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

Abstract

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.

1. Introduction

Let H be a real separable infinite-dimensional Hilbert space endowed with inner product $⟨·,·⟩$ and norm $|·|=\sqrt{⟨·,·⟩}$. Let B denote the completion of H with a measurable norm ${\parallel ·\parallel }_0$ on H with respect to the Gaussian cylinder set measure ${\nu }_0$ on H. Then we have a triple $B^{\ast }\subset H^{\ast }\approx H\subset B$ with $⟨x,y⟩=(x,y)$ for all x in H and y in $B^{\ast }$, where $(·,·)$ denotes the natural dual pairing between B and $B^{\ast }$, and $B^{\ast }$ and $H^{\ast }$ are topological duals of B and H. In the research results of Gross [1], ${\nu }_0\circ i^{-1}$ admits a unique countably additive extension $\nu$ to the Borel $\sigma$-algebra $\mathcal{B}\left(B\right)$ of B, where i is the natural injection from H to B. In this case $(B,H,\nu )$ is called an abstract Wiener space [1,2,3,4,5,6,7].

In [7], Lee defined an integral transform ${\mathcal{F}}_{\gamma ,\beta }$ of analytic functionals on abstract Wiener space and showed that the integral transform ${\mathcal{F}}_{1/c,i},c\in C\setminus \left\{0\right\}$ can be used to obtain the solution of a differential equation known as Cauchy problems. Moreover, several well-known transforms, including the Fourier–Wiener transform, the modified Fourier–Wiener transform, the Fourier–Feynman transform, and the Gauss transform, can be realized as special cases of ${\mathcal{F}}_{\gamma ,\beta }$ for suitable choices of the parameters $\gamma$ and $\beta$.

Following the introduction of ${\mathcal{F}}_{\gamma ,\beta }$, integral transforms on abstract Wiener space have been extensively investigated together with their connections to convolution products, first variations, and other related topics for various classes of functionals in [2,8,9,10,11,12]. In particular, these studies have mainly focused on integral transforms and their conditional versions for functionals on a single abstract Wiener space. More recently, double integral transforms were introduced in [13,14], where several fundamental relationships and related formulas were established. However, the conditional theory of double integral transforms on product abstract Wiener spaces has not been fully developed.

Motivated by these earlier works, the main novelty of this work lies in combining the double-transform structure with conditional integration on product abstract Wiener space. Thus, the results obtained here are not merely direct extensions of the existing theory but provide a broader framework in which several previously studied transforms and their associated formulas can be recovered as special cases. In particular, the existence and relationship results established in this paper provide a foundation for further investigations of conditional function space integrals, convolution products, and variational formulas, with potential applications in stochastic analysis and mathematical physics. Conditional Wiener transforms are important tools in the analysis of Wiener functionals because they incorporate additional information through conditioning and provide a refined framework for studying stochastic systems under prescribed conditions. They extend the classical Wiener transform by retaining the dependence on the conditioning variable and consequently allow for the derivation of more general identities and relationships. This conditional structure is particularly important when investigating functionals associated with diffusion processes, partial differential equations, and mathematical physics. From this perspective, the study of conditional double integral transforms is not merely a formal generalization of the classical theory; rather, it provides a natural framework for capturing interactions between multipleWiener variables under conditional constraints. This motivates the development of the conditional double integral transform and the conditional double convolution product considered in the present paper. The multivariable conditional transform provides a more flexible framework for analyzing functionals that depend on several variables and their interactions under conditioning. In contrast to the one-variable setting, it allows for the dependence among multiple variables to be retained while incorporating additional information through the conditioning parameters. This structure makes it possible to investigate more complicated relationships among transformed functionals and to derive identities that cannot be fully captured by a single-variable conditional transform.

In particular, the multivariable framework enables the simultaneous treatment of several components of a stochastic system and provides a natural setting for studying their interactions under prescribed conditions. It also reveals structural properties arising from the interplay between the transform variables and the conditioning variables. Consequently, the multivariable conditional transform is not merely a formal extension of the one-variable theory but provides a broader framework for describing and analyzing conditional structures on product abstract Wiener spaces.

2. Definitions and Preliminaries

In this section, we introduce the definitions, notation, and preliminary results that will be used throughout the paper.

First, let ${\left\{e_j\right\}}_{j=1}^{\infty }$ be a complete orthonormal set in H with $e_j\in B^{\ast }$. For each $h\in H$ and $x\in B$, we define a stochastic inner product ${(h,x)}^{\sim }$ by

$${(h,x)}^{\sim }=\left\{\begin{array}{cc}\underset{n\to \infty }{\lim }{\displaystyle {\sum }_{j=1}^n}⟨h,e_j⟩(x,e_j),\hfill & \mathrm{if}\mathrm{the}\mathrm{limit}\mathrm{exists}\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right..$$

Then for every $h(\ne 0)$ in H, ${(h,·)}^{\sim }$ exists for all $x\in B$ and is a Gaussian random variable on B with mean zero and variance ${\left|h\right|}^2$. Also, it is essentially independent of the choice of the complete orthonormal set used in its definition. Also, if both h and x are in H, then Parseval’s identity gives ${(h,x)}^{\sim }=⟨h,x⟩$ and ${(h,\lambda x)}^{\sim }={(\lambda h,x)}^{\sim }=\lambda {(h,x)}^{\sim }$ for all $\lambda \in \mathbb{R}$, $h\in H$ and $x\in B$. Furthermore, if $\{h_1,\dots ,h_n\}$ is an orthonormal set in H, then the random variables ${(h_j,x)}^{\sim }$’s are independent [2,10,14,15].

