# The Structure of Fractional Spaces Generated by a Two-dimensional Difference Operator in a Half Plane

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We consider a difference-operator approximation \( {A}_h^x \) of the differential operator

$$ {A}^xu(x)=-{a}_{11}(x){u}_{x_1{x}_1}(x)-{a}_{22}(x){u}_{x_2{x}_2}(x)+\sigma u(x),\kern1em x=\left({x}_1,{x}_2\right), $$

defined in the region ℝ^{+} × ℝ with the boundary condition

$$ u\left(0,{x}_2\right)=0,\kern1em {x}_2\in \mathbb{R}. $$

Here, the coefficients *a*_{ii}(*x*)*, i* = 1*,* 2*,* are continuously differentiable, satisfy the condition of uniform ellipticity \( {a}_{11}^2(x)+{a}_{22}^2(x)\ge \delta >0 \), and *σ* > 0. We study the structure of the fractional spaces generated by the analyzed difference operator. The theorems on well-posedness of difference elliptic problems in a Hölder space are obtained as applications.

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