Geodesic
Exponential Estimates for the Conjugate Function on Locally Compact Abelian Groups
Canadian mathematical bulletin, Tome 33 (1990) no. 1, pp. 34-44

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Let G be a locally compact Abelian group, with character group X. Suppose that X contains a measurable order P. For the conjugate function of f is the function whose Fourier transform satisfies the identity for almost all χ in X where sgnp(χ) = - 1 , 0, 1, according as We prove that, when f is bounded with compact support, the conjugate function satisfies some weak type inequalities similar to those of the Hilbert transform of a bounded function with compact support in R. As a consequence of these inequalities, we prove that possesses strong integrability properties, whenever f is bounded and G is compact. In particular, we show that, when G is compact and f is continuous on G, the function is integrable for all p > 0.
DOI : 10.4153/CMB-1990-006-9
Mots-clés : 43A17 
Asmar, Nakhlé Habib. Exponential Estimates for the Conjugate Function on Locally Compact Abelian Groups. Canadian mathematical bulletin, Tome 33 (1990) no. 1, pp. 34-44. doi: 10.4153/CMB-1990-006-9
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     title = {Exponential {Estimates} for the {Conjugate} {Function} on {Locally} {Compact} {Abelian} {Groups}},
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     pages = {34--44},
     year = {1990},
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     doi = {10.4153/CMB-1990-006-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1990-006-9/}
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