Strongly singular integrals along curves

Abstract

We obtain L 2 bounds for strongly singular integral operators along curves in ‫ޒ‬ d . Our results both generalize and extend to higher dimensions the planar results of Chandarana. In addition, we show that these operators are bounded from L log L to weak L 1 at the critical exponent α = 0.

New results on restriction of Fourier multipliers

“…We have seen in the previous section that the associated multiplier is normalized. Hence the result follows by combining [21,Theorem 3] and Theorem 8.…”

“…, t j d ) with 1 ≤ j 1 < · · · < j d being positive integers. See the works of Laghi and Lyall [21] and Chandarana [12] and the references therein for further details on this type of operators. By [21,Theorem 3] whenever αp ≤ 2β d+1 and 1 < p ≤ 2,…”

Strongly singular integrals along curves on α-modulation spaces

“…See [17, 18] for details. According to [17] (Proposition 3.1), to every smooth well-curved there exists a constant nonsingular matrix M such that is of standard type; that is, approximately homogeneous, taking the form for with . Following the ideas in [17], combining with Theorem 3.3, we can get…”

Strongly singular Radon transforms on the Heisenberg group and folding singularities

“…In Section 7 we shall indicate how the techniques introduced to study operators of the form (4) can be employed to revisit and generalise these results. We however point out that this approach is not exactly necessary and that one can also obtain the result below by simply appealing to van der Corput's lemma; see [11].…”