Metrics and smoothing of translation-invariant Radon transforms along curves

In this paper new <f>Lαp→Lβq</f> estimates are proved for translation-invariant Radon transforms along curves for <f>α&les;β</f> and <f>p<q</f>. For a fixed <f>α</f> and <f>β</f>, if <f>p</f> is sufficiently close to 2 the best possible <f>q</f> is obtained, up to <f>&epsiv;</f>. The method is related to that of Greenblatt (Ph.D. Thesis, Princeton University, 1998). [Copyright &y& Elsevier]