University of Illinois Urbana-Champaign

On Certain Maximal Operators On H(p) Classes, 0 Less Than P Less Than or Equal to 1 (Hardy, Fourier)

Hashimi, Jamil Rasool

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Description

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
Issue Date
1984
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(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/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.
Graduation Semester
1984
Type of Resource
text
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http://hdl.handle.net/2142/71227

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