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1. On the Maximal Directional Hilbert Transform in Three Dimensions

   https://doi.org/10.1093/imrn/rny138  

   Di Plinio, Francesco; Parissis, Ioannis (June 2018, International Mathematics Research Notices)

   Abstract We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $$H_\Omega $$ along finite subsets of a finite order lacunary set of directions $$\Omega \subset \mathbb{R}^3$$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of 2D angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting. 

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2. The C∞ -isomorphism property for a class of singularly-weighted x-ray transforms

   https://doi.org/10.1088/1361-6420/aca8cb  

   Mishra, Rohit Kumar; Monard, François; Zou, Yuzhou (December 2022, Inverse Problems)

   Abstract We study a one-parameter family of self-adjoint normal operators for the x-ray transform on the closed Euclidean disk D , obtained by considering specific singularly weighted L 2 topologies. We first recover the well-known singular value decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of C ∞ ( D ) . As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the x-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior. 

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3. On Harmonic Hilbert Spaces on Compact Abelian Groups

   https://doi.org/10.1007/s00041-023-09992-4  

   Das, Suddhasattwa; Giannakis, Dimitrios (February 2023, Journal of Fourier Analysis and Applications)

   Abstract Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted$$L^2$$ $L^2$spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach$$^*$$ ${}^{\ast }$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian groupGto have the same spectrum as the$$C^*$$ $C^{\ast }$-algebra of continuous functions onG. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on$$L^2(G)$$ $L^2\left(G\right)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation. 

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4. Schatten Classes and Commutator in the Two Weight Setting, I. Hilbert Transform

   https://doi.org/10.1007/s11118-023-10072-x  

   Lacey, Michael; Li, Ji; Wick, Brett D. (April 2023, Potential Analysis)

   Abstract We characterize the Hilbert–Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten classS_presult for dyadic paraproducts, where$$1< p < \infty $$ $1<p<\infty$. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform. 

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5. A modulation invariant Carleson embedding theorem outside local L^2

   Di Plinio, Francesco; Ou, Yumeng (April 2016, Journal d’analyse mathématique)

   The article arXiv:1309.0945 by Do and Thiele develops a theory of Carleson embeddings in outer Lp spaces for the wave packet transform of functions in L^p, in the 2≤p≤∞ range referred to as local L^2. In this article, we formulate a suitable extension of this theory to exponents 1<2, answering a question posed in arXiv:1309.0945. The proof of our main embedding theorem involves a refined multi-frequency Calder\'on-Zygmund decomposition. We apply our embedding theorem to recover the full known range of Lp estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation. 

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