Activity for LL 3.14‭

Type	On...	Excerpt	Status	Date
Edit	Post #291014	Initial revision	—	about 1 year ago
Answer	—	A: The Fourier transform of $1/p^3$ In short, in ${\mathbb{R}}^d$, if I define the Fourier transform as $\mathcal{F}(f)(x)=\int {\mathbb{R}}^{d}f(y)e^{-2i\pi xy}\mathrm{d}y$ the result is $$\boxed{{\displaystyle \mathcal{F}\left(\frac{1}{\text{omegad}|x{|}^d}\right)=\frac{\psi (d/2)-\gamma }{2}-\ln (|\pi x|)}}$$ where $\omegad = \frac{2\,\pi^{d/2}}{\Gam... (more)	—	about 1 year ago
Edit	Post #291012	Post edited:	—	about 1 year ago
Edit	Post #291012	Initial revision	—	about 1 year ago
Question	—	Besov or Triebel-Lizorkin spaces versus Lorentz spaces At the $0$ order of derivatives of Sobolev spaces, we find Besov spaces ${\dot{B}}^{0}p,q$, Triebel Lizorkin spaces ${\dot{F}}^{0}p,q$ and Lorentz spaces $L^{p,q}$, with in particular if $p\ge 2$ $$ \dot{B}^0{p,1} ⊂ \dot{B}^0{p,2} ⊂ L^p = \dot{F}^0{p,2} \subset \dot{F}^0{p,p} =\dot{B}^0{p,p} ⊂ \dot{B}^0{... (more)	—	about 1 year ago