Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with $L^\infty$ coefficients

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Abstract: Let

O m e g a s u b s e t m a t h b b R^{n + 1}

,

n g e q 1

, be an open set with

s

-Ahlfors regular boundary

p a r t i a l O m e g a

, for some

s i n (0, n]

, such that either

s = n

and

O m e g a

is a corkscrew domain with the pointwise John condition, or

s < n

and

O m e g a = m a t h b b R^{n + 1} s e t m i n u s E

, for some

s

-Ahlfors regular set

E s u b s e t m a t h b b R^{n + 1}

. In this paper we provide a unifying method to construct Varopoulos' type extensions of

L^{p}

and

B M O

boundary functions. In particular, we show that a) if

f i n L^{p} (p a r t i a l O m e g a)

,

1 < p l e q i n f t y

, there exists

F i n C^{i} n f t y (O m e g a)

such that the non-tangential maximal functions of

F

and

as well as the Carleson functional of

d i s t (c d o t, O m e g a^{c})^{s - n} a b l a F

are in

L^{p} (p a r t i a l O m e g a)

with norms controlled by the

L^{p}

-norm of

f

, and

F o f

in some non-tangential sense

m a t h c a l H^{s} |_{p a r t i a l O m e g a}

-almost everywhere; b) if

, there exists

such that

is uniformly bounded in

O m e g a

and the Carleson functional of

as well the the sharp non-tangential maximal function of

are uniformly bounded on

p a r t i a l O m e g a

with norms controlled by the

B M O

-norm of

, and

in a certain non-tangential sense

m a t h c a l H^{s} |_{p a r t i a l O m e g a}

-almost everywhere. If, in addition, the boundary function is Lipschitz with compact support, then both

F

and

can be constructed so that they are also Lipschitz on

o v e r l i n e O m e g a

and converge to the boundary data continuously. The latter results hold without the additional assumption of the pointwise John condition.

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