  In this paper, the notion of singular backward stochastic Volterra integral
equations (singular BSVIEs for short) in infinite dimensional space is
introduced, and the corresponding well-posedness is carefully established. A
class of singularity conditions are proposed, which not only cover that of
fractional kernel, Volterra Heston model kernel, completely monotone kernels,
to mention a few, but also happen to be used in the forward stochastic Volterra
integral with new conclusions arising. Motivated by mathematical physics
problem such as the viscoelasticity/thermoviscoelasticity of materials, heat
conduction in materials with memory, optimal control problems of abstract
stochastic Volterra integral equations (including fractional stochastic
evolution equations and stochastic evolutionary integral equations) are
presented. At last, our BSVIEs are surprisingly used in maximum principle of
controlled stochastic delay evolution equations. One advantage of this new
standpoint is that the final cost functional can naturally depend on the past
state for the first time.
