Maximum principle in potential theory and imbedding

MAXIMUM PRINCIPLE IN POTENTIAL THEORY AND IMBEDDING THEOREMS FOR AN!SOTROPIC SPACES OF DIFFERENTIABLE FUNCTIONS S. K. Vodop'yanov

UDC 517.5:517.968

The applicability of the ideas and methods of classical potential theory [1-3] and its nonlinear generalization [4-11] depends on the properties of the potentials ( U ~ ) ( x ) = S K

lq--l. y) d~t(y) (or(UIc,u)(x)=l~nK(x,Y) [ [I. K(Y,z)d[~(z)

dy

Inn possible to preserve in the more general i s t h e g e n e r a l i z e d maximum p r i n c i p l e : if c is a positive constant.

(x,

Rn in the nonlinear

theory.

case,

q~(t,

oo)), w h i c h i t

is

One o f t h e b a s i c p r o p e r t i e s of potentials on s u p p ~ t h e n (U~lx)(x) 0. 2.2. In what follows we consider only the case of Euclidean space R ~ with the group of dilations 8t=tA=exp(Alnt), t > 0, where A is a real n • n-matrix with eigenvalues kj~ Re Xj > 0 and t r A = v, defined on it. Let A' be the matrix conjugate to A, 6 t = t A , and p ~ C ~ ( R ~ \ { 0 } ) be a 5~-homogeneous metric connected with the group ~ . The space of anisotropic Bessel potentials L~ is defined as the convolutions of functions from Lv(R: ) with kernel Gy whose Fourier transform Gv(~) = ~ G~(x) e-~xdx is equal to (I +p~(~))-~/z (here ~x = ~x~ + Rn

~zxz+...+~'x,~ is the scalar product of the vectors x and ~). Thus, L~={/~Lp:

/=G~,g,g~Lp},

l I

Gv (x) g

[ h e r e 2k>max~f/a#, aj a r e t h e e l e m e n t s o f t h e m a t r i x A, A~/(x) i s t h e c e n t e r e d d i f f e r e n c e J t h e f u n c t i o n f o f o r d e r 2k w i t h v e c t o r s t e p t and w i t h c e n t e r a t t h e p o i n t x ) .

of

It is established i n [32] t h a t t h e f u n c t i o n / E P ( L v ) i f and o n l y i f ]~Lq(R'~), t / q = l / p - y / n and T T / ~ L v ( R ~ ) . Comparing t h e s e two d e s c r i p t i o n s , and c o n s i d e r i n g t h e c o m p a c t n e s s o f

[[/I]L~M[I/]IIY(Lp ), where the constant the function f. The imbedding L~p(Rn)-+IV(Lp), L~-+Lq, I/q= I/p-7/n, and consequently the lower

the support of-the function f, we conclude that M depends on the diameter of the support of also follows from these descriptions, since bound follows. THEOREM 4.

Let the matrix A be diagonal,

f~L~p

and

t r A = n.

oo

S cap ( { I / [ > t}, L~) dt p t/N},n~).

of the semiadditivity

of the capacity

h a v e cap({]/[~t},L~)
p we con"-Lq(~), q > 1 is considered in point 2.4. 2.3 . Considering the Besov space B p,p~ l (R n~j, p > 1 , | ~ R ~ , as the space of traces on a hyperplane, of functions from suitably chosen spaces of aniso~tropic Bessel potentials, we get from Theorems 4-7, corresponding assertions for the Besov spaces (cf. the definition in [43]). Let the hyperplane R = in the space R =+~ be defined by fixing the last coordinate: {x ~ R TM :

L e t m ~ R ? ~ +~, x = 1 It

is proved

t >0, pmn+l

in [34]

that

l~= •

for

i = 1, 2, ..., n.

any function/~C~(R

=)

[I/ll )lv,v(Rn ~ inf IIext /IIL~,(~n+~ ), w h e r e e x t / ~ C ~ ( R "+~) i s a n y e x t e n s i o n taken over all such extensions. It

where cap (E,B~,v(Rn)) = inf

p ~ (1, ~ ) ,

o f t h e f u n c t i o n f t o t h e s p a c e R "+~, and t h e i n f i m u m i s follows in particular from this that for any set E~R =

cap(E,

BIv,v(Rn))..-.cap(E, L?* (R'~+i)),

{[[ul]~[,v(Rn) p~ , u >j i on g}.

This equivalence lets us establish,

the space B~,v(Rn), properties 1-6 of the capacity, a number of propositions and theorems.

l*p>n,

Proposition 12.

If

Proposition 13.

If E c R

for

formulated in point 1.2, and lets us prove

In what follows we choose the component ran+l so that l * - n / p = m * - ( n + i ) / p . the metric r(x) is defined by a smoothness vector I.

As above,

then for any compact set e ~ r with diana e~< I one has

cap (e,

1

~ is a Borel set, then I E p-vz*)/n < c cap (E,

where

R n=

x.+~ = 0).

By,v), ' t < p < n/l*,

]El is the Lebesgue measure of the set E. Proposition

14.

IfE~R"

is a Borel set, l~c(ln 2n ~l-p ]--fT/

Proposition

15.

If i < p < n / l *

Proposition

16.

If l