View source for Imbedding theorems

Theorems concerning a kind of problems involved in the study of inequalities between the norms of the same function in different classes (normed spaces). One is usually concerned with two classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502302.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502303.png" /> is a part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502304.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502305.png" />), such that an inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502306.png" /></td> </tr></table>

is satisfied for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502308.png" /> is a constant which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i0502309.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023011.png" /> are the norms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023013.png" />, respectively. Under these conditions one speaks of an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023014.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023015.png" /> or one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023016.png" /> is imbeddable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023017.png" />, written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023018.png" /> (cf. also [[Imbedding of function spaces|Imbedding of function spaces]]). The studies connected with imbedding theorems constitute a branch of the theory of functions, but their main paths of development concern the boundary value problems of mathematical physics, in particular direct variational methods. For this reason a systematic theory of imbeddings of classes of differentiable functions of several variables has been developed during the past three decades.

The following problems are examples of problems solved by imbedding theorems. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023019.png" /> be known to have, usually generalized (cf. [[Generalized derivative|Generalized derivative]]), partial derivatives of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023020.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023021.png" />-th powers are integrable on a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023022.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023023.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023024.png" />. The questions are: 1) How many continuous derivatives does this function have on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023025.png" />? 2) If the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023026.png" /> has a sufficiently smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023027.png" />, is it possible to determine in some sense the trace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023028.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023029.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023030.png" />, i.e. the limit values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023031.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023032.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023033.png" />, and what are the differentiability properties of this trace? Such properties should often be known exactly enough such that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023034.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023035.png" /> and possessing these properties can be extended from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023037.png" /> in such a way that the extended function has generalized derivatives of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023038.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023039.png" />-th powers are integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023040.png" />. It will be seen from the facts given below that these limits (in the sense of almost-everywhere convergence) for the determination of the trace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023042.png" /> and of the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023043.png" /> can be accompanied by inequalities between the norms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023044.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023046.png" />, which are used in the theory of boundary value problems.

The multi-dimensional theory of imbeddings of classes of differentiable functions originated in the 1930s in the studies of S.L. Sobolev in the context of problems in mathematical physics. He is to be credited with fundamental imbedding theorems for the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023047.png" /> (the Sobolev spaces, cf. [[Sobolev space|Sobolev space]]) which play an important role in analysis. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023048.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023051.png" /> if it is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023052.png" /> and has a finite norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

and the sum is extended over all possible (Sobolev-generalized) partial derivatives

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023056.png" /></td> </tr></table>

of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023057.png" />.

Sobolev's fundamental theorem (with completions by V.I. Kondrashov and V.P. Il'in) for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023058.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023061.png" />, the following imbedding is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023063.png" /> is the integer part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023064.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023065.png" />, this means that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023066.png" /> has a trace (see below) on any coordinate hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023067.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023068.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023069.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023070.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023071.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023072.png" /> [[#References|[6]]], [[#References|[7]]].

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023073.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023074.png" /> has a trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023076.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023077.png" />-dimensional (coordinate) subspace of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023078.png" /> with fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023079.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023080.png" /> can be modified on some set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023081.png" />-dimensional measure zero, so that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023082.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023083.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023084.png" /></td> </tr></table>

holds for the modified function (which is again denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023085.png" />).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023086.png" /> is a set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023087.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023088.png" />, the problem of describing the properties of the traces of these functions on a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023089.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023090.png" />) is said to be the trace problem for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023091.png" />.

Theorem (4) is a final theorem in terms of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023092.png" />. Its further strengthening is possible only if new classes are introduced.

In the one-dimensional case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023093.png" />, where the trace problem does not occur, theorem (4) is due to G.H. Hardy and J.E. Littlewood.

