Riesz inequality

Let $ \{ \phi _ {n} \} $ be an orthonormal system of functions on an interval $ [ a, b] $ and let $ | \phi _ {n} | \leq M $ almost everywhere on $ [ a, b] $ for any $ n $.

a) If $ f \in L _ {p} [ a, b] $, $ 1 < p \leq 2 $, then its Fourier coefficients with respect to $ \{ \phi _ {n} \} $,

$$ c _ {n} = \int\limits _ { a } ^ { b } f \overline \phi \; _ {n} dx $$

satisfy the Riesz inequality

$$ \| \{ c _ {n} \} \| _ {q} \leq M ^ {2/p-} 1 \| f \| _ {p} ,\ \ \frac{1}{p} + \frac{1}{q} = 1. $$

b) For any sequence $ \{ c _ {n} \} $ with $ \| \{ c _ {n} \} \| _ {p} < \infty $, $ 1 < p \leq 2 $, there exists a function $ f \in L _ {q} [ a, b] $ with $ c _ {n} $ as its Fourier coefficients and satisfying the Riesz inequality

$$ \| f \| _ {q} \leq M ^ {2/p-} 1 \| \{ c _ {n} \} \| _ {p} ,\ \ \frac{1}{p} + \frac{1}{q} = 1. $$

If $ f \in L _ {p} [ 0, 2 \pi ] $, $ 1 < p \leq \infty $, then the conjugate function $ \overline{f}\; \in L _ {p} [ 0, 2 \pi ] $ and the Riesz inequality

$$ \| \overline{f}\; \| _ {p} \leq A _ {p} \| f \| _ {p} $$

holds, where $ A _ {p} $ is a constant depending only on $ p $.

Assertion 1) was for the first time proved by F. Riesz [1]; particular cases of it were studied earlier by W.H. Young and F. Hausdorff. Assertion 2) was first proved by M. Riesz [2].

References

[1]	F. Riesz, "Ueber eine Verallgemeinerung der Parsevalschen Formel" Math. Z. , 18 (1923) pp. 117–124
[2]	M. Riesz, "Sur les fonctions conjuguées" Math. Z. , 27 (1927) pp. 218–244
[3]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[4]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)

Comments

For 2) see also Interpolation of operators (it is a consequence of the Marcinkiewicz interpolation theorem and the weak type $ ( 1, 1) $ of the conjugation operator) and [a3].

References

[a1]	P.L. Butzer, R.J. Nessel, "Fourier analysis and approximation" , 1 , Birkhäuser (1971) pp. Chapt. 8
[a2]	F. Hausdorff, "Eine Ausdehnung des Parsevalschen Satzes über Fourier-reihen" Math. Z. , 16 (1923) pp. 163–169
[a3]	E.M. Stein, G. Weiss, "Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1975) pp. Chapt. VI, §5