NEW CHARACTERIZATIONS OF WEIGHTED MORREY-CAMPANATO SPACES

Let alpha is an element of (0, infinity), q is an element of [1, infinity], s be a nonnegative integer, omega is an element of A(1)(R-n) (the class of Muckenhoupt's weights). In this paper, the authors introduce the weighted Morrey-Campanato space L(alpha, q, s, omega; R-n) and obtain its equivalence on different q is an element of [1, infinity] and integers s >= left perpendicularn alpha right perpendicular (the integer part of n alpha). The authors then introduce the weighted Lipschitz space Lambda(alpha, omega; R-n) and prove that Lambda(alpha, omega; R-n) = L(alpha, q, s, omega; R-n) when alpha is an element of (0, infinity), s >= left perpendicularn alpha right perpendicular and q is an element of [1, infinity]. Using this, the authors further establish a new characterization of L(alpha, q, s, omega; R-n) by using the convolution phi t(B) * f to replace the minimizing polynomial P(B)(s)f on any ball B of a function f in its norm when alpha is an element of (0, infinity), s >= left perpendicularn alpha right perpendicular, omega is an element of A(1) (R-n) boolean AND RH1+1/alpha(R-n) and q is an element of [1, infinity], where phi is an appropriate Schwartz function, t(B) denotes the radius of the ball B and phi t(B)(.) equivalent to t(B)(-n)phi(t(B)(-1)).