Discrete harmonic analysis associated with ultraspherical expansions

We study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted l^p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by certain difference operator. We also prove weighted l^p-boundedness properties of transplantation operators associated to the system of ultraspherical functions. In order to show our results we previously establish a vector-valued local Calder\'on-Zygmund theorem in our discrete setting

Conical square functions associated with Bessel, Laguerre and Schr\"odinger operators in UMD Banach spaces

In this paper we consider conical square functions in the Bessel, Laguerre and Schr\"odinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hyt\"onen, van Neerven and Portal, in order to define our conical square functions, we use $\gamma$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions

Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

We consider the Weinstein type equation $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times (0,\infty )$, where $\mathcal{L}_\lambda=\partial _t^2+\partial _x^2-\frac{\lambda (\lambda -1)}{x^2}$, with $\lambda >1$. In this paper we characterize the solutions of $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times(0,\infty )$ representable by Bessel-Poisson integrals of BMO-functions as those ones satisfying certain Carleson properties

BMO functions and Balayage of Carleson measures in the Bessel setting

By $BMO_o(R)$ we denote the space consisting of all those odd and bounded mean oscillation functions on R. In this paper we characterize the functions in $BMO_o(R)$ with bounded support as those ones that can be written as a sum of a bounded function on $(0,\infty )$ plus the balayage of a Carleson measure on $(0,\infty )\times (0,\infty )$ with respect to the Poisson semigroup associated with the Bessel operator $B_\lambda =-x^{-\lambda }Dx^{2\lambda }Dx^{-\lambda}$, $\lambda >0$. This result can be seen as an extension to Bessel setting of a classical result due to Carleson

Square functions in the Hermite setting for functions with values in UMD spaces

In this paper we characterize the Lebesgue Bochner spaces $L^p(\mathbb{R}^n,B)$, $1<p<\infty$, by using Littlewood-Paley $g$-functions in the Hermite setting, provided that $B$ is a UMD Banach space. We use $\gamma$-radonifying operators $\gamma (H,B)$ where $H=L^2((0,\infty),\frac{dt}{t})$. We also characterize the UMD Banach spaces in terms of $L^p(\mathbb{R}^n,B)$-$L^p(\mathbb{R}^n,\gamma (H,B))$ boundedness of Hermite Littlewood-Paley $g$-functions

γ\gamma-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space \B. If we denote by $H$ the Hilbert space L^2((0,\infty),dt/t),\gamma(H,\B) represents the space of $\gamma$-radonifying operators from $H$ into \B. We prove that the Hermite square function defines bounded operators from BMO_\mathcal{L}(\R,\B) (respectively, H^1_\mathcal{L}(\R, \B)) into BMO_\mathcal{L}(\R,\gamma(H,\B)) (respectively, H^1_\mathcal{L}(\R, \gamma(H,\B))), where $BMO_\mathcal{L}$ and $H^1_\mathcal{L}$ denote $BMO$ and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO_\mathcal{L}(\R, \B) and H^1_\mathcal{L}(\R,\B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.Comment: 31 page

LpL^p-boundedness properties for the maximal operators for semigroups associated with Bessel and Laguerre operators

In this paper we prove that the generalized (in the sense of Caffarelli and Calder\'on) maximal operators associated with heat semigroups for Bessel and Laguerre operators are weak type (1,1). Our results include other known ones and our proofs are simpler than the ones for the known special cases.Comment: 8 page

Immunophenotypic Comparison of Testicular Sclerosing Sertoli Cell Tumors and Sertoli Cell Tumors Not Otherwise Specified

Testicular Sertoli cell tumors (SCTs) are rare, and most fall into the category of SCT–not otherwise specified (SCT-NOS). Only a few additional types of SCT are recognized. Sclerosing SCT (S-SCT), originally described in 1991, comprises a small fraction of SCTs and was considered a specific entity until the 2016 revision of the World Health Organization classification of non–germ cell tumors, where it was classified as a morphologic variant of SCT-NOS. In a recent study, differences in expression of PAX2/PAX8, inhibin, androgen receptor, and S100 protein between SCT-NOS and S-SCT were noted in a small number of cases. In this interinstitutional study, we compared the expression of these markers and β-catenin in 11 cases each of SCT-NOS and S-SCT to determine if differences exist that could justify keeping a separate classification of these neoplasms. PAX2/PAX8 cocktail was the only marker that was significantly overexpressed in S-SCT. Expression of androgen receptors was strong in S-SCT and variable in SCT-NOS but did not reach statistical significance. Expression of β-catenin was common in both, whereas inhibin was infrequent. The available material was insufficient for a conclusive evaluation of S100 protein expression. Overall, our results support the inclusion of S-SCT as a morphologic variant of SCT-NOS. Expression of PAX2/PAX8 in S-SCT may reflect an overactive epithelial-to-mesenchymal transition as has been shown in experimental models of acute and chronic seminiferous tubular injury and might be related to the process generating the stroma in these tumors