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- W2337227203 abstract "We provide elementary proofs that the 2-variation Carleson operator $V_2$ along with explicit bilinear multipliers adapted to ${xi_1 + xi_2 = 0}$ satisfy no $L^p$ estimates. Furthermore, we obtain $L^p rightarrow L^p$ estimates when $2 < p <infty$ for a smooth restricted variant of $V_2$ that is defined a priori on Schwartz functions by the formula begin{eqnarray*} mathcal{V}^{res}_2 : f mapsto sup_{R in mathbb{R}_+} ~~sup_{0 leq alpha < R} ~~left(sum_{j in mathbb{Z}} left|f*mathcal{F}^{-1} left[ tilde{1}_{[alpha + j R, alpha + (j+1)R]}right] right|^2 right)^{1/2} end{eqnarray*} where $tilde{1}_{I} (x) := tilde{1}(|I|^{-1} (x-c_I))$ for all intervals $I = [c_I - |I|/2, c_I + |I|/2] subset mathbb{R}$ and $tilde{1} in C^infty([-1/2, 1/2])$. We then study bi-sublinear variants of $mathcal{V}_2^{res}$ before showing that multipliers, which are adapted to ${xi_1 + xi_2=0}$ and periodically discretized along each frequency scale, map $L^{p_1}(mathbb{R}) times L^{p_2}(mathbb{R}) rightarrow L^{p_1 p_2 / (p_1 + p_2)}(mathbb{R})$ provided $2 leq p_1, p_2 <infty$ and $frac{1}{p_1} + frac{1}{p_2} <1$." @default.
- W2337227203 created "2016-06-24" @default.
- W2337227203 creator A5024538360 @default.
- W2337227203 date "2016-01-18" @default.
- W2337227203 modified "2023-09-27" @default.
- W2337227203 title "Restricted Carleson Variations at Endpoint and Discretized Hilbert Transforms in the Plane" @default.
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- W2337227203 hasPublicationYear "2016" @default.
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