Near-Endpoint Carleson Embedding of $$\mathcal {Q}_s$$ and F(p, q, s) into Tent Spaces, and its Applications to Compositions of Analytic Paraproducts

Abstract This paper aims to study the $$\mathcal {Q}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> and F ( p , q , s ) Carleson embedding problems near endpoints. We first show that $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> is an s -Carleson measure if and only if $$id: \mathcal {Q}_t \mapsto \mathcal {T}_{s, 2}^2(\mu )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>d</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>↦</mml:mo> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is bounded for any $$0&lt;t&lt;s \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>s</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Using the same idea, we also prove a near-endpoint Carleson embedding for $$F(p, p\alpha -2, s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mi>α</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for $$\alpha &gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Our method is different from the previously known approach, which involves a delicate study of Carleson measures (or logarithmic Carleson measures) on weighted Dirichlet spaces. As some byproducts, the corresponding compactness results are established. Moreover, we completely characterize the boundedness and compactness of a class of g -operators generated by the analytic paraproducts acting on various F ( p , q , s ) spaces. Finally, we compare the near-endpoint Carleson embedding with the existing solutions of Carleson embedding problems proposed by Xiao, Pau, Zhao, Zhu, etc. Our results assert that a “tiny-perturbed" version of a conjecture on the $$\mathcal {Q}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> Carleson embedding problem due to Liu, Lou, and Zhu is true. We also answer an open question by Pau and Zhao on the F ( p , q , s ) Carleson embedding near endpoints.