DIFFERENTIAL PROPERTIES OF LOGARITHMIC POTENTIALS

Аннотация

In this paper, Luzin’s Ck properties are proved. Logarithmic potential in the space 30 n , i.e. it is proved that outside some open set with small Lebesgue measure, the logarithmic potential is n  times continuously differentiable. Key words: potential, measure, capacity, singular integral, theorem Cartan, theorem Kishi, logarithmic potentials Let us consider the logarithmic potential in the space n U l ( x ) l n x y d ( y ) , x n ,       where  is a finite Borel measure with compact support S n   . In Cartan’s work [1] (see also [2, p. 231]) the following analogue of Luzin’s C property (see [3, p. 309] ) was proved for Riesz potentials: U n p  x  d x  y y , x n , 0 p n . Theorem 1 (Cartan). For any 0   there exists an open set O n  with capacity C n р ( O )   such that the potential U n р ( x )  is continuous in the complement n\O . In [4] M. Kishi proved Cartan’s theorem in the case of a more general potential U K ( x ) K ( x , y ) d ( y )     , x  n . Here, also the measure  with compact support and kernel K ( x , y ) , (x,y) n  n, has the following properties: 1) K(x,y) a positive-valued polycontinuous from below a function such that K(x,x)  , x n and continuous in ( n  n ) \ ( x , x ) , x  n , 2) K(x,y) is symmetric, i.e. K(x,y) K(y,x). According to condition 2) the reciprocity law is always satisfied: for any вс  and  measure and with compact supports, ACADEMIC RESEARCH IN MODERN SCIENCE International scientific-online conference K K A measure  is called admissible on a compact set E  n, if 31 S E  and Ux1 everywhere in K n . The family of all admissible measures on E is denoted by M  E  . To each compact set E we associate a number C K  E  , defined as sup ( E )  for all ( E ) .   M Using this set function C K ( E ) , we define the inner and outer capacities of an arbitrary set мы M as follows: the inner capacity C iK ( M ) is equal to s u p C K ( E ) for all compact sets E  M, and the outer capacity C eK ( M ) is equal to i n f C iK ( G ) for all open sets G  M . It follows immediately that Ci (E) C (E) for any compact set K K E , C iK ( M )  C eK ( M ) for any set M and Ci (G)Ce (G) for any open set G. When Ci (M) Ce (M) we K K K K say that M is capacious and we denote by M the common value of these two capacities by C K ( M ) which call the capacity of M . Theorem 2 (Kishi). For any 0 there exists an open set O  n with capacity C (O ) such that the potential of the continuous  K  U K ( x )  complement n\O . Note that in particular the potential Ux can be a Riesz potential or a K logarithmic potential with kernel 2R Kx,yln , x,yB0,RB0,R, x y where B(0,R) x n : x  R (see, for example, [2],[4]). In this work, we study the Luzin C k  properties of logarithmic potentials. Main result. Theorem 3. For any 0 there exists an open set O n with Lebesgue measure m ( O )   such that the logarithmic potential U(x) l belongs to the class C n ( n \ O )  continuously differentiable functions.