{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71227"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71227","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On Certain Maximal Operators On H(p) Classes, 0 Less Than P Less Than or Equal to 1 (Hardy, Fourier)","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","abstract_html":"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","abstract_has_math":false,"creators":["Hashimi, Jamil Rasool"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:11Z","date_published":"2014-12-16T06:18:11Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8502165"],"render_values":[{"text":"(UMI)AAI8502165","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71227","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Hashimi, Jamil Rasool"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:11Z","10000-01-01","1984"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71227","(UMI)AAI8502165"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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."]},{"key":"dc:title","label":"Title","values":["On Certain Maximal Operators On H(p) Classes, 0 Less Than P Less Than or Equal to 1 (Hardy, Fourier)"]}]}],"canonical_facts":{"dc:creator":["Hashimi, Jamil Rasool"],"dc:date":["2014-12-16T06:18:11Z","10000-01-01","1984"],"dc:description":["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). 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