WEIGHTED NORM ESTIMATES FOR GRADIENTS OF HALF-SPACE EXTENSIONS

Let mu(x, y) = fphi(y)(x), with f in boolean OR(1 less than or equal to p < infinity) L(p)(R(d)) and phi a smooth convolution kernel with d ecay at infinity. We prove sufficient conditions on positive measures mu, and nonnegative weights upsilon which ensure that (alpha) (integra l(R+d+1) \del mu\(q) d mu)(1/q) less than or equal to (integral(Rd) \f \(p) upsilon dx)(1/p) holds for all suitable functions f. We prove nec essary conditions in the case where phi = the Poisson kernel. We also consider a discrete (i.e., martingale) analogue to (alpha), and we pro ve sufficient (and necessary) conditions in that case as well.