Conditional Double Integral Transforms with Related Topics on Product Abstract Wiener Space

In this paper, we introduce a conditional double integral transform and a conditional double convolution product for a broad class of functionals defined on product abstract Wiener space. We first establish the existence of the conditional double integral transform, the conditional double convolution product, and the first variation of the associated functionals under appropriate conditions. We then investigate their fundamental properties and derive several relationships among these three concepts, thereby extending the corresponding theory of integral transforms and convolution products on Wiener space. Furthermore, by combining these results, we obtain a unified analytical framework that clarifies the interplay between the conditional double integral transform, the conditional double convolution product, and the first variation. These relationships lead to several new identities and provide a systematic approach to the analysis of functionals on product abstract Wiener space. Finally, we present several observations and illustrative consequences concerning the conditional double integral transform and the conditional double convolution product, which demonstrate the applicability of the proposed framework and suggest directions for further research in Wiener analysis and related areas of mathematical physics.