Rl. Wheeden et Jm. Wilson, WEIGHTED NORM ESTIMATES FOR GRADIENTS OF HALF-SPACE EXTENSIONS, Indiana University mathematics journal, 44(3), 1995, pp. 917-969
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.