Non-localized oscillatory fractional integral operators with polynomial phase functions

초록

We investigate the LP decay estimates of a class of non-localized oscillatory fractional integral operators of the form T-lambda-(alpha)f (x) = integral(R)e(lambda)(i)p((x-y))|x -y|(-alpha)f(y)dy, where 0 <= alpha< 1, >= 1 and P is a real polynomial of degree n >= 2 Our goal is to characterize the sharp decay rate & in the estimate || T(lambda,-alpha||)Lp -> L-p less than or similar to lambda(-delta)i n terms of the parameters alpha and p. To this end, we introduce a hexagonal region H(y, 1-(alpha)/n) in the ((1)/p,(delta))-plane that precisely describes the admissible range for decay, where y = min {1-alpha/l+2, (1)/(2=+m}) Here, & is the order of vanishing of P" at the origin, and m is the maximum multiplicity among nonzero real roots of P"(t) = 0 We prove that the decay estimate holds if and only if (8) lies inside this hexagon. Our approach decomposes the operator into localized parts near singular points and a non-localized remainder. We apply the van der Corput-type estimates to handle L-2 decay near singularities and derive Hardy space bounds to interpolate to the full L-P range. For the non-localized parts, we establish decay estimates using multiplier techniques adapted to translation-invariant kernels. This yields a complete description of the L-P behavior of T-lambda-(alpha)in terms of both singular structure and kernel decay.

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