A subset E of a product abstract Wiener space $B^2\equiv B\times B$ is said to be scale-invariant measurable provided $\{({\rho }_1{x}_1,{\rho }_2{x}_2):(x_1,x_2)\in E\}$ is measurable for every ${\rho }_1>0$ and C, and a scale-invariant measurable set N of $B^2$ is said to be scale-invariant null provided $(\nu \times \nu )\left(\{({\rho }_1{x}_1,{\rho }_2{x}_2):(x_1,x_2)\in N\}\right)=0$ for every ${\rho }_1>0$ and ${\rho }_2>0$. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). A functional F on $B^2$ is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set and $F({\rho }_1·,{\rho }_2·)$ is measurable for every ${\rho }_1>0$ and ${\rho }_2>0$. If two functionals F and G on $B^2$ are equal s-a.e., i.e., for each ${\rho }_1>0$ and ${\rho }_2>0$, $(\nu \times \nu )\left(\{(x_1,x_2)\in B\times B:F({\rho }_1{x}_1,{\rho }_2{x}_2)\ne G({\rho }_1{x}_1,{\rho }_2{x}_2)\}\right)=0$, then we write $F\approx G$ [16].

Next, we state some concepts of the conditional abstract Wiener integral.

Let $\{g_1,\dots ,g_n\}$ be an orthonormal set in H, and let $X:B\to {\mathbb{R}}^n$ be a $\mathcal{B}\left(B\right)$-measurable functional defined by

$$X\left(x\right)=({(g_1,x)}^{\sim },\dots ,{(g_n,x)}^{\sim }).$$

(1)

Let $F:B\to \mathbb{C}$ be a $\nu$-integrable functional. For $\overrightarrow{\eta }\in {\mathbb{R}}^n$, $E\left[F\right|X]\left(\overrightarrow{\eta }\right)$ denotes the conditional abstract Wiener integral of F given X. Then

$$E\left[F\right|X]\left(\overrightarrow{\eta }\right)={\int }_{B}F\left(x-{\displaystyle \sum _{k=1}^n}{(g_k,x)}^{\sim }{g}_k+{\displaystyle \sum _{k=1}^n}{\eta }_k{g}_k\right)d\nu \left(x\right)$$

(2)

for a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$. Equation (2) is called a simple formula for conditioning function X, see [17,18,19].

We use the following notation to simplify various expressions. For $\overrightarrow{\gamma }=({\gamma }_1,{\gamma }_2)\in C^2\setminus \left\{(0,0)\right\}$ and $\overrightarrow{x}=(x_1,x_2)\in B^2$, let $\overrightarrow{\gamma }\circ \overrightarrow{x}=({\gamma }_1{x}_1,{\gamma }_2{x}_2).$ For $\overrightarrow{X}=(X\left(x_1\right),X\left(x_2\right))$, and a given orthonormal set $\{g_1,\dots ,g_n\}$ in H and $\overrightarrow{\eta }\in {\mathbb{R}}^n$, let

$$\left[x_j\right]=x_j-{\displaystyle \sum _{k=1}^n}{(g_k,x_j)}^{\sim }{g}_k+{\displaystyle \sum _{k=1}^n}{\eta }_k{g}_k,j=1,2.$$

Also, let $\overrightarrow{\left[x\right]}=(\left[x_1\right],\left[x_2\right])\in B^2$.

Finally, we give an integration formula that will be used several times in this paper. For each $\alpha \in \mathbb{C}$ and for $h\in H$,

$${\int }_B\exp \left\{\alpha {(h,x)}^{\sim }\right\}d\nu \left(x\right)=\exp \left\{{\displaystyle \frac{{\alpha }^2}{2}}{\left|h\right|}^2\right\}.$$

(3)

We now give the definitions of the CDIT (conditional double integral transform), the CDCP (conditional double convolution product) and the first variation of functionals on ${\tilde{B}}^2\equiv \tilde{B}\times \tilde{B}$, where $\tilde{B}$ denotes the complexification of B.

Definition 1. 

Let F and G be functionals defined on ${\tilde{B}}^2$, and let $\overrightarrow{X}$ be given by (1). For each pair of nonzero complex numbers $\overrightarrow{\gamma }=({\gamma }_1,{\gamma }_2)$ and $\overrightarrow{\beta }=({\beta }_1,{\beta }_2)$, the CDIT ${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})$ of F, the CDCP $({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})$ with respect to $\overrightarrow{\gamma }$ of F and G, and the first variation $\delta F$ of F are given by the formulas

$${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })={\int }_{B^2}F\left(\overrightarrow{\gamma }\circ \overrightarrow{\left[x\right]}+\overrightarrow{\beta }\circ \overrightarrow{y}\right)d\overrightarrow{\nu }\left(\overrightarrow{x}\right),$$

(4)

$$({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })={\int }_{B^2}F\left({\displaystyle \frac{\overrightarrow{y}(·)+\overrightarrow{\gamma }\circ \overrightarrow{\left[x\right]}}{\sqrt{2}}}\right)G\left({\displaystyle \frac{\overrightarrow{y}(·)-\overrightarrow{\gamma }\circ \overrightarrow{\left[x\right]}}{\sqrt{2}}}\right)d\overrightarrow{\nu }\left(\overrightarrow{x}\right)$$

(5)

for $\overrightarrow{y}\in {\tilde{B}}^2$ and a.e. $\overrightarrow{\eta }\in R^n$ and

$$\delta F\left(\overrightarrow{x}\right|\overrightarrow{u})={\displaystyle \frac{\partial }{\partial k_1}}F(x_1+k_1{u}_1,x_2){|}_{k_1=0}+{\displaystyle \frac{\partial }{\partial k_2}}F(x_1,x_2+k_2{u}_2){|}_{k_2=0},$$

(6)

for $k_1,k_2\in \mathbb{R}$ and $\overrightarrow{x},\overrightarrow{u}\in {\tilde{B}}^2$ if they exist.

