Symmetric conditions for a weighted Fourier transform inequality

Abstract

We discuss conditions on weight functions, necessary or sufficient, so that the Fourier transform is bounded from one weighted Lebesgue space to another. The sufficient condition and the primary necessary condition presented are similar, one being phrased is terms of arbitrary measurable sets and the other in terms of cubes. We believe that the symmetry amongst the two conditions helps frame how a single condition, necessary and sufficient, might appear.We discuss necessary conditions and a sufficient condition on nonnegative functions u and v such that the following weighted norm inequality holds for the Fourier transform: there exists a constant C such that

Pitt inequalities and restriction theorems for the Fourier transform

“…(3) To prove this part, we use ideas similar to [4]. In order to consider translations of the sets A and c n A * by the vectors x 0 and ξ 0 correspondingly, it is enough to consider the function g(x) = f (x − x 0 )e −ixξ 0 so that |g(x)| = |f (x−x 0 )| and | g(ξ)| = | g(ξ −ξ 0 )|.…”

“…Note that in this theorem we do not assume q ≥ p. Part (2) of the theorem is known with a smaller constant c; see the proof of Theorem 3.1 in [16]. Moreover, part (3) generalizes the following necessary condition (see [4,Th. 3]):…”

The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality

“…Translation Property [5]. If the weighted Fourier inequality is satisfied then all translations of the weights also satisfy the same inequality.…”

“…The technique used to prove that (i) ⇒ (iii) originates with Benedetto and Heinig [3, p. 253], where it was used in a different setting. The same technique was applied in Berndt [5] to find a necessary condition similar to the one in (iv) above; and, this technique was used very recently by De Carli, Gorbachev, and Tikhonov [7] to prove (iv) as stated.…”

“…Further, we discuss that this class of weights is essentially the same if E and F are replaced with a rectanguloid and its polar, or an ellipsoid and its polar. Previously in [5], we examined this inequality where E and F were cubes of reciprocal measure. The current condition is a generalization because having reciprocal measure and being polar are essentially the same for cubes.…”