The next stages in the development of this theory were Nikol'skii's imbedding theorems for generalized Hölder classes (cf. [[Hölder space|Hölder space]]) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023094.png" />-classes). These classes form a scale with continuously varying parameters which characterize the smoothness of the functions. They are anisotropic in the sense that their functions usually display different differentiability properties in different directions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023095.png" /> be the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023096.png" /> at distance from the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023097.png" /> greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023098.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i05023099.png" /> be a positive vector (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230100.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230101.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230103.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230104.png" />.

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230105.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230109.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230110.png" /> and if for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230111.png" /> a generalized partial derivative

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230112.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

exists which satisfies the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230113.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230114.png" /> denotes the second-difference operator of the function with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230115.png" />, with step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230116.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230117.png" /> is a constant which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230118.png" />.

The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230119.png" /> forms a [[Banach space|Banach space]] with norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230120.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230121.png" /> is the smallest constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230122.png" /> for which the inequalities (7) are satisfied. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230123.png" />, the respective (isotropic) class is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230124.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230125.png" /> is an integer, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230126.png" /> is close to the Sobolev class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230127.png" />, with an accuracy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230128.png" />, in the sense that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230129.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

Nikol'skii's imbedding theorems are valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230130.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230131.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230132.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230133.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230134.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230138.png" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230139.png" /></td> </tr></table>

(cf. [[#References|[5]]]).

Theorem (9) is the anisotropic analogue of theorem (4), with the advantage that the (vectorial) superscripts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230140.png" /> of the classes appearing in it may vary in a continuous manner. Moreover, it includes the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230141.png" />. However, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230142.png" /> it is not valid, unlike (4). Hardy and Littlewood demonstrated the theorem for the one-variable case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230143.png" /> with non-integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230144.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230145.png" />.

The imbedding (10) with the upper arrow is also given by a special case of theorem (9), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230146.png" />. It states that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230147.png" /> has a trace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230148.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230149.png" /> and that also

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230150.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230151.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230152.png" />. The reverse statement, symbolized by the lower arrow, is also true, and should be understood in the following sense: Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230153.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230154.png" /> may be extended to the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230155.png" /> so that the resulting function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230156.png" /> (with trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230157.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230158.png" />) belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230159.png" /> and satisfies the inequality (reverse to (11)):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230160.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230161.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230162.png" />.

The mutually inverse imbeddings (10) represent a complete solution to the trace problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230163.png" />-classes, in terms of these classes.

Theorem (9) is transitive, which means that the transition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230164.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

from the first class in the chain (12) to the second, and then from the second to the third, where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230165.png" /> are computed by the formulas in (9), may be replaced by a direct transition from the first to the third class, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230166.png" /> being calculated by the same formulas.

Subsequently (cf. [[#References|[14]]]) a solution was given for the trace problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230167.png" />-classes, which are in general anisotropic. This resulted in the introduction of a new family of classes of differentiable functions of several variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230168.png" />, which depend on the vector parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230169.png" /> and on two scalar parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230170.png" /> which satisfy the inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230171.png" />. This family was completely determined by O.V. Besov, who also studied its fundamental properties.

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230172.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230175.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230176.png" /> is an integer vector, if it has finite meaningful norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230177.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230178.png" /></td> </tr></table>

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230179.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230182.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230183.png" /> is an arbitrary, not necessarily integer, vector, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230185.png" />, if it has finite norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230186.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230187.png" /></td> </tr></table>

where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230188.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230189.png" /> were defined above.

It is natural to regard the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230190.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230191.png" /> as identical with the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230192.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230193.png" />). One usually writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230194.png" /> rather than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230195.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230196.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230198.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230199.png" /> are Banach spaces for any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230200.png" />.

The imbedding theorems (9) and (10) are valid if the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230201.png" /> in them are replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230202.png" />. There also exist a mutually inverse imbedding:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230204.png" /> is an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230207.png" />, which completely solves the trace problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230208.png" />-classes, and does not interfere with mutually inverse imbeddings, completely expressed in the language of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230209.png" />-classes:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230210.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>

The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230211.png" /> corresponding to the parameter values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230212.png" /> are usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230213.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230214.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230215.png" />, the imbeddings (14) may also be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230216.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>

Classes whose definition involves the concept of a Liouville fractional derivative (cf. [[Fractional integration and differentiation|Fractional integration and differentiation]]) are natural extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230217.png" />-classes.

