Nikol'skii space

A Banach space $ H _ {p} ^ {r} ( \Omega ) $ consisting of functions defined on an open set $ \Omega $ of an $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $ and having certain difference-differentiability properties characterized by a vector $ r = ( r _ {1} \dots r _ {n} ) $, $ r _ {i} > 0 $, $ i = 1 \dots n $, in the $ L _ {p} $- metric, $ 1 \leq p \leq \infty $. The concept was introduced by S.M. Nikol'skii.

The Nikol'skii space $ H _ {p} ^ {r} ( \Omega ) $ can be described in terms of properties of the partial derivatives of order $ r _ {i} ^ {*} $ in the variable $ x _ {i} $, where $ r _ {i} = r _ {i} ^ {*} + \alpha _ {i} $, $ r _ {i} ^ {*} $ is an integer, $ 0 < \alpha _ {i} \leq 1 $, $ i = 1 \dots n $; if $ \Delta _ {h _ {i} } ^ {s} $ denotes the difference of order $ s = 1 , 2 \dots $ and of step $ h _ {i} $ with respect to $ x _ {i} $ of a function $ f $, then

$$ f \in H _ {p} ^ {r} ( M _ {1} \dots M _ {n} ; \Omega ) ,\ \ M _ {i} > 0 , $$

if and only if $ f $ has in $ \Omega $ generalized partial derivatives

$$ f _ {x _ {i} } ^ { ( r _ {i} ^ {*} ) } = \ \frac{\partial ^ {r _ {i} ^ {*} } f }{\partial x _ {i} ^ {r _ {i} ^ {*} } } , $$

$ i = 1 \dots n $, and if for $ 0 < \alpha _ {i} < 1 $,

$$ \| \Delta _ {h _ {i} } ^ {1} f _ {x _ {i} } ^ { ( r _ {i} ^ {*} ) } \| _ {L _ {p} ( \Omega _ {| h _ {i} | } ) } \leq \ M _ {i} | h _ {i} | ^ {\alpha _ {i} } , $$

while for $ \alpha _ {i} = 1 $,

$$ \| \Delta _ {h _ {i} } ^ {2} f _ {x _ {i} } ^ { ( r _ {i} ^ {*} ) } \| _ {L _ {p} ( \Omega _ {2 | h _ {i} | } ) } \leq \ M _ {i} | h _ {i} | , $$

where $ \Omega _ \eta $ is the set of points $ x \in \Omega $ that are distant by more than $ \eta > 0 $ from the boundary of $ \Omega $ and $ h _ {i} $ is arbitrary.

The space $ H _ {p} ^ {r} ( \Omega ) = H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ is defined as the union of all $ H _ {p} ^ {r} ( M _ {1} \dots M _ {n} ; \Omega ) $ for all $ M _ {i} > 0 $, $ i = 1 \dots n $.

If $ \Omega \neq \emptyset $, then for any $ r _ {i} > 0 $, $ i = 1 \dots n $, $ 1 \leq p \leq \infty $, the Nikol'skii space $ H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ is not empty and contains functions that do not belong to $ H _ {p} ^ {r _ {1} \dots r _ {i-} 1 , r _ {i} + \epsilon , r _ {i+} 1 \dots r _ {n} } $ for any $ \epsilon > 0 $ and any $ i = 1 \dots n $.

When $ p = \infty $, the $ r _ {i} $ are not integers and the relevant derivatives are continuous, then a Nikol'skii space is a Hölder space. The concept of a Nikol'skii space generalizes to the case of functions that are defined on sufficiently smooth manifolds (see [2]).

There is a description of the Nikol'skii space $ H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ in terms of properties of the differences of the partial derivatives of orders less than $ r _ {i} ^ {*} $; in particular, of those of sufficiently high order of the function itself.

