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Title:  On Certain Maximal Operators On H(p) Classes, 0 Less Than P Less Than or Equal to 1 (Hardy, Fourier) 
Author(s):  Hashimi, Jamil Rasool 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  In this thesis we give a generalization of a theorem of C. Fefferman and E. M. Stein on maximal operators on the Hardy classes of tempered distributions H('p)((//R)('n)) for 0 < p (LESSTHEQ) 1. Fix p, 0 < p (LESSTHEQ) 1 and let N = n/pn . Let (phi) (ELEM) C('N)((//R)('n)) have compact support and suppose (phi) satisfies the Dinitype condition (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) This thesis is divided into four chapters. In Chapter 1 we survey the literature leading to this problem, including Coifman and Latter's atomic H('p) theory. In Chapter II we define the notion of quasiregularity for moduli of continuity, show that it is more general than regularity conditions used by others, and use it to construct a kernel for S with a prescribed N('th) modulus of continuity. Chapter III contains the strongtype results for 0 < p (LESSTHEQ) 1 as well as the proof of the existence of the function b mentioned above (which is actually an atom). In Chapter IV we present the weaktype results for 0 < p < 1 and give examples to show that they cannot be extended to the case p = 1. where (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) We call (omega) the N('th) modulus of continuity of (phi), and (alpha) is a multiindex. Then the maximal operator S for f (ELEM) H('p)((//R)('n)) by (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where (phi)(,t)(x) = (phi)(x/t)/t('n) is bounded from H('p)((//R)('n)) in L('p)((//R)('n)). This result is best possible in the sense that if (eta) is any continuous, increasing function such that (eta)(0) = 0, (eta) fails condition (*) and (eta) satisfies a mild regularity condition, then there exists a kernel (phi) (ELEM) ((//R)('n)) whose N('th) modulus of continuity is essentially (eta) and a function b (ELEM) H('p)((//R)('n)) such that (VBAR)(VBAR)Sb(VBAR)(VBAR)(,L('p)) = (INFIN). Fefferman and Stein originally proved this in case p = 1 using entirely different methods. We also give a weaktype version of this result. Fix p, 0 < p < 1. Let (phi) be as above except that instead of condition (*) assume that (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) Then the operator S is a bounded map from H('p)((//R)('n)) into weakL('p)((//R)('n)). This result is also best possible in a sense similar to the strong case. The proofs of all these results make use of the atomic decomposition of H('p)((//R)('n)) given by R. R. Coifman and R. Latter. 
Issue Date:  1984 
Type:  Text 
Description:  50 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1984. 
URI:  http://hdl.handle.net/2142/71227 
Other Identifier(s):  (UMI)AAI8502165 
Date Available in IDEALS:  20141216 
Date Deposited:  1984 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois