Regularization of some classes of ultradistribution semigroups and sines

Abstract

We systematically analyze regularization of different kinds of ultradistribution semigroups and sines, in general, with nondensely defined generators and contemplate several known results concerning the regularization of Gevrey type ultradistribution semigroups. We prove that, for every closed linear operator A which generates an ultradistribution semigroup (sine), there exists a bounded injective operator C such that A generates a global differentiable C-semigroup (C-cosine function) whose derivatives possess some expected properties of operator valued ultradifferentiable functions. With the help of regularized semigroups, we establish the new important characterizations of abstract Beurling spaces associated to nondensely defined generators of ultradistribution semigroups (sines). The study of regularization of ultradistribution sines also enables us to perceive significant ultradifferentiable properties of higher-order abstract Cauchy problems.

Differential and Analytical Properties of Semigroups of Operators

“…) is an ultradistribution fundamental solution for A, then G is a pre-(UDSG) of * -class in E. (iii) [67] Suppose, in addition, that (M p ) satisfies (M.3) and that A generates a (UDSG) of (M p )-class. Then the abstract Cauchy problem (ACP ) :…”

“…has a unique solution for all x ∈ E (Mp) (A), where the abstract Beurling space of (M p ) class associated to a closed linear operator A is defined as follows [18,67]:…”

“…Furthermore, for every compact set K ⊆ [0, ∞) and h > 0, the solution u of (ACP ) satisfies sup t∈K,p∈N0 [67] Suppose, in addition, that (M p ) satisfies (M.3) and that A generates an (EUDSG) of * -class. Then there exists an injective operator C ∈ L(E) such that A generates an exponentially bounded C-regularized semigroup (S(t)) t≥0 that is infinitely differentiable in t ≥ 0.…”

“…The above example also shows that a large number of (pseudo-)differential operators acting in L p (R n ) (the case p = 2 is also allowed) generate (local) convoluted K a,b -semigroups of class C L . Then G is a non-dense (EUDSG) generated by A [67] and G(δ t ) = 0, t ≥ 1. Suppose now t ∈ (0, 1).…”

“…Suppose now t ∈ (0, 1). Then it can be simply verified [63,67] that, for every s ∈ [0, 1] and f ∈ D(G(δ t )), (G(δ t )f )(s) = 0, 0 ≤ s ≤ t, f (s − t) , 1 ≥ s > t.…”