3. Existence Theorems

Let $\mathcal{M}\left(H\right)$ denote the space of all $\mathbb{C}$-valued countably additive Borel measures on H. Under the total variation norm $\parallel ·\parallel$ and with convolution as multiplication, $\mathcal{M}\left(H\right)$ is a commutative Banach algebra with identity.

Lemma 1. 

For a $f\in \mathcal{M}\left(H\right)$ and a complex number α with $Re\left({\alpha }^2\right)\le 0$,

$${\int }_B{\int }_H\exp \left\{\alpha {(h,x)}^{\sim }\right\}df\left(h\right)d\nu \left(x\right)={\int }_H\exp \left\{{\displaystyle \frac{{\alpha }^2}{2}}{\parallel h\parallel }^2\right\}df\left(h\right)$$

and

$$\left|{\int }_B{\int }_H\exp \left\{\alpha {(h,x)}^{\sim }\right\}df\left(h\right)d\nu \left(x\right)\right|\le \parallel f\parallel <\infty .$$

Proof. 

Using Equation (3) and the fact that $Re\left({\alpha }^2\right)\le 0$, the proof of Lemma 1 is established. □

Let $A_1$ and $A_2$ be bounded, nonnegative self-adjoint operators on H. For complex numbers ${\alpha }_1$ and ${\alpha }_2$ with $Re\left({\alpha }_j^2\right)\le 0,j=1,2$, the class ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$ is defined to be the space of all functionals F on $B^2$ of the form

$$F(x_1,x_2)={\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}h,x_j)}^{\sim }\right\}df\left(h\right)$$

(7)

for some $f\in \mathcal{M}\left(H\right)$. More precisely, since we identify functionals that coincide s-a.e. on $B^2$, ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$ can be regarded as the space of all s-equivalence classes of functionals F of the form (7).

Remark 1. 

When ${\alpha }_1$ and ${\alpha }_2$ are purely imaginary and the range of $A_1+A_2$ is dense in H, the map $f\mapsto \left[F\right]$ defined by (7) establishes an algebra isomorphism between $\mathcal{M}\left(H\right)$ and ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$. Hence, ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$ is a Banach algebra under the norm $\left|F\right|=\left|f\right|$, see [13,15,20,21]. This class contains several important function classes as special cases. If ${\alpha }_1={\alpha }_2=i$, it essentially reduces to the generalized Fresnel class ${\mathcal{F}}_{A_1,A_2}$ in [20]. If, in addition, $A_1$ is the identity operator and $A_2\equiv 0$, it reduces to the Fresnel class $\mathcal{F}\left(B\right)$ and, more generally, is related to the class $\mathcal{S}{\alpha }_1$ introduced in [22] and used in [13,23]. Thus, ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$ provides a unified framework encompassing several classes previously studied in the literature.

We first establish the following preliminary observations, which will be used repeatedly in the proofs of our main results. We observe that

$$\overrightarrow{\gamma }\circ \overrightarrow{\left[x\right]}\left(t\right)=\left({\gamma }_1(x_1-{\displaystyle \sum _{k=1}^n}{(g_k,x_1)}^{\sim }{g}_k+{\displaystyle \sum _{k=1}^n}{\eta }_k{g}_k),{\gamma }_2(x_2-{\displaystyle \sum _{k=1}^n}{(g_k,x_2)}^{\sim }{g}_k+{\displaystyle \sum _{k=1}^n}{\eta }_k{g}_k)\right).$$

For $F\in {\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$, as we will see below theorems and formulas, when evaluating the CDIT, the CDCP and the first variation we encounter the stochastic inner product

$${(A_j^{1/2}h,\left[x_j\right])}^{\sim }={(A_j^{1/2}h,x_j-{\displaystyle \sum _{k=1}^n}{(g_k,x)}^{\sim }{g}_k+{\displaystyle \sum _{k=1}^n}{\eta }_k{g}_k)}^{\sim }$$

for $j=1,2$. Now, let $A_j^{1/2}{b}_h=A_j^{1/2}h-{\displaystyle \sum _{k=1}^n}⟨A_j^{1/2}h,g_k⟩g_k$ for $h\in H$. Then $A_j^{1/2}{b}_h$ is an element of H, and for $j=1,2$ we have

$$\begin{array}{cc}\hfill {(A_j^{1/2}h,x_j-{\displaystyle \sum _{k=1}^n}{(g_k,x_j)}^{\sim }{g}_k+{\displaystyle \sum _{k=1}^n}{\eta }_k{g}_k)}^{\sim }& ={(A_j^{1/2}{b}_h,x_j)}^{\sim }+{\displaystyle \sum _{k=1}^n}{\eta }_k⟨A_j^{1/2}h,g_k⟩\hfill \\ & ={(A_j^{1/2}{b}_h,x_j)}^{\sim }+{⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{R^n}\hfill \end{array}$$

where ${⟨·,·⟩}_{{\mathbb{R}}^n}$ denotes the inner product on ${\mathbb{R}}^n$.

Remark 2. 

