Weighted norm inequalities for a family of one-sided minimal operators

Transmission of Convergence

“…Ê Ñ Ö º In the special cases where dµ j = χ Ej |E j | dx, the differentiation of the integral case, or dµ j = φ εj dx, the approximate identity case, this type of problem was already examined in [2], [3] with a more restrictive hypothesis.…”

“…Our analysis will be on R = R 1 . Let µ, ν be two positive Borel measures on R, and let mf (x) = inf x∈I 1 |I| I f , where I is an interval in R (see [2], [3]). We consider the following three conditions.…”

Norm inequalities for the minimal and maximal operator, and differentiation of the integral

“…We also examined the one-weight, weighted norm inequalities which it satisfies. Further, we have used the minimal operator to study one-sided reverse Hölder inequalities and one-sided (A p ) weights [4], and to study the convergence of convolution operators with "bad" kernels [5]. In a forthcoming paper [3] we apply the weighted norm inequalities for the minimal operator (including the results in this paper) to study the weighted norm inequalities for the geometric maximal operator…”