Estimation of the essential supremum of a regression function

Given an independent and identically distributed sample of the distribution of an Rd×RRd×R-valued random vector (X,Y)(X,Y), the problem of estimation of the essential supremum of the corresponding regression function m(x)=E{Y|X=x} is considered. Estimates are constructed, which converge almost surely to this value whenever the dependent variable YY satisfies some weak integrability condition.