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Abstract We study the$$L^p$$regularity of the Bergman projectionPover the symmetrized polydisc in$$\mathbb C^n$$. We give a decomposition of the Bergman projection on the polydisc and obtain an operator equivalent to the Bergman projection over antisymmetric function spaces. Using it, we obtain the$$L^p$$irregularity ofPfor$$p=\frac {2n}{n-1}$$which also implies thatPis$$L^p$$bounded if and only if$$p\in (\frac {2n}{n+1},\frac {2n}{n-1})$$. 

We prove that the weak- $L^p$norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim <\#comment/>(p-<\#comment/>1{)}^{-<\#comment/>1}$as $p\to <\#comment/>1^{+}$. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak- $L^p$type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space ${\mathrm{Q}\mathrm{A}}_{\mathrm{\infty <\#comment/>}}\left(w\right)$, which contains the weighted Antonov space $L\log <\#comment/>L\log <\#comment/>\log <\#comment/>\log <\#comment/>L(\mathbb{T};w)$, converge almost everywhere whenever $w\in <\#comment/>A_1$. 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 $w$must be Lebesgue measure. The backbone of our treatment is a new, sharply quantified near- $L^1$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 $p=1$, 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. 

Paraproducts are a special subclass of the multilinear Calderón-Zygmund operators, and their Lebesgue space estimates in the full multilinear range are characterized by the norm of the symbol. In this note, we characterize the Sobolev space boundedness properties of multilinear paraproducts in terms of a suitable family of Triebel-Lizorkin type norms of the symbol. Coupled with a suitable wavelet representation theorem, this characterization leads to a new family of Sobolev space T(1)-type theorems for multilinear Calderón-Zygmund operators. 

We quantify the Sobolev space norm of the Beltrami resolvent \((I- \mu S)^{-1}\), where \(S\) is the Beurling–Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation \(\mu\) in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in \(W^{1,p}\), \(p \ge 2\). Our proof strategy is then adapted to yield quantitative estimates for the resolvent \((I-\mu S_\Omega)^{-1}\) of the Beltrami equation on a sufficiently regular domain \(\Omega\), with \(\mu\in W^{1,p}(\Omega)\). Here, \(S_\Omega\) is the compression of \(S\) to a domain \(\Omega\). Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón–Zygmund operators to domains, recently obtained by the authors, to extend the Astala–Iwaniec–Saksman technique to higher regularities.