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Two-parameter Hardy-Littlewood inequalities

Ferenc Weisz (1996)

Studia Mathematica

The inequality (*) ( | n | = 1 | m | = 1 | n m | p - 2 | f ̂ ( n , m ) | p ) 1 / p C p ƒ H p (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space H p on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in L p whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series...

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