When we evaluate the CDIT, the CDCP and the first variation of functionals in ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$, it is necessary to establish the existence of the following integrals;

(i) 

First we could consider the following integral,

$${\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}\left[{\alpha }_j{(A_j^{1/2}h,x_j)}^{\sim }+\zeta {⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right]\right\}df\left(h\right),\zeta \in C.$$

(8)

If we assume that

$${\int }_H\exp \left\{{\displaystyle \sum _{k=1}^n}|\zeta {\eta }_k\left|\right|A_j^{1/2}h\left|\right|g_k|\right\}\left|df\left(h\right)\right|<\infty$$

(9)

for all $j=1,2$ and complex numbers ζ, then

$${\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}h,x_j)}^{\sim }\right\}df\left(h\right)\mathit{and}{\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}\zeta {⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right\}df\left(h\right)$$

exist. However, the integral (8) might not exist because the product of $L_1$-functionals might not be in $L_1$.

(ii) 

Therefore, an additional condition on $f\in \mathcal{M}\left(H\right)$ is required to ensure the existence of (8). Throughout this paper, we assume that, whenever $f\in \mathcal{M}\left(H\right)$ satisfies (9), the integral (8) exists.

(iii) 

In particular, when ζ is purely imaginary, the integral (8) exists. More general sufficient conditions for the existence of (8) can be found in [13,15,20].

Definition 2. 

For each $\alpha \in \mathbb{C}$, let

$$E_{\alpha }=\left\{(\gamma ,\beta )\in \mathbb{C}\times \mathbb{C}:\mathit{Re}\left({\alpha }^2{\gamma }^2\right)\le 0\mathit{and}\mathit{Re}\left({\alpha }^2{\beta }^2\right)\le 0\right\}.$$

In our first theorem, we obtain the formulas for the CDIT and the CDCP of functionals from ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$.

Theorem 1. 

Let $F\in {\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$ be given by Equation (7). Let $G\in {\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$ be given by

$$G(x_1,x_2)={\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}w,x_j)}^{\sim }\right\}dg\left(w\right)$$

for s-a.e. $\overrightarrow{x}\in B^2$, where g is an element of $M\left(H\right)$. Then for all $({\gamma }_j,{\beta }_j)\in E_{{\alpha }_j},j=1,2$, the CDIT ${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})$ of F and the CDCP $({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})$ of F and G given $\overrightarrow{X}$ exist and are given by the formulas

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })& ={\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\alpha }_j{\beta }_j{(A_j^{1/2}h,y_j)}^{\sim }\hfill \\ & +{\displaystyle \sum _{j=1}^2}\left[{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{2}}{\left|A_j^{1/2}{b}_h\right|}^2+{\alpha }_j{\gamma }_j{⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right]\}df\left(h\right)\hfill \end{array}$$

(10)

and

$$\begin{array}{c}({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })\hfill \\ ={\int }_H{\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j}{\sqrt{2}}}{(A_j^{1/2}(h+w),y_j)}^{\sim }\hfill \\ +{\displaystyle \sum _{j=1}^2}\left[{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{4}}{\left|A_j^{1/2}{b}_{h-w}\right|}^2+{\displaystyle \frac{{\alpha }_j{\gamma }_j}{\sqrt{2}}}{⟨X\left(A_j^{1/2}(h-w)\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right]\}df\left(h\right)dg\left(w\right)\hfill \end{array}$$

(11)

for s-a.e. $\overrightarrow{y}\in B^2$ and a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$.

Proof. 

Using Equations (4) and (3), it follows that for s-a.e. $\overrightarrow{y}\in B^2$ and a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$,

$$\begin{array}{c}{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })\hfill \\ ={\int }_{B^2}{\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\alpha }_j{\gamma }_j{(A_j^{1/2}{b}_h,x_j)}^{\sim }\hfill \\ +{\displaystyle \sum _{j=1}^2}{\alpha }_j{\gamma }_j{⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}+{\displaystyle \sum _{j=1}^2}{\alpha }_j{\beta }_j{(A_j^{1/2}h,y_j)}^{\sim }\}df\left(h\right)d\overrightarrow{\nu }\left(\overrightarrow{x}\right)\hfill \\ ={\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\alpha }_j{\beta }_j{(A_j^{1/2}h,y_j)}^{\sim }\hfill \\ +{\displaystyle \sum _{j=1}^2}\left[{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{2}}{\left|A_j^{1/2}{b}_h\right|}^2+{\alpha }_j{\gamma }_j{⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right]\}df\left(h\right)\hfill \end{array}$$

(12)

and so Equation (10) is established. On the other hand, using Equations (5) and (3) it follows that for s-a.e. $\overrightarrow{y}\in B^2$ and a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$

$$\begin{array}{c}({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })\hfill \\ ={\int }_{B^2}{\int }_H{\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\displaystyle (\frac{{\alpha }_j{\gamma }_j}{\sqrt{2}}}(A_j^{1/2}{b}_h-A_j^{1/2}{b}_w),x_j{)}^{\sim }\hfill \\ +{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j{\gamma }_j}{\sqrt{2}}}{⟨X\left(A_j^{1/2}(h-w)\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}+{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j}{\sqrt{2}}}{(A_j^{1/2}(h+w),y_j)}^{\sim }\}df\left(h\right)dg\left(w\right)d\overrightarrow{\nu }\left(\overrightarrow{x}\right)\hfill \\ ={\int }_H{\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j}{\sqrt{2}}}{(A_j^{1/2}(h+w),y_j)}^{\sim }\hfill \\ +{\displaystyle \sum _{j=1}^2}\left[{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{4}}{\left|A_j^{1/2}{b}_{h-w}\right|}^2+{\displaystyle \frac{{\alpha }_j{\gamma }_j}{\sqrt{2}}}{⟨X\left(A_j^{1/2}(h-w)\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right]\}df\left(h\right)dg\left(w\right).\hfill \end{array}$$

(13)

Hence we complete the proof of Theorem 1. □

From Theorem 1, we have the following observations.

