Conditional integral transforms and convolutions for a general vector-valued conditioning functions

We study the conditional integral transforms and conditional  convolutions of functionals defined on $K[0, T]$. We consider a general vector-valued conditioning functions  $X_k (x) = \left(\gamma_1 (x), \ldots, \gamma_k (x)\right)$  where $\gamma_j (x)$ are Gaussian random variables  on the Wiener space which need not depend upon the values of $x$ at only finitely many points in $(0, T]$.  We then obtain several relationships and formulas for the conditioning functions that exist among conditional integral transform,  conditional convolution and first variation of functionals in $E_\sigma$.