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Results tagged with harmonic-analysis
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user 22815
Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
3
votes
0
answers
105
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Boundedness of Calderon-Zygmund type operator
I am trying to prove the following fact.
Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define
$$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$
where P.V. means …
- 633
2
votes
1
answer
184
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Boardline case of $W^{2,p}$ estimates on elliptic equations
Suppose $u$ is a strong solution of
$$\begin{cases}\Delta u =f &\quad \text{in} \quad B_1(0)\\ u=0 &\quad \text{on}\quad \partial B_1\end{cases}$$
The well known $W^{2,p}$ estimates says if $f\in L^p( …
- 633
4
votes
1
answer
144
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Boundedness of Riesz potential on Hardy space
I encounter the following claim in one paper:
If $(-\Delta)^{\frac14}u\in L^{2,\infty}(\mathbb{R})$, then $u\in BMO(\mathbb{R})$. Equivalently in its dual version, if $u\in \mathcal{H}^1(\mathbb{R})$ …
- 633
7
votes
0
answers
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Definition of homogeneous Sobolev spaces
As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \x …
- 633