| Abstract: |
| We establish a dimension-free limiting formula for the norms of negative Sobolev spaces. These spaces play a fundamental role in fluid mechanics, and their norms are conveniently characterized via the heat kernel. Our main result captures a sharp nonlocal-to-local transition: as the fractional order tends to zero, negative Sobolev norm converges to a $L^p$ norm. Our method is based on basic properties of the heat kernel and $L^p$-norm approximation techniques. |
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