Using the terminology of generalized functions (cf. [[Generalized function|Generalized function]]), it is possible to define a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230218.png" /> of test functions such that the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230219.png" /> of generalized functions constructed over it will have the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230220.png" /> for any finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230221.png" />; 2) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230222.png" />, not necessarily an integer, the operation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230223.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230224.png" /> denote, respectively, the direct and the inverse [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230225.png" />, is meaningful; and 3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230226.png" /> is an integer and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230227.png" /> has a Sobolev-generalized derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230228.png" />, then equation (17) applies to it.

In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230229.png" /> is a fraction, operation (17) on infinitely-differentiable functions of compact support is identical with the Liouville fractional differentiation operation. It is natural to call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230230.png" /> the fractional derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230231.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230232.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230233.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230234.png" /> is not an integer.

If an arbitrary vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230235.png" /> is given, one may introduce the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230236.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230237.png" />, which is identical with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230238.png" /> for integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230239.png" />, by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230240.png" /> in (13) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230241.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230242.png" />, one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230243.png" />. The family of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230244.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230245.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230246.png" />, may be regarded as a natural extension of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230247.png" /> to fractional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230248.png" /> —  "natural" , since from the point of view of the present circle of interest the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230249.png" /> display  "all the advantages and all the drawbacks of Wpl" . If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230250.png" /> is substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230251.png" /> in formula (4) (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230252.png" /> may be substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230253.png" />) or in (8) (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230254.png" /> may be a fraction) or in (14), (16) (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230255.png" /> may be a fraction), these formulas will remain valid. The same also applies to formula (9) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230256.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230257.png" />, even under the wider condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230258.png" />, but under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230259.png" />.

In what follows the apparatus of generalized functions will be used, except that these now constitute the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230260.png" />. For any real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230261.png" /> the Bessel–Macdonald operation is meaningful:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230262.png" /></td> </tr></table>

It has the following properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230263.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230264.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230265.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230266.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230267.png" /> is the [[Laplace operator|Laplace operator]].

The isotropic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230268.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230269.png" />, may also be defined as the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230270.png" /> that can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230271.png" /> where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230272.png" /> run through the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230273.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230274.png" />); moreover, up to equivalence,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230275.png" /></td> </tr></table>

This definition of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230276.png" /> is also suitable for negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230277.png" />, but in such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230278.png" /> is a set of (usually generalized) functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230279.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230280.png" />.

The operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230281.png" /> may also be employed as a tool in defining the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230282.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230283.png" />). To do this, one calls a generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230284.png" /> regular in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230286.png" /> or belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230287.png" /> if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230288.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230289.png" />. Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230290.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230291.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230292.png" />, can be defined as a function that is regular in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230293.png" /> and that can be represented as a series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230294.png" /></td> </tr></table>

weakly converging towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230295.png" /> (in the sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230296.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230297.png" /> has spectrum (the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230298.png" />) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230299.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230300.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230301.png" />, has spectrum in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230302.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230303.png" /></td> </tr></table>

and also

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230304.png" /></td> </tr></table>

In particular,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230305.png" /></td> </tr></table>

This definition of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230306.png" /> is automatically extended to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230307.png" />, and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230308.png" /> belonging to classes with negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230309.png" /> will usually be generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230310.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230311.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230312.png" />.

There also exist other, equivalent, definitions of the negative classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230313.png" />, based on the principle of interpolation of function spaces. The definition given above is constructive — each class defined by the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230314.png" /> is defined independently, and it is possible to define constructively linear operations with the aid of which a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230315.png" /> (of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230316.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230317.png" /> and of type 1 if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230318.png" />) is defined in terms of a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230319.png" />.