Remark 3. 

(i) The CDIT ${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})$, as a function of $\overrightarrow{y}$, is an element of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1{\beta }_1,{\alpha }_2{\beta }_2}$. Let ${\varphi }_1^{\overrightarrow{\eta }}$ be a set function defined by formula

$${\varphi }_1^{\overrightarrow{\eta }}\left(E\right)={\int }_E\exp \left\{{\displaystyle \sum _{j=1}^2}\left[{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{2}}{\left|A_j^{1/2}{b}_h\right|}^2+{\alpha }_j{\gamma }_j{⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right]\right\}df\left(h\right)$$

for $E\in \mathcal{B}\left(H\right)$. Then ${\varphi }_1^{\overrightarrow{\eta }}$ is an element of $M\left(H\right)$ since $({\gamma }_j,{\beta }_j)\in E_{{\alpha }_j},j=1,2$, and so the last expression in Equation (12) becomes

$${\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{\beta }_j{(A_j^{1/2}h,y_j)}^{\sim }\right\}d{\varphi }_1^{\overrightarrow{\eta }}\left(h\right).$$

Hence the CDIT ${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})$ is an element of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1{\beta }_1,{\alpha }_2{\beta }_2}$.

(ii) The CDCP $({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})$, as a function of $\overrightarrow{y}$, is also an element of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$. Let ${\varphi }^{\overrightarrow{\eta }}$ be a set function defined by formula

$${\varphi }^{\overrightarrow{\eta }}\left(E\right)={\int }_E\exp \left\{{\displaystyle \sum _{j=1}^2}\left[{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{4}}{\left|A_j^{1/2}{b}_{h-w}\right|}^2+{\displaystyle \frac{{\alpha }_j{\gamma }_j}{\sqrt{2}}}{⟨X\left(A_j^{1/2}(h-w)\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\right]\right\}df\left(h\right)dg\left(w\right)$$

for $E\in \mathcal{B}(H\times H)$ and $\rho :H\times H\to H$ be a function defined by $k(h,w)=(h+w)/\sqrt{2}$. Then ${\varphi }_2^{\overrightarrow{\eta }}={\varphi }^{\overrightarrow{\eta }}\circ k^{-1}$ is an element of $M\left(H\right)$ since $({\gamma }_j,{\beta }_j)\in E_{{\alpha }_j},j=1,2$ and so the last expression in Equation (11) becomes

$${\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}k,y_j)}^{\sim }\right\}d{\varphi }_2^{\overrightarrow{\eta }}\left(k\right)$$

and so the CDCP $({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })$ is an element of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$.

The following observation will be useful in the proofs of our main results. One can see that if $h\in H$ and $x\in H\subset B$, then ${(h,x)}^{\sim }=(h,x)$. For any $w,u\in H$, the Cauchy–Schwarz inequality gives

$$|⟨w,u⟩|\le \left|w\right|\left|u\right|,$$

and consequently,

$${|(w,u)}^{\sim }| \le |w\left|\right|u|.$$

For $F\in {\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$, let $f\in M\left(H\right)$ denote its associated measure. Throughout this paper, we assume that f satisfies

$${\int }_H|{\alpha }_j||A_j^{1/2}h||df\left(h\right)|<\infty ,j=1,2.$$

(14)

In our next theorem, we obtain a formula for the first variation of functionals from ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$ to ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$.

Theorem 2. 

Let F and f be as in Theorem 1 and let $u_1,u_2\in H$. Assume that $F(x_1,x_2)$ has a first variation $\delta F\left(\overrightarrow{x}\right|\overrightarrow{u})$ for all $\overrightarrow{x}\in B^2$ such that for some ${\rho }_1>0$ and ${\rho }_2>0$,

$$\underset{|t_1|\le {\rho }_1}{\sup }|\delta F(x_1+t_1{u}_1,x_2|u_1,u_2)| +\underset{|t_2|\le {\rho }_2}{\sup }\left|\delta F(x_1,x_2+t_2{u}_2|u_1,u_2)\right|$$

is integrable on $B^2$. Then the first variation $\delta F\left(\overrightarrow{x}\right|\overrightarrow{u})$ of F is given by the formula

$$\delta F\left(\overrightarrow{x}\right|\overrightarrow{u})={\displaystyle \sum _{j=1}^2}{\int }_H{\alpha }_j⟨A_j^{1/2}h,u_j⟩\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}h,x_j)}^{\sim }\right\}df\left(h\right)$$

(15)

for s-a.e. $\overrightarrow{x}\in B^2$. Furthermore, as a function of $\overrightarrow{x}$, $\delta F$ is an element of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$. In fact,

$$\delta F\left(\overrightarrow{x}\right|\overrightarrow{u})={\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}h,x_j)}^{\sim }\right\}d{\varphi }_3\left(h\right),$$

where ${\varphi }_3$ is an element of $M\left(H\right)$, defined as in the following proof.

Proof. 

