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The unboundedness of the maximal directional Hilbert transform Marinelli, Alessandro

Abstract

In this dissertation we study the maximal directional Hilbert transform operator associated with a set U of directions in the n-dimensional Euclidean space. This operator shall be denoted by ℋU. We discuss in detail the proof of the (p; p)-weak unboundedness of ℋU in all dimensions n ≥ 2 and all Lebesgue exponents 1 < p < +∞ if U contains infinitely many directions in Rn. This unboundedness result for ℋU is an immediate consequence of a lower estimate for ||ℋU||_Lp(ℝn) → Lp(ℝn) that we prove if the cardinality of U is finite. In this case, we show that ||ℋU||_Lp(ℝn) → Lp(ℝn) is bounded from below by the square root of √log(#U) up to a positive constant depending only on p and n, for any exponent p in the range 1 < p < +∞ and any n ≥ 2. These results were first proved by G. A. Karagulyan ([17]) in the case n = p = 2.

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