  This paper is concerned with a boundedness of trace and extension operators
for Besov spaces and Triebel-Lizorkin spaces on upper half space with variable
exponents. To define trace and extension operators, we introduce a quarkonial
decomposition for Besov spaces and Triebel-Lizorkin spaces with variable
exponents on ${\mathbb R}^n$. Furthermore, we study trace and extension
operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents
on upper half spaces ${\mathbb R}^n_+$.
