Sobolev spaces of infinite order with rapidly decreasing

Abstract: The Banach space $$ W^\infty\big\{a_n,p,r\big\}_{(0,a)}\equiv \bigg\{u(x)\inC^\infty(0,a): \sum^\infty_{n=0}a_n\Vert D^n u\Vert^p_r<\infty\bigg\}, \tag1 $$ \noindent where $a_n\ge0$, $1\le p<\infty$, $r\ge1$, $\|\cdot\|_r$ is the norm in the Lebesgue space, is referred to as a Sobolev space of infinite order.

The problem is to find conditions for existence of a function in the space (1) with given values on the boundary $$ D^n u(0)=b_n,\quad D^n u(a)=c_n,\quad n=0,1,... . \tag2 $$

The sequences $\{b_n\}$ and $\{c_n\}$ are called the traces of the function $u$ on a boundary of the domain $(0,a)$ and the function $u(x)$ is the extension of this trace in the space (1). Naturally, we consider the space (1) as nontrivial, i.e., containing a function $u(x)\not\equiv0$ for which $b_n=c_n=0$ for all $n$.

\proclaim{Theorem 1} For existence of an extension of a trace in any nontrivial space {\rm (1)} it is necessary and sufficient that the trace is analytic, i.e. $$ \overline{\lim_{n\to\infty}}\,\tfrac1n\,|b_{n}|^{\frac1n}=K<\infty. $$ \endproclaim

\proclaim{Theorem 2} In any nontrivial space there is a class of nonanalytic traces, which may be extended in this space. \endproclaim

We cannot describe this class for every nontrivial spaces (1) completely. For general case sufficient conditions for both one-dimensional $(0,a)$ andmulti-dimensional $(0,a)\times R^\nu$, $\nu\ge1$, domains are obtained. In some cases criteria are obtained.

\proclaim{Theorem 3} If the sequence $\{a_n\}$ is rapidly decreasing, i.e., $$ a_{n+1}\le a^q_n,\quad q>1,\quad n=0,1,... , \tag3 $$ then for the existence of an extension of a trace $\{b_n\}$ in the space {\rm (1)} it is necessary and sufficient that $$ \sum^\infty_{n=0}|b_n|^p(M^c_n)^{-1-\frac1r}(M^c_{n+1})^{-\frac1r}<\infty. \tag4 $$ (Here $M^c_n$ is the logarithmically convex regularization of the sequence $\{M_n\}=\{a^{-1}_n\}$.) \endproclaim

\proclaim{Corollary} If $q\ge2$ in {\rm (3)}, then the criterion of the extension of a trace is $$ \sum^\infty_{n=0}|b_n|^pa^{1-\frac1r}_na^{\frac1r}_{n+1}<\infty. $$ \endproclaim

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