A Pointwise Estimate for the Fourier Transform and Maxima of a Function

Abstract

Abstract. We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.

Symmetric conditions for a weighted Fourier transform inequality

“…Theorem 1 shows how a less stringent inequality than (2) is enough for decreasing functions, and Theorem 2 shows how an analogous inequality for decreasing rearrangements is enough for (1) to hold more generally. [6].) If 1 < p q < ∞ and there is a constant C such that…”