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

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.

### 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
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### References:

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