This paper establishes a general mathematical framework defining a parametric family of non-Gaussian processes, termed the degenerate Prabhakar process (dpp), which encompasses classical stochastic models such as fractional Brownian motion and grey Brownian motion. Processes in this family feature self-similarity and stationary increments, parameterized by four real numbers ( λ∈ (0, 1) , α , β , and γ ) subject to specific conditions. By explicitly computing their finite-dimensional probability density functions, we reveal novel characterizations of the dpp. Furthermore, we prove that the dpp almost surely admits a local time that is ν -square-integrable.
We study the boundedness of fractional integral operators with homogeneous kernels and their commutators on Morrey spaces built on rearrangement-invariant quasi-Banach function spaces . Precisely, we first establish the norm inequalities for these operators and their commutators on rearrangement-invariant quasi-Banach function spaces using interpolation. These estimates are then employed to derive the boundedness of such operators on Morrey spaces built on rearrangement-invariant quasi-Banach function spaces.
Let T be a bounded operator on a complex Hilbert space and let x be a unit vector. We determine exactly the possible values of the first moment ⟨ Tx,x⟩ when the second-order data ‖ Tx‖ ^2 , ‖ T^*x‖ ^2 , and ⟨ T^2x,x⟩ are prescribed. The admissible set is an elliptic disk, degenerating to a line segment in the degenerate case. An associated 3× 3 Gram matrix gives a necessary and sufficient condition for realizability, and its rank is shown to be the least dimension of a matrix realization. From this description we obtain the best phase-independent upper bound for | ⟨ Tx,x⟩| ^2 , a sharper phase-sensitive estimate, exact difference formulas, and complete equality conditions. We also prove a sharp Buzano inequality with a prescribed ratio of the two projection magnitudes; in the nondegenerate case, the classical Buzano bound is recovered by optimizing over that ratio. The Hilbert-space results are then transferred to states on unital C^* -algebras and to the standard A-Berezin number through the Hilbert space associated with the seminorm ‖·‖ _A . The A-Berezin estimates retain the necessary kernel factor ‖k_λ‖ _A^2 and contain an explicit correction involving the A-self-commutator. Finite-dimensional examples, multiplier operators, the Bergman shift, and a nonnormal Toeplitz family on the Hardy space illustrate sharpness and quantify the improvement over several established Buzano-type bounds.
Some new sums of series involving products of the Anger and Weber functions, the argument of which depends on the summation index, as well as trigonometric and power functions, are derived using complex analysis.
In this paper, we study the convolution product of modulated Hermite functions and establish connections with two important families of functions, namely the complex Itô-Hermite polynomials and the special Hermite functions. Our analysis is based on the Segal-Bargmann transform and Weyl operators in the setting of classical Fock spaces. The results presented here build on earlier work [4], where related methods were investigated in the context of superoscillations via the short-time Fourier transform.
