MAXIMAL OPERATORS ASSOCIATED WITH SOME SINGULAR SUBMANIFOLDS

Abstract

Let U be a bounded open subset of R-d and let Omega be a Lebesgue measurable subset of U. Let gamma = (gamma(1),...,gamma(n)) : U \ Omega -> R-n be a Lebesgue measurable function, and let mu be a Borel measure on Rd+n defined by <mu, f > = integral(d)(R)f(y,gamma(y))(sic)(y) xU\Omega(y) dy, where (sic) is a smooth function supported in U. In this paper we give some conditions under which the Fourier decay estimates vertical bar(mu) over cap(xi)vertical bar <= C(1+vertical bar xi vertical bar)(-epsilon) hold for some epsilon > 0. As a corollary we obtain the L-p- boundedness properties of the maximal operators M-S associated with a certain class of possibly non-smooth n-dimensional submanifolds of Rd+n, i.e., M(s)f(x) = sup r(-d) integral(vertical bar y vertical bar < r) vertical bar f(x-(y, gamma(y)))vertical bar x(R)d \Omega(sym) dy, r > 0 where Omega(sym) is a radially symmetric Lebesgue measurable subset of R-d,gamma(y) = (gamma(1)(y),...,gamma(n)(y)), gamma(i)(ty) = t(ai) gamma(i)(y) for each t > 0 where a(i) is an element of R, and the function gamma(i) : R-d\Omega(sym) -> R satisfies some singularity conditions over a certain subset of R-d. Also we investigate the endpoint (parabolic H-1, L-1,L-infinity) mapping properties of the maximal operators M-H associated with a certain class of possibly non-smooth hypersurfaces, i.e., M(H)f(x) = sup vertical bar integral(R)d f(x - (y, gamma(y)))r(-d)(sic)(r(-1)y)dy vertical bar , r > 0 where the function gamma : R-d -> R satisfies some singularity conditions over a certain subset of R-d and gamma(ty) = t(m) gamma(ty) for each t > 0 where m > 0.