Exact Traces for Vector-Valued Morrey Spaces
Abstract
We establish an exact trace theorem for vector-valued time-Morrey spaces and complementary continuity theorems for time-BMO evolution classes. For a Banach couple X1↪ X0, 1<p<∞ and 0<λ<1, we prove that the trace of the class u∈ℳp,λ(X1), ut∈ℳp,λ(X0) is exactly (X0,X1)θ,∞, where θ=1−(1−λ)/p. A constructive extension realizes every admissible trace, and counterexamples establish optimality of both the smoothness exponent and the weak fine index. Time-BMO control yields continuous finite-index interpolation traces and, under a mixed boundedness hypothesis, a logarithmic time modulus whose base-space logarithm is necessary. For the quadratic viscous Hamilton--Jacobi equation on the torus, we characterize the bounded initial data generating Morrey-regular solutions by the exact Besov condition u0∈ B2,∞2θ. The proof uses Hopf--Cole linearization and a direct composition argument, and requires no smallness assumption. Lacunary nonlinear solutions realize the sharp trace threshold. A forced Hamilton--Jacobi example exhibits genuinely logarithmic time regularity, and a forward--backward mean-field-game system illustrates initial and terminal trace compatibility. \par\smallskip\noindent\textbf{Keywords:} Morrey spaces; trace spaces; real interpolation; Hamilton--Jacobi equations; bounded mean oscillation.
Related
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BibTeX
@misc{shahmurov2026exact,
title = {{Exact Traces for Vector-Valued Morrey Spaces}},
author = {Rishad Shahmurov and Veli Shahmurov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.29148},
eprint = {2609.29148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29148},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Traces for Vector-Valued Morrey Spaces
AU - Rishad Shahmurov
AU - Veli Shahmurov
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.29148
UR - https://arxiv.org/abs/2609.29148
AB - We establish an exact trace theorem for vector-valued time-Morrey spaces and complementary continuity theorems for time-BMO evolution classes. For a Banach couple X_1↪ X_0, 1<p<∞ and 0<λ<1, we prove that the trace of the class u∈ℳ^p,λ(X_1), u_t∈ℳ^p,λ(X_0) is exactly (X_0,X_1)_θ,∞, where θ=1−(1−λ)/p. A constructive extension realizes every admissible trace, and counterexamples establish optimality of both the smoothness exponent and the weak fine index. Time-BMO control yields continuous finite-index interpolation traces and, under a mixed boundedness hypothesis, a logarithmic time modulus whose base-space logarithm is necessary. For the quadratic viscous Hamilton--Jacobi equation on the torus, we characterize the bounded initial data generating Morrey-regular solutions by the exact Besov condition u_0∈ B_2,∞^2θ. The proof uses Hopf--Cole linearization and a direct composition argument, and requires no smallness assumption. Lacunary nonlinear solutions realize the sharp trace threshold. A forced Hamilton--Jacobi example exhibits genuinely logarithmic time regularity, and a forward--backward mean-field-game system illustrates initial and terminal trace compatibility. \par\smallskip\noindent\textbf{Keywords:} Morrey spaces; trace spaces; real interpolation; Hamilton--Jacobi equations; bounded mean oscillation.
ER -