{"id":{"repo_id":"unm","oai_identifier":"oai:digitalrepository.unm.edu:math_etds-1032"},"canonical_url":"https://search.dev.ndltd.org/etd/unm/oai:digitalrepository.unm.edu:math_etds-1032","repository":{"repo_id":"unm","name":"University of New Mexico","base_url":"https://digitalrepository.unm.edu/do/oai/"},"display":{"title":"Weighted estimates for dyadic operators with complexity","abstract":"We extend the definitions of dyadic paraproduct, dual dyadic paraproduct and $t$-Haar multipliers to dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$ positive integers. We will use the ideas developed by Nazarov and Volberg in \\\\cite{NV} to prove that the weighted $L^2(w)$-norm of a paraproduct with complexity $(m,n)$ and the dual paraproduct associated to a function $b\\\\in BMO$, depends linearly on the $A_2$-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and polynomially in the complexity. Moreover we prove that the $L^2(w)$-norm of the composition of these operators depends linearly on the $A_2$-characteristic of the weight $w$, quadratic on the $BMO$-norm of $b$, and polynomially in the complexity. The argument for the paraproduct provides a new proof of the linear bound for the dyadic paraproduct \\\\cite{Be1} (the one with complexity $(0,0)$). Paraproducts and their adjoints are examples of Haar shift multipliers of type 2 and 3. We adapt the Nazarov and Volberg method to show that for certain Haar shift multipliers of type 4 and complexity $(m,n)$ the same type of bounds in $L^2(w)$ hold. Also we prove that the $L^2$-norm of a $t$-Haar multiplier for any $t$ and weight $w$ depends on the square root of the $C_{2t}$-characteristic of $w$ times the square root of the $A_q$-characteristic of $w^{2t}$ and polynomially in the complexity $(m,n)$, recovering a result of Beznosova \\\\cite{Be} for the $(0,0)$-complexity case. Last, we prove that for a pair of weights $u$ and $v$ and a class of locally integrable function $b$ that satisfies certain conditions, the dyadic paraproduct $\\\\pi_b$ is bounded from $L^2(u)$ into $L^2(v)$ if and only if the weights satisfies the joint $A_2$ condition.","abstract_html":"We extend the definitions of dyadic paraproduct, dual dyadic paraproduct and $t$-Haar multipliers to dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$ positive integers. We will use the ideas developed by Nazarov and Volberg in \\\\cite{NV} to prove that the weighted <span class=\"etd-inline-math\">L<sup>2</sup>(w)</span>-norm of a paraproduct with complexity $(m,n)$ and the dual paraproduct associated to a function $b\\\\in BMO$, depends linearly on the <span class=\"etd-inline-math\">A<sub>2</sub></span>-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and polynomially in the complexity. Moreover we prove that the <span class=\"etd-inline-math\">L<sup>2</sup>(w)</span>-norm of the composition of these operators depends linearly on the <span class=\"etd-inline-math\">A<sub>2</sub></span>-characteristic of the weight $w$, quadratic on the $BMO$-norm of $b$, and polynomially in the complexity. The argument for the paraproduct provides a new proof of the linear bound for the dyadic paraproduct \\\\cite{Be1} (the one with complexity $(0,0)$). Paraproducts and their adjoints are examples of Haar shift multipliers of type 2 and 3. We adapt the Nazarov and Volberg method to show that for certain Haar shift multipliers of type 4 and complexity $(m,n)$ the same type of bounds in <span class=\"etd-inline-math\">L<sup>2</sup>(w)</span> hold. Also we prove that the <span class=\"etd-inline-math\">L<sup>2</sup></span>-norm of a $t$-Haar multiplier for any $t$ and weight $w$ depends on the square root of the <span class=\"etd-inline-math\">C<sub>2t</sub></span>-characteristic of $w$ times the square root of the <span class=\"etd-inline-math\">A<sub>q</sub></span>-characteristic of <span class=\"etd-inline-math\">w<sup>2t</sup></span> and polynomially in the complexity $(m,n)$, recovering a result of Beznosova \\\\cite{Be} for the $(0,0)$-complexity case. Last, we prove that for a pair of weights $u$ and $v$ and a class of locally integrable function $b$ that satisfies certain conditions, the dyadic paraproduct <span class=\"etd-inline-math\">\\&pi;<sub>b</sub></span> is bounded from <span class=\"etd-inline-math\">L<sup>2</sup>(u)</span> into <span class=\"etd-inline-math\">L<sup>2</sup>(v)</span> if and only if the weights satisfies the joint <span class=\"etd-inline-math\">A<sub>2</sub></span> condition.","abstract_has_math":true,"creators":["Moraes, Jean Carlo Pech de"],"institution":null,"degree_name":"Mathematics","degree_level":"Doctoral","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["Pereyra, Maria Cristina","Maria Cristina Pereyra","Matthew D. Blair","Jens Lorenz","Carlos Pérez"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-02-01T08:00:00Z","date_published":"2012-02-01T08:00:00Z","updated_at":"2026-07-24T05:27:37Z","subjects":["Measure theory","Linear operators","Inequalities (Mathematics)","Integrals","Haar","Lebesgue integral."],"languages":["English"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalrepository.unm.edu/math_etds/33"],"render_values":[{"text":"https://digitalrepository.unm.edu/math_etds/33","href":"https://digitalrepository.unm.edu/math_etds/33","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1928/17491","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Pereyra, Maria Cristina","Maria Cristina Pereyra","Matthew D. 