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Unique continuation for the solutions of the laplacian plus a drift

Alberto Ruiz, Luis Vega (1991)

Annales de l'institut Fourier

We prove unique continuation for solutions of the inequality | Δ u ( x ) | ≤ V ( x ) | ∇ u ( x ) | , x ∈ Ω a connected set contained in R n and V is in the Morrey spaces F α , p , with p ≥ ( n - 2 ) / 2 ( 1 - α ) and α < 1 . These spaces include L q for q ≥ ( 3 n - 2 ) / 2 (see [H], [BKRS]). If p = ( n - 2 ) / 2 ( 1 - α ) , the extra assumption of V being small enough is needed.

Universally divergent Fourier series via Landau's extremal functions

Gerd Herzog, Peer Chr. Kunstmann (2015)

Commentationes Mathematicae Universitatis Carolinae

We prove the existence of functions f ∈ A ( 𝔻 ) , the Fourier series of which being universally divergent on countable subsets of 𝕋 = ∂ 𝔻 . The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on 𝕋 ∖ { 1 } .

Upper bounds for certain trigonometric sums involving cosine powers

Anastasios D. Simalarides (2015)

Colloquium Mathematicae

We establish upper bounds for certain trigonometric sums involving cosine powers. Part of these results extend previous ones valid for the sum ∑ m = 1 k - 1 | s i n ( π r m / k ) | / s i n ( π m / k ) . We apply our results to estimate character sums in an explicit and elementary way.

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