Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces
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Bibliographic record
Abstract
We introduce a new method for proving the estimate <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar StartFraction partial-differential squared u Over partial-differential x Subscript i Baseline partial-differential x Subscript j Baseline EndFraction double-vertical-bar Subscript upper C Sub Superscript alpha Subscript Baseline less-than-or-equal-to c double-vertical-bar f double-vertical-bar Subscript upper C Sub Superscript alpha Subscript Baseline comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi> α </mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mi>c</mml:mi> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>f</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi> α </mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \Vert \frac {\partial ^2 u}{\partial x_i \partial x_j} \right \Vert _{C^\alpha }\leq c\|f\|_{C^\alpha },</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> solves the equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta u minus lamda u equals f"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>u</mml:mi> <mml:mo> − </mml:mo> <mml:mi> λ </mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Delta u-\lambda u=f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The method can be applied to the Laplacian on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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