Using Equation (6) it follows that for s-a.e. $\overrightarrow{x}\in B^2$, by a similar method to that used in the proof of Theorem 1, we can easily obtain Equation (15) as follows.

$$\begin{array}{c}\delta F\left(\overrightarrow{x}\right|\overrightarrow{u})\hfill \\ ={\displaystyle \frac{\partial }{\partial t_1}}\left({\int }_H\exp \left\{{\alpha }_1{(A_1^{1/2}h,x_1)}^{\sim }+{\alpha }_2{(A_2^{1/2}h,x_2)}^{\sim }+{\alpha }_1{t}_1⟨A_1^{1/2}h,u_1⟩\right\}df\left(h\right)\right){|}_{t_1=0}\hfill \\ +{\displaystyle \frac{\partial }{\partial t_2}}\left({\int }_H\exp \left\{{\alpha }_1{(A_1^{1/2}h,x_1)}^{\sim }+{\alpha }_2{(A_2^{1/2}h,x_2)}^{\sim }+{\alpha }_2{t}_2⟨A_2^{1/2}h,u_2⟩\right\}df\left(h\right)\right){|}_{t_2=0}\hfill \\ ={\displaystyle \sum _{j=1}^2}{\int }_H{\alpha }_j⟨A_j^{1/2}h,u_j⟩\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}h,x_j)}^{\sim }\right\}df\left(h\right)\hfill \end{array}$$

(16)

and so we can establish Equation (15). Now let ${\varphi }_3$ be a set function defined by

$${\varphi }_3\left(E\right)={\displaystyle \sum _{j=1}^2}{\int }_E{\alpha }_j⟨A_j^{1/2}h,u_j⟩df\left(h\right)$$

for $E\in \mathcal{B}\left(H\right)$. Then ${\varphi }_3$ is an element of $M\left(H\right)$ using Equation (14) and so the last expression in Equation (15) becomes

$${\int }_H\exp \left\{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}h,x_j)}^{\sim }\right\}d{\varphi }_3\left(h\right).$$

Hence $\delta F$ is an element of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$. □

4. Some Relationships

In many previous studies [2,8,10,11,12,23], various relationships have been established among integral transforms and related operators, including the integral transform, the Fourier–Wiener transform, the modified Fourier–Wiener transform, the Gauss transform, the analytic Fourier–Feynman transform, and conditional integral transforms. Motivated by these earlier works, we investigate analogous relationships for the proposed framework. In particular, we derive several fundamental relationships among the CDIT, the CDCP, and the first variation of functionals in ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$.

We begin this section by establishing two fundamental relationships. The first states that the CDIT of the CDCP is the product of the corresponding CDITs, while the second shows that the CDIT of the first variation coincides with the first variation of the CDIT.

Theorem 3. 

Let $F,G,f,g$ and u be as in Theorems 1 and 2. Let $({\gamma }_j,{\beta }_j)\in E_{{\alpha }_j},j=1,2$. Then we have

$$\begin{array}{c}{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_1})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_2})\hfill \\ ={\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}}){\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})\hfill \end{array}$$

(17)

and

$${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(\delta F\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })=\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})\left(\overrightarrow{y}\right|\overrightarrow{u},\overrightarrow{\eta })$$

(18)

for s-a.e. $\overrightarrow{y}\in B^2$ and a.e. $\overrightarrow{{\eta }_1},\overrightarrow{{\eta }_2},\overrightarrow{\eta }\in R^n$. Also, both sides of the expression in Equations (17) and (18) are given by the formula

$$\begin{array}{c}{\int }_H{\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j{\beta }_j}{\sqrt{2}}}{(A_j^{1/2}(h+w),y_j)}^{\sim }+{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{2}}\left[|A_j^{1/2}{b}_h{|}^2+{\left|A_j^{1/2}{b}_w\right|}^2\right]\hfill \\ +{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j{\gamma }_j}{\sqrt{2}}}\left[{⟨X\left(A_j^{1/2}(h+w)\right),\overrightarrow{{\eta }_2}⟩}_{{\mathbb{R}}^n}+{⟨X\left(A_j^{1/2}(h-w)\right),\overrightarrow{{\eta }_1}⟩}_{{\mathbb{R}}^n}\right]\}df\left(h\right)dg\left(w\right)\hfill \end{array}$$

and

$$\begin{array}{c}{\displaystyle \sum _{j=1}^2}{\int }_H{\alpha }_j{\beta }_j⟨A_j^{1/2}h,u_j⟩\exp \{{\displaystyle \sum _{j=1}^2}{\alpha }_j{\beta }_j{(A_j^{1/2}h,y_j)}^{\sim }\hfill \\ +{\displaystyle \sum _{j=1}^2}{\displaystyle \frac{{\alpha }_j^2{\gamma }_j^2}{2}}{\left|A_j^{1/2}{b}_h\right|}^2+{⟨X\left(A_j^{1/2}h\right),\overrightarrow{\eta }⟩}_{{\mathbb{R}}^n}\}df\left(h\right).\hfill \end{array}$$

Proof. 

