Interpolation between noncommutative martingale Hardy and BMO spaces: the case

Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$ . For $0<p <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ . We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$ , then for $1/r=(1-\theta )/p$ , $$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$ with equivalent quasi norms. For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$ , then for $1/r =(1-\theta )/p +\theta /q$ , $$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$ with equivalent quasi norms. These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces. We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

Hermeneutika Hadis tentang “ Hidupkan Saya Bersama Orang Miskin.” (Analisis Kualitas dan Sharh Hadis)

This article discusses the text of the hadith: "Hadith Turns Me With the Poor People" from the side of the hadiths and understands the hadith using the hermeneutic theory approach in the hadith. This research is research based on the library (library research). The approach used is descriptive qualitative which aims to describe or describe the existing or actual facts of the object under study. The results of this study indicate that the quality of the hadiths from the narrative path is dha'if (weak) based on the comments of hadith critics scholars in the books of al-jarh wa al-ta'dil and the hadith scholars conclusion that hadith dha'īf is not including false it can be part of moralities. The real method of hermeneutics to explain hadith texts contributes to enriching the explanation of the explanation hadiths, or sharh hadiths, so the meaning of the hadith text is that the words of miskin( poverty), al-masakin(needy) mean humility, reverence, and humbleness. And the meaning of hadith has been strengthened by presenting other hadiths as witnesses or i'tibar(regard). Keywords: hermeneutics, hadith, poor. Abstrak Artikel ini membahas teks hadits: “Hadits Hidupkan Saya Bersama Orang Miskin” dari sisi sanad/jalur periwayatan hadistnya dan memahami hadits dengan pendekatan teori hermeneutika dalam hadits. Penelitian ini adalah penelitian kualitatif berdasarkan perpustakaan (library research). Pendekatan yang digunakan adalah deskriptif kualitatif yang bertujuan untuk menggambarkan atau menggambarkan kenyataan yang ada atau kenyataan aktual dari objek yang diteliti. Hasil penelitian ini menunjukkan bahwa kulitas hadist dari jalur periwayatnya dha’if atau lemah berdasarkan komentar ulama kritikus hadits dalam kitab-kitab al-Jarh wa al-Ta’dil dan simpulan ulama hadits bahwa hadits dha’if yang bukan hadits palsu dapat dijadikan bagian dari fadha’il al-a’mal. Sesungguhnya metode hermeneutika untuk menjelaskan teks-teks hdits memberi kontribusi memperkaya upaya penjelasan/ syarh hadits/ teks hadits maka makna teks hadits bahwa ucapan kemiskinan / miskinan , al-masakin maknanya adalah kerendahan hati, ke_khusyu’-an dan tawadhu’. Dan makna hadits telah dikuatkan dengan menghadirkan hadits-hadits lain sebagai syaksi/i’tibar. Kata Kunci: hermeneutika, hadits, miskin

L’écospiritualité est-elle une corrélation?

In this article, we want to examine how the method of correlation of Paul Tillich is implemented in the present-day concept of ecospirituality as worked by Michel-Maxime Egger. According to us, more than an attitude, Egger’s ecospirituality can be understood as a real method, in theology, to “answer" the question of the ecological crisis, by avoiding partial and dualist answers and to come to a spiritual theology of nature.

“Rational, Emotional, Affective Learning” and the Use of Innovative Methods in Foreign Language Teaching

Foreign language didactics is a field which, notwithstanding the many centuries of application and development, is still very methodologically problematic. Statistics clearly show that, in certain countries, the educational system provides foreign language instruction that is not apt and does not deliver effective linguistic competence; a few methods, however, have been developed to contravene this problem. In the specific case of Italy, a country which is renowned for its general monolingualism, Associazione Culturale Linguistica Educational (ACLE) has developed an innovative language teaching method (Rational, Emotional, Affective Learning) to attempt to fill the lacunae of the Italian school system, especially in regards to the teaching of English as a foreign language. This paper seeks to briefly outline the REAL method and its applications, describe its affinity to didactic and cognitive theories, and speculate on its potential effectiveness.

On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces

The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of ℓp spaces. In a previous paper, an interpolation space, defined via the real method and using ℓp spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.