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Description of traces of functions in the Sobolev space with a Muckenhoupt weight

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Abstract

We characterize the trace of the Sobolev space W l p (ℝn, γ) with 1 < p < ∞ and weight γ ∈ A loc p (ℝn) on a d-dimensional plane for 1 ≤ d < n. It turns out that for a function φ to be the trace of a function f ∈ W l p (ℝn, γ), it is necessary and sufficient that φ belongs to a new Besov space of variable smoothness, \(\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)\), constructed in this paper. The space \(\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)\) is compared with some earlier known Besov spaces of variable smoothness.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700, Russia

    A. I. Tyulenev

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  1. A. I. Tyulenev
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Original Russian Text © A.I. Tyulenev, 2014, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Vol. 284, pp. 288–303.

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Tyulenev, A.I. Description of traces of functions in the Sobolev space with a Muckenhoupt weight. Proc. Steklov Inst. Math. 284, 280–295 (2014). https://doi.org/10.1134/S0081543814010209

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