Narrow Results

In this paper we establish Fubini theorems for integral transforms and convolution products for functionals on function space.

In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space .

An important connection between the finite-dimensional Gaussian Wick products and Lebesgue convolution products will be proven first. Then this connection will be used to prove an important Hölder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Hölder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Hölder inequality and classic Hölder inequality, between the Nelson hypercontractivity and classic Young inequality with the sharp constant, and between the third more general inequality and an extension by Lieb of the Young inequality with the best constant. Since the Gaussian probability measure exists even in the infinite-dimensional case, the above three inequalities can be extended, via a classic Fatou's lemma argument, to the infinite-dimensional framework.

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

In this work, we introduce some properties of a complex fractional differential operator. The boundedness and some other properties are studied in view of geometric function theory.

In this paper, we prove the boundedness for the maximal and fractional maximal operators and Riesz potential-type operator associated with the Kontorovich–Lebedev transform (KL transform)in the $L^p({ℝ}_{+},x^{-\beta }dx)$ spaces.

In this paper, we first introduce and study the quantum (K₁, K₂)-Gross Laplacian denoted Δ_QG(K₁, K₂). Then, we prove that Δ_QG(K₁, K₂) is a well defined and linear continuous operator acting on the space of continuous operators and has a quantum stochastic integral. Finally, we give an explicit solution of the quantum heat equation associated with Δ_QG(K₁, K₂). Then, under some positive conditions, we give an integral representation of this solution.