The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914235068448768 |
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| author | Wakrim, Mohamed |
| author_facet | Wakrim, Mohamed |
| contents | We introduce a new two-parameter fractional time operator with Volterra structure, denoted by the W-operator, defined through a generalized Laplace symbol. The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime through an additional parameter, leading to regularized memory effects. We develop a complete symbolic and Volterra theory, including explicit Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. We show that the natural factorization of the Laplace symbol does not fit the classical Bernstein product mechanism and that the symbol is not a Bernstein function in general. Despite this non-Bernstein character, we establish well-posedness of abstract 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 influence of the modulation parameter on the relaxation of spectral modes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02876 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol Wakrim, Mohamed Analysis of PDEs Functional Analysis 26A33, 45D05, 35R11 26A33, 45D05, 35R11 We introduce a new two-parameter fractional time operator with Volterra structure, denoted by the W-operator, defined through a generalized Laplace symbol. The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime through an additional parameter, leading to regularized memory effects. We develop a complete symbolic and Volterra theory, including explicit Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. We show that the natural factorization of the Laplace symbol does not fit the classical Bernstein product mechanism and that the symbol is not a Bernstein function in general. Despite this non-Bernstein character, we establish well-posedness of abstract 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 influence of the modulation parameter on the relaxation of spectral modes. |
| title | The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol |
| topic | Analysis of PDEs Functional Analysis 26A33, 45D05, 35R11 26A33, 45D05, 35R11 |
| url | https://arxiv.org/abs/2601.02876 |