The following imbedding theorem is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230320.png" /></td> </tr></table>

This theorem is of the same type as theorem (4), but with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230321.png" />; it is valid for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230322.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230323.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230324.png" />, or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230325.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230326.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230327.png" />, or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230328.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230329.png" />.

On the other hand, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230330.png" />, an arbitrary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230331.png" /> usually has no trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230332.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230333.png" />) unless additional conditions are imposed.

The imbedding theorems formulated above apply to classes of functions defined on the entire <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230334.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230335.png" /> [[#References|[5]]]. In practical applications, however, it is important to have similar theorems for domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230336.png" /> which should be as general as possible. The geometrical structure of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230337.png" /> for which the above imbedding theorems are valid for the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230338.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230339.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230340.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230341.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230342.png" /> must be replaced, respectively, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230343.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230344.png" />, has now been clarified. For the isotropic classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230345.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230346.png" /> the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230347.png" /> must satisfy a cone condition or, which is equivalent to it, its boundary must locally satisfy a [[Lipschitz condition|Lipschitz condition]]. For the anisotropic classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230348.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230349.png" /> the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230350.png" /> must satisfy an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230351.png" />-horn condition or a bent [[Cone condition|cone condition]], and this condition is, in a certain sense, necessary [[#References|[2]]].

Another problem with important practical applications is the trace problem on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230352.png" />-dimensional manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230353.png" />.

This problem has been completely solved for the isotropic classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230354.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230355.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230356.png" /> (see [[#References|[2]]], ). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230357.png" /> is sufficiently differentiable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230358.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230359.png" /> can be substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230360.png" /> in (14), (15) and (16), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230361.png" /> can be substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230362.png" /> in . If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230363.png" /> is piecewise smooth the problem has also been solved completely , [[#References|[22]]]. The conditions for the solution of the problem are expressed by mutually inverse imbeddings on individual smooth pieces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230364.png" /> on one hand, and, on the other hand, by special additional conditions on the behaviour of the functions of the respective classes at the points of contact of these smooth pieces. The solution of the trace problem for anisotropic classes [[#References|[9]]], [[#References|[21]]] is also in an advanced stage. Here major difficulties arise, concerning the characteristics of the trace at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230365.png" /> where the tangent planes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230366.png" /> are parallel to the coordinate axes.

One more problem follows. Given a function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230367.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230368.png" /> denotes one of the classes considered above. What mixed partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230369.png" /> does this function have and what are their properties? A positive answer to this question depends on the magnitude

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230370.png" /></td> </tr></table>

In fact, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230371.png" /> there exists a partial derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230372.png" /> which belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230373.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230374.png" />. This condition may be generalized to the case of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230375.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230376.png" /> (see [[#References|[5]]]).

Yet follows another characteristic theorem, which may perhaps be called a theorem on weak compactness, and which has applications in the theory of direct methods of variational calculus.

Out of the infinite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230377.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230378.png" /> which satisfy the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230379.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230380.png" /> is a known constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230381.png" /> is one of the classes discussed above, it is possible to separate a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230382.png" /> of functions and to indicate a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230383.png" /> with norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230384.png" /></td> </tr></table>

such that, for all bounded domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230385.png" /> and all vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230386.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230387.png" /></td> </tr></table>

[[#References|[5]]]. In this formulation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230388.png" /> may be replaced by a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230389.png" /> if the latter has a sufficiently nice boundary. Only the typical function classes (and the imbedding theorems connected with them) which are most often encountered in practical applications were discussed above. In modern investigations stress is laid [[#References|[2]]] on more general classes, in which more or less arbitrary differential operators play the role of the starting partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230390.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230391.png" />.

Other classes under study comprise the so-called weight classes (cf. [[Weight space|Weight space]]), a typical example of which is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230392.png" />, defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230393.png" /> be the distance between a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230394.png" /> and the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230395.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230396.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230397.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230398.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230399.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230400.png" />, if it has finite norm (see [[#References|[4]]], [[#References|[12]]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230401.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230402.png" /></td> </tr></table>

One result is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230403.png" /> be a sufficiently smooth boundary of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230404.png" />; then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230405.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230408.png" />.