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We will use the ideas developed by Nazarov and Volberg in \\\\cite{NV} to prove that the weighted $L^2(w)$-norm of a paraproduct with complexity $(m,n)$ and the dual paraproduct associated to a function $b\\\\in BMO$, depends linearly on the $A_2$-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and polynomially in the complexity. Moreover we prove that the $L^2(w)$-norm of the composition of these operators depends linearly on the $A_2$-characteristic of the weight $w$, quadratic on the $BMO$-norm of $b$, and polynomially in the complexity. The argument for the paraproduct provides a new proof of the linear bound for the dyadic paraproduct \\\\cite{Be1} (the one with complexity $(0,0)$). Paraproducts and their adjoints are examples of Haar shift multipliers of type 2 and 3. We adapt the Nazarov and Volberg method to show that for certain Haar shift multipliers of type 4 and complexity $(m,n)$ the same type of bounds in $L^2(w)$ hold. Also we prove that the $L^2$-norm of a $t$-Haar multiplier for any $t$ and weight $w$ depends on the square root of the $C_{2t}$-characteristic of $w$ times the square root of the $A_q$-characteristic of $w^{2t}$ and polynomially in the complexity $(m,n)$, recovering a result of Beznosova \\\\cite{Be} for the $(0,0)$-complexity case. Last, we prove that for a pair of weights $u$ and $v$ and a class of locally integrable function $b$ that satisfies certain conditions, the dyadic paraproduct $\\\\pi_b$ is bounded from $L^2(u)$ into $L^2(v)$ if and only if the weights satisfies the joint $A_2$ condition."]},{"key":"dc:title","label":"Title","values":["Weighted estimates for dyadic operators with complexity"]}]}],"canonical_facts":{"dc:contributor":["Pereyra, Maria Cristina","Maria Cristina Pereyra","Matthew D. Blair","Jens Lorenz","Carlos Pérez"],"dc:creator":["Moraes, Jean Carlo Pech de"],"dc:description.abstract":["We extend the definitions of dyadic paraproduct, dual dyadic paraproduct and $t$-Haar multipliers to dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$ positive integers. We will use the ideas developed by Nazarov and Volberg in \\\\cite{NV} to prove that the weighted $L^2(w)$-norm of a paraproduct with complexity $(m,n)$ and the dual paraproduct associated to a function $b\\\\in BMO$, depends linearly on the $A_2$-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and polynomially in the complexity. Moreover we prove that the $L^2(w)$-norm of the composition of these operators depends linearly on the $A_2$-characteristic of the weight $w$, quadratic on the $BMO$-norm of $b$, and polynomially in the complexity. The argument for the paraproduct provides a new proof of the linear bound for the dyadic paraproduct \\\\cite{Be1} (the one with complexity $(0,0)$). Paraproducts and their adjoints are examples of Haar shift multipliers of type 2 and 3. We adapt the Nazarov and Volberg method to show that for certain Haar shift multipliers of type 4 and complexity $(m,n)$ the same type of bounds in $L^2(w)$ hold. Also we prove that the $L^2$-norm of a $t$-Haar multiplier for any $t$ and weight $w$ depends on the square root of the $C_{2t}$-characteristic of $w$ times the square root of the $A_q$-characteristic of $w^{2t}$ and polynomially in the complexity $(m,n)$, recovering a result of Beznosova \\\\cite{Be} for the $(0,0)$-complexity case. Last, we prove that for a pair of weights $u$ and $v$ and a class of locally integrable function $b$ that satisfies certain conditions, the dyadic paraproduct $\\\\pi_b$ is bounded from $L^2(u)$ into $L^2(v)$ if and only if the weights satisfies the joint $A_2$ condition."],"dc:identifier":["http://hdl.handle.net/1928/17491","https://digitalrepository.unm.edu/math_etds/33"],"dc:language":["English"],"dc:subject":["Measure theory","Linear operators","Inequalities (Mathematics)","Integrals","Haar","Lebesgue integral."],"dc:title":["Weighted estimates for dyadic operators with complexity"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Doctoral","Dissertation"],"thesis:degree_name":["Mathematics"]},"updated_at":"2026-07-24T05:27:37Z"}