Let $ H _ {p} ^ {r} ( \Omega ) $ be an isotropic space, that is, $ r _ {1} = \dots r _ {n} = r $. If the domain $ \Omega $ is such that any function $ f $ of class $ H _ {p} ^ {r} ( \Omega ) $ can be extended with preservation of the class to the whole space $ \mathbf R ^ {n} $, that is, in such a way that the extended function belongs to $ H _ {p} ^ {r} ( \mathbf R ^ {n} ) $( this is always the case when the boundary of the domain is sufficiently smooth), then $ f \in H _ {p} ^ {r} ( \Omega ) $ if and only if for any non-negative integers $ k $ and $ s $ such that $ 0 < r - s < k $ the function $ f $ has partial derivatives $ f ^ { ( s) } $ of all orders $ s $ and there is a constant $ M > 0 $ such that

$$ \tag{1 } \| \Delta _ {h} ^ {k} f ^ { ( s) } \| _ {L _ {p} ( \Omega _ {k | h | } ) } \leq M | h | ^ {r-} s , $$

where $ h =( h _ {1} \dots h _ {n} ) $ and $ \Delta _ {h} ^ {k} f ^ { ( s) } $ is the difference of order $ k $ of $ f ^ { ( s) } $ with vectorial step $ h $. Condition (1) is equivalent to the analogous condition for the modulus of continuity of $ f ^ { ( s) } $: There is an $ M > 0 $ such that

$$ \Omega ^ {k} ( f ^ { ( s) } , \delta ) \leq \ M \delta ^ {r-} s ,\ \delta > 0 , $$

where

$$ \Omega ^ {k} ( f ^ { ( s) } , \delta ) = \ \sup _ {| h | = 1 } \ \sup _ {0 \leq t \leq \delta } \ \| \Delta _ {th} ^ {k} f ^ { ( s) } \| _ {L _ {p} ( \Omega _ {kt} ) } \leq M \delta ^ {r-} s . $$

If $ M _ {f} $, for $ f \in H _ {p} ^ {r} ( \Omega ) $, denotes the infimum of all $ M $ for which (1) holds for all $ h \in \mathbf R ^ {n} $ and all partial derivatives of an admissible order $ s $, then

$$ \| f \| = \| f \| _ {L _ {p} ( \Omega ) } + M _ {f} $$

is a norm in $ H _ {p} ^ {r} ( \Omega ) $ and the norms obtained for distinct admissible pairs $ k , s $ are equivalent.

A Nikol'skii space consisting of functions defined on the whole space $ \mathbf R ^ {n} $ can be characterized in terms of best approximations of the functions in this space by entire functions of exponential type. Let $ E _ {v _ {1} \dots v _ {n} } ( f ) _ {p} $ be the best approximation (error) in the $ L _ {p} ( \mathbf R ^ {n} ) $- metric of an $ f \in L _ {p} ( \mathbf R ^ {n} ) $ by entire functions $ q _ {v _ {1} \dots v _ {n} } ( x _ {1} \dots x _ {n} ) \in L _ {p} ( \mathbf R ^ {n} ) $ of exponential type and of order $ v _ {i} $ in $ x _ {i} $, $ i = 1 \dots n $. The following direct and inverse theorems of Bernshtein, Jackson and Zygmund type hold for Nikol'skii functions.

If $ f \in H _ {p} ^ {r _ {1} \dots r _ {n} } ( M _ {1} \dots M _ {n} ; \mathbf R ^ {n} ) $, then for any $ v _ {i} > 0 $,

$$ \tag{2 } E _ {v _ {1} \dots v _ {n} } ( f ) \leq c \sum _ { i= } 1 ^ { n } \frac{M _ {i} }{v _ {i} ^ {r _ {i} } } $$

(the constant $ c > 0 $ does not depend on $ f $).

