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

We introduce a new two-parameter fractional time operator with Volterra structure, denoted by ${}^W{D}_t^{\alpha ,\beta }$, defined through the Laplace symbol $${\mathrm{\Phi }}_{\alpha ,\beta }(s)=\frac{s^{\alpha }}{(1+(1-\alpha )s^{\alpha -1}{)}^{\beta }},0<\alpha <1,\beta \ge 0.$$ The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime via $\beta$. 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 ${\mathrm{\Phi }}_{\alpha ,\beta }(s)=s^{\alpha }{h}_{\alpha }(s{)}^{\beta }$ does not fit the classical Bernstein product mechanism for any $\beta >0$. Nevertheless, by a direct complete-monotonicity argument on ${\mathrm{\Phi }}_{\alpha ,\beta }^{\mathrm{\prime }}$, we prove the exact Bernstein threshold $${\mathrm{\Phi }}_{\alpha ,\beta }\in \mathcal{B}\mathcal{F}⟺0\le \beta \le 1.$$ where $\mathcal{B}\mathcal{F}$ denotes the class of Bernstein functions \noindent For $\beta >1$, the Bernstein property fails by a low-frequency asymptotic convexity obstruction. This shows that the Bernstein nature of the natural range $0\le \beta \le 1$ 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 $\beta$ on the relaxation of spectral modes.