We first note that for all $h,w\in H$,

$${|h+w|}^2{ + |h-w|}^2={2\left(\right|h|}^2+{\left|w\right|}^2)$$

and for each $j=1,2$

$$\begin{array}{c}{⟨X\left(A_j^{1/2}h\right),{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}}⟩}_{{\mathbb{R}}^n}+{⟨X\left(A_j^{1/2}w\right),{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}}⟩}_{{\mathbb{R}}^n}\hfill \\ ={⟨X\left(A_j^{1/2}(h+w)\right),{\displaystyle \frac{\overrightarrow{{\eta }_2}}{\sqrt{2}}}⟩}_{{\mathbb{R}}^n}+{⟨X\left(A_j^{1/2}(h-w)\right),{\displaystyle \frac{\overrightarrow{{\eta }_1}}{\sqrt{2}}}⟩}_{{\mathbb{R}}^n}\hfill \end{array}$$

for a.e $\overrightarrow{{\eta }_1},\overrightarrow{{\eta }_2}\in {\mathbb{R}}^n$. The proof follows by applying (3)–(6), together with (10)–(12), and (15), to both sides of (17) and (18). This completes the proof of Theorem 3. □

We summarize the following formulas in tabular form without proofs, since their existence can be established by arguments analogous to those used in the previous results and formulas presented in this paper. Indeed, each identity listed in the table follows directly from repeated applications of Equations (3)–(18). For example, the relationship R3 is obtained by the formula

$$\begin{array}{c}{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left(\delta ({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})\right)(\overrightarrow{·},\overrightarrow{{\eta }_1})\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_2})\hfill \\ =\delta \left[{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(\left(({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})\right)(\overrightarrow{·},\overrightarrow{{\eta }_1})\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_2})\right]\left(\overrightarrow{y}\right|\overrightarrow{u}).\hfill \end{array}$$

(19)

as elements of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1{\beta }_1,{\alpha }_2{\beta }_2}$. Furthermore using relationship $R1$, we get a relationship

$$\begin{array}{c}{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left(\delta ({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})\right)(\overrightarrow{·},\overrightarrow{{\eta }_1})\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_2})\hfill \\ =\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|{\displaystyle \frac{\overrightarrow{u}}{\sqrt{2}}}){\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})\hfill \\ +{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}})\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|{\displaystyle \frac{\overrightarrow{u}}{\sqrt{2}}}).\hfill \end{array}$$

(20)

as elements of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1{\beta }_1,{\alpha }_2{\beta }_2}$.

In the table, all relationships are explained as follows;

(R1)

Formulas for the first variation of CDIT of the CDCP.

(R2)

Formulas for the CDIT with respect to the first argument of the first variation of CDCP 1.

(R3)

Formulas for the CDIT with respect to the first argument of the first variation of CDCP 2.

(R4)

Formulas for the CDIT of the CDCP with respect to the first argument of the first variation.

(R5)

A formula for the CDCP of the CDIT of the first variation.

(R6)

Formulas for the first variation of the CDIT of the product functional.

Remark 4. 

From Theorems 1–3, all expressions in the Table 1 are elements of ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1{\beta }_1,{\alpha }_2{\beta }_2}$.

Table 1. Various relationships.

	Relationships
$R1$.
	$\delta [{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left(({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})\right)(\overrightarrow{·},\overrightarrow{{\eta }_1})\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_2})]\left(\overrightarrow{y}\right|\overrightarrow{u})$
	$=\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|{\displaystyle \frac{\overrightarrow{u}}{\sqrt{2}}}){\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})$
	$+{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}})\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|{\displaystyle \frac{\overrightarrow{u}}{\sqrt{2}}})$
$R2$.
	${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left(\delta ({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})\right)(\overrightarrow{·},\overrightarrow{{\eta }_1})\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_2})$
	$=\delta \left[{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(\left(({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})\right)(\overrightarrow{·},\overrightarrow{{\eta }_1})\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_2})\right]\left(\overrightarrow{y}\right|\overrightarrow{u})$
$R3$.
	${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left(\delta ({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})\right)(\overrightarrow{·},\overrightarrow{{\eta }_1})\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_2})$
	$=\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|{\displaystyle \frac{\overrightarrow{u}}{\sqrt{2}}}){\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})$
	$+{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}})\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|{\displaystyle \frac{\overrightarrow{u}}{\sqrt{2}}})$
$R4$.
	${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(({(\delta F\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\ast \delta G\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u}))}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_1})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_2})$
	$={\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(\delta F\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}}){\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(\delta G\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})$
	$=\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}+\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|\overrightarrow{u})\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})(\overrightarrow{·},{\displaystyle \frac{\overrightarrow{{\eta }_2}-\overrightarrow{{\eta }_1}}{\sqrt{2}}})\left({\displaystyle \frac{\overrightarrow{y}}{\sqrt{2}}}\right|\overrightarrow{u})$
$R5$.
	$({\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}{((\delta F\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_1})\ast {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(\delta G\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_2}))}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_3})$
	$={((\delta \left({\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_1})\right)\left(\overrightarrow{·}\right|\overrightarrow{u})\ast \delta \left({\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(G\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_2})\right))\left(\overrightarrow{·}\right|\overrightarrow{u})\parallel \overrightarrow{X})}_{\overrightarrow{\gamma }})(\overrightarrow{y},\overrightarrow{{\eta }_3})$
$R6$.
	${\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(\delta F\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})G\left(\overrightarrow{·}\right)\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })+{\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(F\left(\overrightarrow{·}\right)\delta G\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })$
	$={\mathcal{F}}_{\gamma ,\beta }(\delta \left(FG\right)\left(\overrightarrow{·}\right|\overrightarrow{\beta }\circ \overrightarrow{u})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })$
	$=\delta {\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}(FG\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{\eta })\left(\overrightarrow{y}\right|\overrightarrow{u})$

5. Some Observations

In this section, we present three remarks that provide additional observations and related results concerning the CDIT and the CDCP.

