Weighted Inequalities for $t$-Haar multipliers
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arXiv
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| Format: | Preprint |
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2023
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| author | Chung, Daewon Huang, Weiyan Moraes, Jean Carlo Pereyra, María Cristina Wick, Brett D. |
| author_facet | Chung, Daewon Huang, Weiyan Moraes, Jean Carlo Pereyra, María Cristina Wick, Brett D. |
| contents | In this paper, we provide necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multipliers $T^t_{w,σ}$, $t\in \R$, %defined in \cite{P} when $σ=1$, are uniformly (on the choice of signs $σ$) bounded from $L^2(u)$ into $L^2(v)$. These dyadic operators have symbols $s(x,I)=σ_I\,(w(x)/\langle w\rangle_I)^t$ which are functions of the space variable $x\in\R$ and the frequency variable $I\in \mathcal{D}$, making them dyadic analogues of pseudo-differential operators. Here $\mathcal{D}$ denotes the dyadic intervals, $σ_I=\pm1$, and $\langle w\rangle_I$ denotes the integral average of $w$ on $I$. When $w\equiv 1$ we have the martingale transform and our conditions recover the known two-weight necessary and sufficient conditions of Nazarov, Treil and Volberg. %We will discuss some relations between the three weights inequality for these operators given the inequality for other dyadic operators. We also show how these conditions are simplified when $u=v$. In particular, the martingale one-weight and the $t$-Haar multiplier unsigned and unweighted (corresponding to $σ_I\equiv 1$ and $u=v\equiv 1$) known results are recovered or improved. We also obtain necessary and sufficient testing conditions of Sawyer type for the two-weight boundedness of a single variable Haar multiplier similar to those known for the martingale transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14556 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weighted Inequalities for $t$-Haar multipliers Chung, Daewon Huang, Weiyan Moraes, Jean Carlo Pereyra, María Cristina Wick, Brett D. Classical Analysis and ODEs In this paper, we provide necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multipliers $T^t_{w,σ}$, $t\in \R$, %defined in \cite{P} when $σ=1$, are uniformly (on the choice of signs $σ$) bounded from $L^2(u)$ into $L^2(v)$. These dyadic operators have symbols $s(x,I)=σ_I\,(w(x)/\langle w\rangle_I)^t$ which are functions of the space variable $x\in\R$ and the frequency variable $I\in \mathcal{D}$, making them dyadic analogues of pseudo-differential operators. Here $\mathcal{D}$ denotes the dyadic intervals, $σ_I=\pm1$, and $\langle w\rangle_I$ denotes the integral average of $w$ on $I$. When $w\equiv 1$ we have the martingale transform and our conditions recover the known two-weight necessary and sufficient conditions of Nazarov, Treil and Volberg. %We will discuss some relations between the three weights inequality for these operators given the inequality for other dyadic operators. We also show how these conditions are simplified when $u=v$. In particular, the martingale one-weight and the $t$-Haar multiplier unsigned and unweighted (corresponding to $σ_I\equiv 1$ and $u=v\equiv 1$) known results are recovered or improved. We also obtain necessary and sufficient testing conditions of Sawyer type for the two-weight boundedness of a single variable Haar multiplier similar to those known for the martingale transform. |
| title | Weighted Inequalities for $t$-Haar multipliers |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2303.14556 |