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<p>This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> spaces with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(A_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> weights. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy’s inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.</p>

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On the trace theorem to Volterra-type equations with local or non-local derivatives

  • 期刊论文

  • 发表日期：

    2026年2月25日

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  • Jae-Hwan Choi,
  • Jin Bong Lee,
  • Jinsol Seo,
  • Kwan Woo

摘要

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued \(L_p\) L p spaces with \(A_p\) A p weights. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy’s inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.