(i)

In [2,5,7,8,10,13,22], inverse transforms have been established for several classes of integral transforms. For example, the inverse analytic Fourier–Feynman transform $T_q,q\in \mathbb{R}$ is $T_{-q}$, and the inverse integral transform ${\mathcal{F}}_{\gamma ,\beta }$ is ${\mathcal{F}}_{i\gamma /\beta ,1/\beta }$. In particular, the inverse double integral transform is given by the formula

$${\mathcal{F}}_{\overrightarrow{{\gamma }_0},\overrightarrow{{\beta }_0}}\left({\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left(F\right)\right)\left(\overrightarrow{y}\right)=F\left(\overrightarrow{y}\right)={\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left({\mathcal{F}}_{\overrightarrow{{\gamma }_0},\overrightarrow{{\beta }_0}}\left(F\right)\right)\left(\overrightarrow{y}\right)$$

for s-a.e. $\overrightarrow{y}\in B^2$, where $\overrightarrow{{\gamma }_0}=(i{\displaystyle \frac{{\gamma }_1}{{\beta }_1}},i{\displaystyle \frac{{\gamma }_2}{{\beta }_2}})$ and $\overrightarrow{{\beta }_0}=({\displaystyle \frac{1}{{\beta }_1}},{\displaystyle \frac{1}{{\beta }_2}})$, see [13]. However, the corresponding inverse transform does not exist for the CDIT in general. Indeed,

$$\begin{array}{c}({\mathcal{F}}_{\overrightarrow{{\gamma }_0},\overrightarrow{{\beta }_0}}({\mathcal{F}}_{\overrightarrow{\gamma },\overrightarrow{\beta }}\left(F\right)\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_1})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_0})\hfill \\ ={\int }_H\exp \{{\displaystyle \sum _{j=1}^2}{\alpha }_j{(A_j^{1/2}h,y_j)}^{\sim }\hfill \\ +{\displaystyle \sum _{j=1}^2}{\alpha }_j{\gamma }_j\left({⟨X\left(A_j^{1/2}h\right),\overrightarrow{{\eta }_1}⟩}_{R^n}+i{⟨X\left(A_j^{1/2}h\right),\overrightarrow{{\eta }_0}⟩}_{{\mathbb{R}}^n}\right)\}df\left(h\right)\hfill \\ \ne F\left(\overrightarrow{y}\right)\hfill \end{array}$$

since a.e. $\overrightarrow{{\eta }_0},\overrightarrow{{\eta }_1}\in {\mathbb{R}}^n$ and so $\overrightarrow{{\eta }_1}+i\overrightarrow{{\eta }_0}\ne \overrightarrow{0}$.

(ii)

The CDIT satisfies a commutative property under appropriate conditions. More precisely,

$$\begin{array}{c}({\mathcal{F}}_{\overrightarrow{{\gamma }_1},\overrightarrow{{\beta }_1}}({\mathcal{F}}_{\overrightarrow{{\gamma }_2},\overrightarrow{{\beta }_2}}\left(F\right)\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_2})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_1})\hfill \\ =({\mathcal{F}}_{\overrightarrow{{\gamma }_2},\overrightarrow{{\beta }_2}}({\mathcal{F}}_{\overrightarrow{{\gamma }_1},\overrightarrow{{\beta }_1}}\left(F\right)\parallel \overrightarrow{X})(\overrightarrow{·},\overrightarrow{{\eta }_1})\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{{\eta }_2})\hfill \end{array}$$

if ${\gamma }_{1j}{\eta }_{1m}={\gamma }_{2j}{\eta }_{2m}$ and ${\gamma }_{jk}^2+{\beta }_{jk}^2=1$ for $j=1,2,m=1,2\dots ,n,k=1,2$.

(iii)

For many classical integral transforms, including the integral transform and the analytic Fourier–Feynman transform, the corresponding convolution products are commutative. In contrast, conditional convolution products generally do not possess the same commutativity property. In particular, the conditional convolution associated with the CDCP satisfies

$$({(F\ast G)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{y},\overrightarrow{\eta })=({(G\ast F)}_{\overrightarrow{\gamma }}\parallel \overrightarrow{X})(\overrightarrow{y},-\overrightarrow{\eta })$$

for s-a.e. $\overrightarrow{y}\in B^2$ and a.e. $\overrightarrow{\eta }\in {\mathbb{R}}^n$. This identity shows that, although the conditional convolution product does not generally preserve the usual commutativity of the convolution product, its lack of commutativity is precisely reflected by a change in the conditioning parameter from $\overrightarrow{\eta }$ to $-\overrightarrow{\eta }$. Thus, the result reveals a structural effect of conditioning rather than being merely a formal extension of the corresponding double-transform theory. Thus, the CDCP does not generally yield a commutative conditional convolution structure. Nevertheless, the CDCP satisfies the usual additive properties described in the theorems and formulas of Section 3 and Section 4.

6. Conclusions

In Section 3, we established the existence of the CDIT, the CDCP, and the first variation for functionals in the class ${\mathcal{H}}_{A_1,A_2}^{{\alpha }_1,{\alpha }_2}$. We then derived several fundamental relationships among these notions and presented additional observations concerning the CDIT and the CDCP. The results obtained in this paper extend and generalize those of previous studies [2,8,10,11,12,23]. In particular, the corresponding results and formulas in these works can be recovered as special cases of our results. Thus, the present framework provides a unified and more general setting for studying conditional integral transforms, convolution products, and first variations on product abstract Wiener space.

These results provide a basis for further investigations of conditional function space integrals and their related structures. They may also be useful in future studies of generalized integral transforms and their applications in stochastic analysis and mathematical physics.