On the weak behaviour of partial sums of Legendre series

Chanillo, S.

doi:10.1090/S0002-9947-1981-0632534-1

On the weak behaviour of partial sums of Legendre series
HTML articles powered by AMS MathViewer

by S. Chanillo

Trans. Amer. Math. Soc. 268 (1981), 367-376

DOI: https://doi.org/10.1090/S0002-9947-1981-0632534-1

PDF | Request permission

Abstract:

We show that the partial sum operator associated with the Legendre series is restricted weak type, but not weak type, on the ${L^p}$ spaces when $p = 4$.

References

Richard A. Hunt, On $L(p,\,q)$ spaces, Enseign. Math. (2) 12 (1966), 249–276. MR 223874
Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
Jerome Newman and Walter Rudin, Mean convergence of orthogonal series, Proc. Amer. Math. Soc. 3 (1952), 219–222. MR 47811, DOI 10.1090/S0002-9939-1952-0047811-2
Harry Pollard, The mean convergence of orthogonal series. I, Trans. Amer. Math. Soc. 62 (1947), 387–403. MR 22932, DOI 10.1090/S0002-9947-1947-0022932-1
Harry Pollard, The convergence almost everywhere of Legendre series, Proc. Amer. Math. Soc. 35 (1972), 442–444. MR 302973, DOI 10.1090/S0002-9939-1972-0302973-7

Orthogonal polynomials

A. Zygmund, Trigonometric series. Vol. I, II, Cambridge University Press, Cambridge-New York-Melbourne, 1977. Reprinting of the 1968 version of the second edition with Volumes I and II bound together. MR 0617944