The weak-type Carleson theorem via wave packet estimates

Abstract

We prove that the weak-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norms, and in fact the sparse <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norms, of the Carleson maximal partial Fourier sum operator are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equivalent-to left-parenthesis p minus 1 right-parenthesis Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mo>≲</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\lesssim (p-1)^{-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p right-arrow 1 Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:msup> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">p\to 1^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norms bound imply new and stronger results at the endpoint <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Q normal upper A Subscript normal infinity Baseline left-parenthesis w right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Q</mml:mi> <mml:mi mathvariant="normal">A</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {QA}_{\infty }(w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which contains the weighted Antonov space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L log upper L log log log upper L left-parenthesis double-struck upper T semicolon w right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>L</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo>;</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L\log L\log \log \log L(\mathbb T; w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, converge almost everywhere whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w element-of upper A 1"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">w\in A_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is an extension of the results of Antonov [Proceedings of the XXWorkshop on Function Theory (Moscow, 1995), 1996, pp. 187–196] and Arias De Reyna, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be Lebesgue measure. The backbone of our treatment is a new, sharply quantified near-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Carleson embedding theorem for the modulation-invariant wave packet transform. The proof of the Carleson embedding relies on a newly developed smooth multi-frequency decomposition which, near the endpoint <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, outperforms the abstract Hilbert space approach of past works, including the seminal one by Nazarov, Oberlin and Thiele [Math. Res. Lett. 17 (2010), pp. 529–545]. As a further example of application, we obtain a quantified version of the family of sparse bounds for the bilinear Hilbert transforms due to Culiuc, Ou and the first author.

Locations

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