The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications. (English) Zbl 1435.47045

Ashyralyev, Allaberen; Taskin, Abdulgafur

Adv. Oper. Theory 4, No. 1, 140-155 (2019).

In this paper, a densely defined operator \(B\) in a Banach space \(E\) is called positive if its spectrum lies in a sector \(|\arg\lambda| \le \varphi,\) \(0 \le \varphi < \pi\) and \[ \| (\lambda I - B)^{-1} \| \le \frac{M}{1 + |\lambda|}, \quad | \arg \lambda| > \varphi \, . \] The operator to which this definition is applied is \(B = -A,\) where \(A\) is the neutron transport operator \[ Au(x, y, \omega_1, \omega_2) = \omega_1 \frac{\partial u}{\partial x}+ \omega_2 \frac{\partial u}{\partial y} \] in \(\mathbb{R}^2,\) with \(\omega_1, \omega_2\) direction cosines. The first result is positivity of \(B\) in the spaces \(C(\mathbb{R}^2)\) and \(L^p(\mathbb{R}^2),\) \(1 \le p < \infty.\) Then, the space \(E_\alpha(C(\mathbb{R}^2), B)\) is introduced in terms of the growth of \(\|B(\lambda I + B)f\|_{C({\mathbb{R}^2})}\) as \(\lambda \to \infty\) and \(B\) is proved positive in \(E_\alpha(C(\mathbb{R}^2), B),\) hence in a Hölder space \(C^\alpha(\mathbb{R})\) with equivalent norm. Finally, the same results are shown for the space \(E_\alpha^p (L^p(\mathbb{R}^2), B)\) defined as \(E_\alpha(C(\mathbb{R}^2), B)\) but replacing \(C(\mathbb{R}^2)\) by \(L^p(\mathbb{R}^2),\) and for a Sobolev-Slobodecki space \(W^p_\alpha(\mathbb{R}^2)\) of equivalent norm.

Reviewer: Hector O. Fattorini (Los Angeles)

MSC:

47B65	Positive linear operators and order-bounded operators
47F05	General theory of partial differential operators
47B38	Linear operators on function spaces (general)
47B01	Operators on Banach spaces
35A35	Theoretical approximation in context of PDEs
35K30	Initial value problems for higher-order parabolic equations
34B27	Green’s functions for ordinary differential equations
35Q49	Transport equations
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

neutron transport operator; fractional space; Slobodeckij space; positive operator; Hölder spaces; Sobolev-Slobodeckiĭ spaces; positive operators

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