Activity for LL 3.14
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #291014 | Initial revision | — | 8 months ago |
| Answer | — |
A: The Fourier transform of $1/p^3$ In short, in $\mathbb{R}^d$, if I define the Fourier transform as $\mathcal{F}(f)(x) = \int{\mathbb{R}^d} f(y) \,e^{-2iπxy}\,\mathrm{d}y$ the result is $$ \boxed{\mathcal{F}\left(\frac{1}{\omegad\,|x|^d}\right) = \frac{\psi(d/2)-\gamma}{2} - \ln(|πx|)} $$ where $\omegad = \frac{2\,\pi^{d/2}}{\Gam... (more) |
— | 8 months ago |
| Edit | Post #291012 |
Post edited: |
— | 8 months ago |
| Edit | Post #291012 | Initial revision | — | 8 months 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 $p≥ 2$ $$ \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) |
— | 8 months ago |