Singular integrals with highly oscillating kernels on the product domains

Singular integrals with highly oscillating kernels on product spaces

“…See also Lemma 1 of [6] and [7]. Second we wish to point out that the L p boundedness, 1 < p < ∞, of the operator (1) is still an open problem since the proof in [2] is inconclusive.…”

“…The paper is organized as follows: in Section 1 we prove the Theorem; in Section 2, concerning the L p theory, we summarize the known results and point out the unproven claims of [2].…”

“…dx ′ first and dy ′ second, is reversed. The next step, in [2], is more easily seen by considering (2) with M 1 (x) = 0 for all x and…”

“…To evaluate the L p (dx) norm of the above operator, the boundedness of the Hilbert transform, acting on x ′ , cannot be used (the inner core-the operator acting on y ′ -varies arbitrarily with x, due to the phase M (x), and the variable x is saturated in the integration). In [2] (line 7 from below on p. 299, line 4 from below on p. 300 and (6)), apparently, to dominate from above the L p (dx) norm of the operator (1), the phase M (x) is replaced by M (x ′ ) to be integrated together with f (x ′ , •) in dx ′ (which is equivalent to keeping M (x), replacing f (x ′ , •) by f (x, •) and integrating in dx, precisely as in [2]).…”