Conversely, if (2) holds for a function $ f \in L _ {p} ( \mathbf R ^ {n} ) $ for $ v _ {i} = a _ {i} ^ {k} $, $ k = 0 , 1 \dots $ $ a _ {i} > 1 $, $ i = 1 \dots n $, and if $ q $ is an entire function of order 1 in each variable $ x _ {1} \dots x _ {n} $ for which

$$ \| f - q \| _ {L _ {p} ( \mathbf R ^ {n} ) } \leq c \sum _ { i= } 1 ^ { n } M _ {i} $$

(which exist for $ k = 0 $, by (2)), then

$$ f - q \in \ H _ {p} ^ {r _ {1} \dots r _ {n} } ( M _ {1} ^ {*} \dots M _ {n} ^ {*} ; \mathbf R ^ {n} ) , $$

where

$$ \tag{3 } M _ {i} ^ {*} = c _ {i} \sum _ { j= } 1 ^ { n } M _ {j} , $$

and the constants $ c > 0 $ in (2) and $ c _ {i} > 0 $ in (3) do not depend on $ M _ {i} $, $ i = 1 \dots n $.

If $ f $ is periodic in all variables, then a similar description of a Nikol'skii space can be given by means of best approximations of the functions by trigonometric polynomials instead of entire functions of exponential type (see [1], [4]).

Nikol'skii spaces can be described by means of a Bessel–Macdonald operator applied to some class of generalized functions (see Imbedding theorems).

For the space $ H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ Nikol'skii has proved transitive imbedding theorems for various dimensions and metrics (see [3] and Imbedding theorems), which were subsequently carried over to more general classes of functions. These theorems show that Nikol'skii spaces form a closed system relative to the boundary values of the functions occurring in them: The traces of functions in Nikol'skii spaces on smooth manifolds can in a certain sense be completely described in terms of Nikol'skii spaces.

The properties of Nikol'skii spaces make it possible to obtain necessary and sufficient conditions for the solvability of the Dirichlet problem in appropriate Nikol'skii spaces in terms of membership of the boundary function to a certain Nikol'skii space: A harmonic function $ u $ belongs to the class $ H _ {p} ^ {r} ( \Omega ) $, $ r > 1 / p $, where $ \Omega $ is a bounded domain in $ \mathbf R ^ {n} $ with a sufficiently smooth boundary $ \partial \Omega $, if and only if the boundary values $ u \mid _ {\partial \Omega } $ belong to the class $ H _ {p} ^ {r - 1/p } ( \partial \Omega ) $. This implies for $ p = 2 $, in particular, that if $ u \mid _ {\partial \Omega } \in H _ {2} ^ \rho ( \partial \Omega ) $, $ \rho > 1/2 $, then the Dirichlet integral $ D ( u) $ of $ u $ over $ \Omega $ is finite, therefore, the Dirichlet problem can be solved by a direct variational method. From imbedding theorems for Nikol'skii spaces it follows that if the Dirichlet integral of $ u $ over $ \Omega $ is finite, then $ u \mid _ {\partial \Omega } \in H _ {2} ^ {1/2} ( \partial \Omega ) $( see [6]). A generalization of Nikol'skii spaces are the Besov spaces $ B _ {p \theta } ^ {r _ {1} \dots r _ {n} } $.

References

[1]	S.M. Nikol'skii, "Inequalities for entire functions of finite order and their application to the theory of differentiable functions in several variables" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 244–278 (In Russian)
[2]	S.M. Nikol'skii, "Properties of certain classes of functions of several variables on a differentiable manifold" Mat. Sb. , 33 : 2 (1953) pp. 261–326 (In Russian)
[3]	S.M. Nikol'skii, "Imbedding theorems for functions with partial derivatives, considered in differential metrics" Dokl. Akad. Nauk SSSR , 118 : 1 (1958) pp. 35–37 (In Russian)
[4]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[5]	O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)
[6]	S.M. Nikol'skii, "On the solution of the polyharmonic equation by a variational method" Dokl. Akad. Nauk SSSR , 88 : 3 (1953) pp. 409–411 (In Russian)