Example. The use of imbedding theorems presents a complete solution of the problem of conditions on the boundary function under which the [[Dirichlet principle|Dirichlet principle]] is applicable. In fact, take the partial derivatives in the generalized sense and assume, for the sake of simplicity, that the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230409.png" /> (the boundary of a three-dimensional domain) is bounded and is twice differentiable, and that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230410.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230411.png" /> has been given. For this function the [[Dirichlet integral|Dirichlet integral]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230412.png" /> and also, in accordance with the imbedding theorem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230413.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230414.png" /> has a trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230415.png" /> (the fact that a trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230416.png" /> exists can be established with the aid of coarser imbedding theorems). Denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230417.png" /> the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230418.png" /> with the same trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230419.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230420.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230421.png" />, the Dirichlet principle may be formulated as follows: The minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230422.png" /> over the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230423.png" /> is attained for a unique function which is also harmonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230424.png" />. It follows from the imbedding theorem above that the Dirichlet principle is applicable if and only if the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230425.png" /> is non-empty, i.e. when the boundary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230426.png" />.

In justifying the Dirichlet principle, the first step is to prove that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230427.png" /> exists and is unique, and the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230428.png" /> is a [[Generalized solution|generalized solution]] of the [[Dirichlet problem|Dirichlet problem]]. A special method is then used to successively establish that the generalized solution belongs to the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230429.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230430.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230431.png" /> is an arbitrary closed sphere. In particular, from the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230432.png" />, applying the imbedding theorem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230433.png" /></td> </tr></table>

(cf. [[#References|[2]]], [[#References|[5]]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230434.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230435.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230436.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230437.png" />, one deduces that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230438.png" /> may be modified on a set of three-dimensional measure zero so that the function thus obtained is twice continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230439.png" />. It can then readily be proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230440.png" /> is harmonic.

