WEIGHTED SOLYANIK ESTIMATES FOR THE STRONG MAXIMAL FUNCTION

Let M-s denote the strong maximal operator on R-n and let w be a non-negative, locally integrable function. For alpha is an element of (0, 1) we define the weighted Tauberian constant C-S,C-w associated with M-S by C-s,C-w(alpha) := sup(E subset of Rn0<w(E)<+infinity) 1/w(E)w({x is an element of R-n : M-s(1(E))(x) > alpha}). We show that lim(alpha -> 1) - C-s,C-w (alpha) = 1 if and only if w is an element of A(infinity)(*), that is if and only if w is a strong Muckenhoupt weight. This is quantified by the estimate C-s,C-w(alpha) - 1 less than or similar to n (1-a)((en[w])A(infinity)(*))(-1), as alpha -> 1(-), where c > 0 is a numerical constant independent of n; this estimate is sharp in the sense that the exponent 11(cn[w]A*(infinity)) can not be improved in terms of [w]A*(infinity). As corollaries, we obtain a sharp reverse Holder inequality for strong Muckenhoupt weights in R-n as well as a quantitative imbedding of A(infinity)(*) into A(p)(*). We also consider the strong maximal operator on R-n associated with the weight w and denoted by M-s(w). In this case the corresponding Tauberian constant C-S(w) is defined by C-s(w)(alpha) := sup(E subset of Rn0<w(E)<+infinity) 1/w(E)w({x is an element of R-n : M-s(1(E))(x) > alpha}). We show that there exists some constant c(w,n) > 0 depending only on w and the dimension n such that C-s(w) (alpha) - 1 less than or similar to(w,n) (1-alpha)(cw,n) as alpha -> 1(-) whenever w is an element of A*(infinity) is a strong Muckenhoupt weight.