Scale-deviating operators of Riesz type and the spaces of variable dimensions

Abstract

The article introduces the scale-deviating operator. The scale-deviating differential operator comprises the parameters to designate the operator order and the parameters to define the dimension of space. The operator order depends on the characteristic length κ. There are two types of linear scale-deviating operators. For the distances r, which are much less than κ, the scale-deviating operator of the first type $A$ reduces to the common operators. For the distances, which exceed the length κ, this operator reduces to the fractional Riesz operator. The second type of the scale-deviating operator $B$ behaves oppositely. For the distances r, which are much higher than κ, the scale-deviating operator of the second type reduces to the common operators. Finally, for the distances, which below the length κ, this operator lessens to the fractional Riesz operator. These linear, isotropic operators possess the order, less than two. The solutions of new scale-deviating equations and the shell theorem for these operators are provided closed form.