferential operator \(\operatorname {Op}(a)\) on the weighted variable Lebesgue space \(L^{p(\cdot )}(\mathbb {R},\varrho )\) .