On relative boundedness of a class of degenerate differential operators in the Lebesgue space
In the space $L_p(\Omega)$, where $1<p<+\infty$ and $\Omega$ is an arbitrary (bounded or unbounded) domain in $R^n$, we investigate relative boundedness for a class of higher order partial differential operators in non-divergent form. These operators have nonpower degeneracy on the whole boundary of $\Omega$ and degeneracy with respect to each of independent variables is characterized by different functions.
In the earlier published papers in this direction, as a rule, firstly the operator is defined in $\Omega$ and then functions characterizing degeneracies of the operator’s coefficients are defined in this domain.
In contrast to that, here we define $\Omega$ and these functions related to each other while fulfilling the “immersion condition” introduced by P. I. Lizorkin in . In addition, differentiability of the functions by which we define degeneracy of the investigated operator is not required. Study of relative boundedness of differential operators is one of the modern directions in such operators theory with results theory of differentiable functions of many variables, the separation theory of differential operators, the spectral theory of differential operators, etc.
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