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We prove unique continuation for solutions of the inequality , a connected set contained in and is in the Morrey spaces , with and . These spaces include for (see [H], [BKRS]). If , the extra assumption of being small enough is needed.
We prove the existence of functions , the Fourier series of which being universally divergent on countable subsets of . The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on .
We establish upper bounds for certain trigonometric sums involving cosine powers. Part of these results extend previous ones valid for the sum
.
We apply our results to estimate character sums in an explicit and elementary way.
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