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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.
Item Metadata
Title |
The unboundedness of the maximal directional Hilbert transform
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2018
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Description |
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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Genre | |
Type | |
Language |
eng
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Date Available |
2018-04-13
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0365609
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2018-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International