Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights

In the settings of The Real Numbers: Standing Notation and Background and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation. The variance sequence $c$ is a weight sequence: its terms are positive by Variance Sequences and Their Truncations §variances, and each term $c_{k}$ is at most the sum of the convergent series $\sum_{j=1}^{\infty}c_{j}$ of nonnegative terms by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. In this item The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis, The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability, The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space, Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space and The Gaussian Sobolev Space of the Noise Gradient are all read with the weight sequence $c$ in place of the noise weights $a$. Thus $X^{c}$ is the noise space of $c$, with the pairing $\langle\cdot,\cdot\rangle_{c}$ and norm $|\cdot|_{c}$ of The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product; by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §space and The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §space it is the Cameron-Martin space $H_{c}$, and $|h|_{c}^{2}$ is the Cameron-Martin square of $h\in H_{c}$, the two series being the same. $\mathbb{D}^{1,2}_{c}$ and $\nabla_{c}$ are the Gaussian Sobolev space and its noise gradient so read, and $\nabla_{c}F$ is a class in $L^{2}(\gamma_{c};X^{c})$ for $F\in\mathbb{D}^{1,2}_{c}$.

(Cameron-Martin partial derivatives)¶ For $F\in\mathbb{D}^{1,2}_{c}$, a map $g:X\to X^{c}$ in the class $\nabla_{c}F$, and $k\in\mathbb{N}$, the $k$-th Cameron-Martin partial derivative $\partial_{k}F$ of $F$ is the class in $L^{2}(\gamma_{c})$ of the function $x\mapsto c_{k}^{-1}\langle g(x),e_{k}\rangle$. This function is Borel by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable; it is square-integrable, since $c_{k}^{-1}\langle g(x),e_{k}\rangle^{2}\le|g(x)|_{c}^{2}$ by the series form of $|\cdot|_{c}^{2}$ in The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, so that $(c_{k}^{-1}\langle g(x),e_{k}\rangle)^{2}\le c_{k}^{-1}|g(x)|_{c}^{2}$; and its class does not depend on the choice of $g$, two such maps agreeing $\gamma_{c}$-almost everywhere. For $F\in\mathcal{F}C^{1}_{b}(X)$, the set of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical, it is the class of the partial derivative $\partial_{k}F$ of that clause, by The Gaussian Sobolev Space of the Noise Gradient §gradient and the formula of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient. A Borel function in the class $\partial_{k}F$ is again written $\partial_{k}F$.