We establish estimates for restrictions to certain curves in ${{\mathbb{R}}^{2}}$ of the Fourier transforms of some fractal measures.

We establish a mixed norm estimate for the Radon transform in ${{\mathbb{R}}^{2}}$ when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.

We consider the Fourier restriction operators associated to certain degenerate curves in ℝd for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves.

We establish a sharp Fourier restriction estimate for a measure on a $k$-surface in ${{\mathbb{R}}^{n}}$, where $n\,=\,k\left( k\,+\,3 \right)/2$

This paper contains a geometric proof of an estimate for a restricted x-ray transform. The result complements one of A. Greenleaf and A. Seeger.

Let S be a smooth hypersurface in ℝn with surface area measure ds and Gaussian curvature κ(s). Define the convolution operator T by

formula here

for suitable functions f on ℝn. We are interested in the Lp − Lq mapping properties of T. Write [Sscr ] for the type set of T, the set

formula here

It is well known (see, e.g. [O1]) that [Sscr ] is contained in the closed triangle [Tscr ] with vertices (0, 0), (1, 1) and (n/(n+1), 1/(n+1)). This paper is concerned with estimates of the form

formula here

The estimate (1) is interesting because it serves as a weak substitute for the L(n+1)/n − Ln+1 boundedness of T. For example, if S is compact and (1) holds, then well-known arguments show that [Sscr ] differs from the full triangle [Tscr ] by at most the point (n/(n + 1), 1/(n + 1)). Our main result is a condition sufficient to imply (1). Its statement requires the following definition.

We study convolution properties ofmeasures on the curves $({{t}^{{{a}_{1}}}},{{t}^{{{a}_{2}}}},{{t}^{{{a}_{3}}}})$ in ${{\mathbb{R}}^{3}}$.

We prove an (n + l)-linear inequality which generalizes the classical bilinear inequality of Young concerning the LP norm of the convolution of two functions.

This note contains a strengthened version of the following well-known theorem: there exists a continuous function whose Fourier series diverges at a point.

Let H1(U2) be the Hardy space of the bidisc as described in (3). Each function f ∈ H1(U2) has a Taylor expansion of the form . For 0<p<∞, a doubly-indexed sequence is said to be a multiplier of H1(U2) into lp if

This paper is concerned with the cases p = 2 and p = 1. Theorem 1 characterises the multipliers of into l2 and is an analogue in two variables of an old result of Hardy and Littlewood. Theorem 2 characterises the sequences (an)n≥0 such that (an+m)n,m≥0 is a multiplier of H1(U) into l1

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.