The W-Operator: A Volterra Fractional Time Operator with Sharp Bernstein Threshold and Regularized Memory

Abstract

We introduce a new two-parameter fractional time operator with Volterra structure, denoted by ${}^W{D}_t^{\hat{I}\pm ,{\hat{I}}^2}$, defined through the Laplace symbol \[ Φ_{α,β}(s) = \frac{s^α}{\bigl(1+(1-α)s^{α-1}\bigr)^β}, \qquad 0<α<1, \ β\ge0. \] The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime via ${\hat{I}}^2$. We develop an explicit symbolic/Volterra theory: Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. A central contribution is a sharp clarification of the Bernstein structure of the symbol. We show that the natural factorization $\hat{I}{¦}_{\hat{I}\pm ,{\hat{I}}^2}(s)=s^{\hat{I}}\pm h_{\hat{I}}\pm (s{)}^2$ does not fit the classical Bernstein product mechanism for any ${\hat{I}}^2>0$. Nevertheless, by a direct complete-monotonicity argument on $\hat{I}{¦}_{\hat{I}\pm ,{\hat{I}}^2}^{\prime }$, we prove the exact Bernstein threshold \[ Φ_{α,β}\in\mathcal{BF} \quad\Longleftrightarrow\quad 0\leβ\le1. \] where $\mathcal{B}\mathcal{F}$ denotes the class of Bernstein functions \noindent For ${\hat{I}}^2>1$, the Bernstein property fails by a low-frequency asymptotic convexity obstruction. This shows that the Bernstein nature of the natural range $0\le {\hat{I}}^2\le 1$ is genuine but is not produced by the standard product mechanism. We then establish well-posedness of abstract W-fractional Cauchy problems with sectorial generators by resolvent estimates and Laplace inversion, yielding a W-resolvent family with temporal regularity and smoothing properties. As an illustration, we apply the theory to a W-fractional diffusion model and discuss the effect of ${\hat{I}}^2$ on the relaxation of spectral modes.

30 pages, 3 figures. Submitted to Fractional Calculus and Applied Analysis (FCAA)