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Hashimi, Jamil Rasool 2014-12-16T06:18:11Z 2014-12-16T06:18:11Z 10000-01-01 1984 1984 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/p-n . Let (phi) (ELEM) C('N)((//R)('n)) have compact support and suppose (phi) satisfies the Dini-type 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 quasi-regularity 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 strong-type 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 weak-type results for 0 < p < 1 and give examples to show that they cannot be extended to the case p = 1. where We call (omega) the N('th) modulus of continuity of (phi), and (alpha) is a multi-index. Then the maximal operator S for f (ELEM) H('p)((//R)('n)) by 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 weak-type version of this result. Fix p, 0 < p < 1. Let (phi) be as above except that instead of condition (*) assume that Then the operator S is a bounded map from H('p)((//R)('n)) into weak-L('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. Made available in DSpace on 2014-12-16T06:18:11Z (GMT). No. of bitstreams: 1 8502165.pdf: 1124208 bytes, checksum: afb3425bfb39ba5eeac2bf17914d934e (MD5) Previous issue date: 1984 Embargo set by: Seth Robbins for item 71393 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 50 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984. http://hdl.handle.net/2142/71227 (UMI)AAI8502165 Mathematics On Certain Maximal Operators On H(p) Classes, 0 Less Than P Less Than or Equal to 1 (Hardy, Fourier) text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.