BOUNDEDNESS AND COMPACTNESS OF THE HARDY TYPE OPERATOR WITH VARIABLE UPPER LIMIT IN WEIGHTED LEBESGUE SPACES

Let 0 < alpha < 1. The operator of the form K(alpha,phi)f(x) = integral(phi(x))(a) f(t)w(t)dt/(W(x) - W(t))((1-alpha)), x > 0, is considered, where the real weight functions v(x) and w(x) are locally integrable on I := (a, b), 0 <= a < b <= infinity and dW(x)/dx equivalent to w(x). In this paper we derive criteria for the operator K-alpha,K-phi, 0< alpha < 1, 0 < p; q < infinity, p > 1/alpha to be bounded and compact from the spaces L-p,L- w to the spaces L-q,L- v.