On a generalization of the Fefferman–Stein theorem

R. V. Shanin

We investigate the conditions under which membership of the maximal function of C. Fefferman and E. M. Stein to the class $\varphi (L)$ implies that the Hardy--Littlewood function belong to the same class. We obtain the best possible in some sense, conditions on the function $\varphi$ defining these classes.

1. Hardy G.H., Littlewood J.E. A maximal theorem with function-theoretic applications// Acta Math. – 1930. – V.54. – P. 81–116.

2. Fefferman C., Stein E.M. Hp spaces of several variables// Acta Math. – 1972. – V.129. – P. 137–139.

3. Korenovskii A.A. Properties of functions defined in terms of the mean oscillations: dis. . . kand. phis.-math. nauk: 01.01.01 – Odessa, 1988. – 121 p. (in Russian)

4. Kolyada V.I. Rearrangements of functions and embedding theorems// Usp. Mat. Nauk. – 1989. – V.44, №5(269). – P. 61–95. (in Russian)

5. Bennett C., Sharpley R. Weak-type inequalities for $H^p$ and BMO// Proc. Sympos. Pure Math. - 1979. - V.35, Part1. – P. 201–229.

6. Kolyada V.I. On imbedding in classes '(L)// Isv. AN SSSR. – 1975. – V.39, №2. – P. 418–437. (in Russian)

	On a generalization of the Fefferman–Stein theorem (in Russian)
Author	R. V. Shanin ruslanshanin@gmail.com Odessa I. I. Mechnikov National University
Abstract	We investigate the conditions under which membership of the maximal function of C. Fefferman and E. M. Stein to the class $\varphi (L)$ implies that the Hardy--Littlewood function belong to the same class. We obtain the best possible in some sense, conditions on the function $\varphi$ defining these classes.
Keywords	Hardy–Littlewood maximal function; Fefferman–Stein maximal function; equimeasurable rearrangement
Reference	1. Hardy G.H., Littlewood J.E. A maximal theorem with function-theoretic applications// Acta Math. – 1930. – V.54. – P. 81–116. 2. Fefferman C., Stein E.M. Hp spaces of several variables// Acta Math. – 1972. – V.129. – P. 137–139. 3. Korenovskii A.A. Properties of functions defined in terms of the mean oscillations: dis. . . kand. phis.-math. nauk: 01.01.01 – Odessa, 1988. – 121 p. (in Russian) 4. Kolyada V.I. Rearrangements of functions and embedding theorems// Usp. Mat. Nauk. – 1989. – V.44, №5(269). – P. 61–95. (in Russian) 5. Bennett C., Sharpley R. Weak-type inequalities for $H^p$ and BMO// Proc. Sympos. Pure Math. - 1979. - V.35, Part1. – P. 201–229. 6. Kolyada V.I. On imbedding in classes '(L)// Isv. AN SSSR. – 1975. – V.39, №2. – P. 418–437. (in Russian)
Pages	209-219
Volume	42
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On a generalization of the Fefferman–Stein theorem (in Russian)