WEAK ESTIMATES FOR THE MAXIMAL AND RIESZ POTENTIAL OPERATORS ON NON-HOMOGENEOUS CENTRAL MORREY TYPE SPACES IN L1 OVER METRIC MEASURE SPACES

In a metric measure space (X, d, mu), our first aim in this paper is to discuss the weak estimates for the maximal and Riesz potential operators in the non-homogeneous central Morrey type space M-1,M-q,M-a(X) (about x(0) is an element of X) of all measurable functions f on X satisfying parallel to f parallel to(M1,q,a(X)) = (integral(infinity)(1) (r(-a)parallel to f parallel to(L1(B(x0,r))))(q) dr/r)(1/q) < infinity for a >= 0 and 0 < q < infinity; when q = infinity, we apply a necessary modification. To do this, we consider the family WM phi,q,a (X) of all measurable functions f is an element of L-loc(1)(X) such that parallel to f parallel to(WM phi,q,a(X)) = sup(lambda>0) lambda (integral(infinity)(1) (r(-q)phi(-1) (integral(B(x0,r)) chi(Ef(lambda)) (x) d mu(x)))(q) dr/r)(1/q) < infinity where phi is a general function satisfying certain conditions and chi(Ef (lambda)) denotes the characteristic function of E-f(lambda) = {x is an element of X : vertical bar f (x)vertical bar > lambda}. In connection with M-1,M-q,M-a(X), we treat the complementary space N-infinity,N-q,N-a(X) of all measurable functions f on X satisfying parallel to f parallel to(N infinity,q,a(X)) = parallel to f parallel to(L infinity(B(x0,2))) + (integral(infinity)(1) (r(a)parallel to f parallel to(L1(X\B(x0,r))))(q) dr/r)(1/q) < infinity.