This example may be generalized to include certain functionals with partial derivatives of different orders, raised to a power usually distinct from 2 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230441.png" />; it is then necessary to use imbedding theorems for more general, usually anisotropic, classes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O.V. Besov,   et al.,   "The theory of imbedding classes of differentiable functions of several variables" , ''Partial differential equations'' , Moscow  (1970)  pp. 38–63  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O.V. Besov,   V.P. Il'in,   S.M. Nikol'skii,   "Integral representations of functions and imbedding theorems" , Wiley  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Burenkov,   "Imbedding and extension theorems for classes of differentiable functions of several variables in the whole space"  ''Itogi Nauk. Mat. Anal. 1965''  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,   "On imbedding, continuation and approximation theorems for differentiable functions of several variables"  ''Russian Math. Surveys'' , '''16''' :  5  (1961)  pp. 55–104  ''Uspekhi Mat. Nauk'' , '''16''' :  5  (1961)  pp. 63–114</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.L. Sobolev,   "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.L. Sobolev,   "Introduction to the theory of cubature formulas" , Moscow  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  O.V. Besov,   "Investigation of a class of function spaces in connection with imbedding and embedding theorems"  ''Trudy Mat. Inst. Steklov.'' , '''60'''  (1961)  pp. 42–81  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  Ya.S. Bugrov,   "Boundary properties of functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230442.png" /> on domains with corner points"  ''Sibirsk. Mat. Zh.'' , '''5''' :  5  (1964)  pp. 1007–1026  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  V.P. Il'in,   "On an inclusion theorem for a limiting exponent"  ''Dokl. Akad. Nauk SSSR'' , '''96''' :  5  (1954)  pp. 905–908  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  V.I. Kondrashov,   "Sur certaines propriétés des fonctions dans l'espace"  ''Dokl. Akad. Nauk SSSR'' , '''48'''  (1945)  pp. 535–538</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  L.D. Kudryavtsev,   "Direct and inverse imbedding theorems. Applications to solutions of elliptic equations by variational methods"  ''Trudy Mat. Inst. Steklov.'' , '''55'''  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  P.I. Lizorkin,   "Boundary properties of functions from  "weight"  classes"  ''Soviet Math. Dokl.'' , '''1''' :  3  (1960)  pp. 589–593  ''Dokl. Akad. Nauk SSSR'' , '''132''' :  3  (1960)  pp. 514–517</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  P.I. Lizorkin,   "Generalized Liouville differentiation and the function spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230443.png" />. Imbedding theorems"  ''Mat. Sb.'' , '''60 (102)''' :  3  (1963)  pp. 325–353  (In Russian)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  S.M. Nikol'skii,   "Inequalities for entire functions of finite degree and their application to the theory of differentiable functions in several variables"  ''Trudy Mat. Inst. Steklov.'' , '''38'''  (1951)  pp. 244–278  (In Russian)</TD></TR><TR><TD valign="top">[16a]</TD> <TD valign="top">  S.M. Nikol'skii,   "Properties of certain classes of functions of several variables on a differentiable manifold"  ''Mat. Sb.'' , '''33 (75)''' :  2  (1953)  pp. 261–326  (In Russian)</TD></TR><TR><TD valign="top">[16b]</TD> <TD valign="top">  S.M. Nikol'skii,   "Boundary properties of functions defined in a region with corner points"  ''Mat. Sb.'' , '''43 (85)''' :  1  (1957)  pp. 127–144  (In Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  S.L. Sobolev,   "Le problème de Cauchy dans l'espace des fonctionnelles"  ''Dokl. Akad. Nauk SSSR'' , '''3''' :  7  (1935)  pp. 291–294</TD></TR><TR><TD valign="top">[18a]</TD> <TD valign="top">  S.L. Sobolev,   "A new method for solving the Cauchy problem for partial differential equations of normal hyperbolic type"  ''Mat. Sb.'' , '''1 (43)''' :  1  (1936)  pp. 39–72  (In Russian)</TD></TR><TR><TD valign="top">[18b]</TD> <TD valign="top">  S.L. Sobolev,   "On a theorem in functional analysis"  ''Mat. Sb.'' , '''4 (46)''' :  3  (1938)  pp. 471–497  (In Russian)  (French abstract)</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top">  L.N. Slobodetskii,   "S.L. Sobolev's spaces of fractional order and their application to boundary value problems for partial differential equations"  ''Dokl. Akad. Nauk SSSR'' , '''118''' :  2  (1958)  pp. 243–246  (In Russian)</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top">  S.V. Uspenskii,   "Imbedding theorems for classes with weights"  ''Trudy Mat. Inst. Steklov.'' , '''60'''  (1961)  pp. 282–303  (In Russian)</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top">  S.V. Uspenskii,   "Boundary properties of a class of differentiable funtions in smooth regions"  ''Soviet Math. Dokl.'' , '''6''' :  5  (1965)  pp. 1299–1302  ''Dokl. Akad. Nauk SSSR'' , '''164''' :  4  (1965)  pp. 750–752</TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top">  G.N. Yakovlev,   "Boundary properties of a class of functions"  ''Trudy Mat. Inst. Steklov.'' , '''60'''  (1961)  pp. 325–349  (In Russian)</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top">  E. Gagliardo,   "Caratterizzazioni delle trace sulla frontiera relative ad alcune classi di funzioni in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050230/i050230444.png" /> variabli"  ''Rend. Sem. Mat. Univ. Padova'' , '''27'''  (1957)  pp. 284–305</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   "A convergence criterion for Fourier series"  ''Math. Z.'' , '''28'''  (1928)  pp. 612–634</TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top">  J.L. Lions,   E. Magenes,   "Non-homogenous boundary value problems and applications" , '''1–2''' , Springer  (1972)  (Translated from French)</TD></TR></table>