A note on Hausdorff-Young inequalities in function spaces

The classical Hausdorff-Young inequalities for the Fourier transform acting between appropriate $L_p$ spaces are cornerstones of Fourier analysis. Here we extend it to weighted spaces of Besov or Sobolev type where the weight has the form $w(x)=(1+|x|^2)^{\alpha/2}$. This note is not a paper or draft but a sketchy complement to some earlier results where we dealt with mapping properties of the Fourier transform

Hardy inequalities in function spaces

summary:Let $\Omega$ be a bounded $C^\infty$ domain in $\Bbb{R}^n$. The paper deals with inequalities of Hardy type related to the function spaces $B^s_{pq}(\Omega)$ and $F^s_{pq}(\Omega)$

A New Approach to Function Spaces on Quasi-Metric Spaces

Sin resumenA d-space X = (X, , µ) is a compact set X with respect to a quasi-metric and a Borel measure µ such that the measure of a ball of radius r is equivalent to rd , where d > 0. The paper deals with spaces Bp (X ; H ) of Besov type where s 1 < p < and s R. Here H is a bi-Lipschitzian map of the snowflaked version (X, , µ) of X for some 0 < < 1, onto a fractal d/-set = H X in s/ some Rn , reducing the spaces Bp (X ; H ) to the better known spaces Bp ().

Traces of some weighted function spaces and related non‐standard real interpolation of Besov spaces

We study traces of weighted Triebel–Lizorkin spaces F p , q s ( R n , w ) $F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes R n − k $\mathbb {R}^{n-k}$ , where the weight is of Muckenhoupt type. We concentrate on the example weight w α ( x ) = | x n | α $w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when | x n | ≤ 1 $\big\vert x_n\big\vert \le 1$ , x ∈ R n $x\in \mathbb {R}^n$ , and w α ( x ) = 1 $w_\alpha (x)=1$ otherwise, where α > − 1 $\alpha >-1$ . Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space ( B p 1 , p 1 s 1 ( R n − k ) , B p 2 , p 2 s 2 ( R n − k ) ) θ , r $\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big ), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big )\big )}_{\theta ,r}$ , 0 0$sufficiently large, 0 < θ < 1$0<\theta <1$, 0 < r ≤ ∞$

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756

The Weiss conjecture and weak norms

In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators Re(\lambda)^\einhalb C(\lambda+A)^{-1}, \qquad Re(\lambda)>0 on the complex right half plane and weak Lebesgue $L^{2,\infty}$--admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the 'Weiss conjecture' within the scale of Lorentz spaces $L^{p,q}$

Paclitaxel plus Eftilagimod Alpha, a Soluble LAG-3 Protein, in Metastatic, HR⁺ Breast Cancer:Results from AIPAC, a Randomized, Placebo Controlled Phase IIb Trial

Purpose: Eftilagimod alpha (efti), a soluble lymphocyte activation gene (LAG-3) protein and MHC class II agonist, enhances innate and adaptive immunity. Active Immunotherapy PAClitaxel (AIPAC) evaluated safety and efficacy of efti plus paclitaxel in patients with predominantly endocrine-resistant, hormone receptor–positive, HER2-negative metastatic breast cancer (ET-resistant HR+ HER2– MBC). Patients and Methods: Women with HR+ HER2– MBC were randomized 1:1 to weekly intravenous paclitaxel (80 mg/m2) and subcutaneous efti (30 mg) or placebo every 2 weeks for six 4-week cycles, then monthly subcutaneous efti (30 mg) or placebo maintenance. Primary endpoint was progression-free survival (PFS) by blinded independent central review. Secondary endpoints included overall survival (OS), safety/tolerability, pharmacokinetics/pharmacodynamics, and quality of life. Exploratory endpoints included cellular biomarkers. Results: 114 patients received efti and 112 patients received placebo. Median age was 60 years (91.6% visceral disease, 84.1% ET-resistant, 44.2% with previous CDK4/6 inhibitor treatment). Median PFS at 7.3 months was similar for efti and placebo. Median OS was not significantly improved for efti (20.4 vs. 17.5 months; HR, 0.88; P = 0.197) but became significant for predefined exploratory subgroups. EORTC QLQC30-B23 global health status was sustained for efti but deteriorated for placebo. Efti increased absolute lymphocyte, monocyte and secondary target cell (CD4, CD8) counts, plasma IFNg and CXCL10 levels. Conclusions: Although the primary endpoint, PFS, was not met, AIPAC confirmed expected pharmacodynamic effects and demonstrated excellent safety profile for efti. OS was not significantly improved globally (2.9-month difference), but was significantly improved in exploratory biomarker subgroups, warranting further studies to clarify efti’s role in patients with ET-resistant HER2– MBC.</p