Abstract
We consider a family \(\{L_t,\, t\in [0,T]\}\) of closed operators generated by a family of regular (non-symmetric) Dirichlet forms \(\{(B^{(t)},V),t\in [0,T]\}\) on \(L^2(E;m)\). We show that a bounded (signed) measure \(\mu \) on \((0,T)\times E\) is smooth, i.e. charges no set of zero parabolic capacity associated with \(\frac{\partial }{\partial t}+L_t\), if and only if \(\mu \) is of the form \(\mu =f\cdot m_1+g_1+\partial _tg_2\) with \(f\in L^1((0,T)\times E;\mathrm{d}t\otimes m)\), \(g_1\in L^2(0,T;V')\), \(g_2\in L^2(0,T;V)\). We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator \(\frac{\partial }{\partial t}+L_t\) and a functional from the dual \({{\mathcal {W}}}'\) of the space \({{\mathcal {W}}}=\{u\in L^2(0,T;V):\partial _t u\in L^2(0,T;V')\}\) on the right-hand side of the equation.
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This work was supported by Polish National Science Centre (Grant No. 2016/23/B/ST1/01543).
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Klimsiak, T., Rozkosz, A. Smooth measures and capacities associated with nonlocal parabolic operators. J. Evol. Equ. 19, 997–1040 (2019). https://doi.org/10.1007/s00028-019-00500-0
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DOI: https://doi.org/10.1007/s00028-019-00500-0