p-Helson curves in the plane (Q1820351)

The author denotes by \(A_ p({\mathbb{R}}^ n)\) (p\(\geq 1)\) the class of functions whose Fourier transform is in \(L_ p({\mathbb{R}}^ n)\). A closed set E in \({\mathbb{R}}^ n\) is called p-Helson set if every continuous function on E can be extended to a continuous function in \(A_ p\). A p- Helson set is p'-Helson for every \(p'>p\). A. M. Olevskij has constructed a set of measure 0 in \({\mathbb{R}}^ 1\) which is not p-Helson for any \(p<2\) (1975). McGehee and Woodward showed that if f is a function in \(C_ 1\) or strictly convex then its graph is not Helson (\(\equiv 1\)-Helson) in \({\mathbb{R}}^ 2\) (1982). The present author proves that \(E\subset {\mathbb{R}}^ n\) is not p-Helson for \(p\in (1,2)\) if there exists a measure \(\mu\in M(E)\) such that \(\| {\hat \mu}\|_ q<\infty\) \((1/p+1/q=1)\). Therefrom he deduces that if f is a nonlinear function in \(C_ 2\) in an interval [a,b] then the graph of f is not p-Helson (in \({\mathbb{R}}^ 2)\) for any \(p<4/3\). The remaining main results are as follows: (1) If f is of bounded variation in [a,b] and \(4/3<p\) then the graph of f is p-Helson. (2) If a continuous broken line in \({\mathbb{R}}^ 2\) is the graph of a function in [a,b] then it is a p-Helson set for every \(p>1\). (3) There exists a strongly convex function in \(C_ 1[0,1]\) whose graph is p-Helson for every \(p>1.\) The main lemma on which the proofs of these results are based reads: Let E be a closed bounded set in \({\mathbb{R}}^ 2\) and \(\beta_ p(E)=\inf \| \hat {\mathbf{1}}_ G\|_ p\) where inf is taken over all open sets \(G\supset E\). If for some \(p>1\) and \(\epsilon\in (0,p-1)\) one has \(\beta_{p-\epsilon}(E)=0\) then E is p-Helson. (In the proof of this lemma ''\(f(x,y)-\phi_{n-1}(x,y)\)'' should be replaced by ''\(f(x,y)- \sum^{n-1}_{i=1}\phi_ i(x,y)\)''.)