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In many aspects of dispersive PDEs, the "optimal" function spaces are those adapted to the symbol of the linear evolution. They were introduced by Bourgain for the nonlinear Schroedinger equation (and …

The first two are equivalent, as the $H^1(\Omega)$ norm and $H^1(\mathbb{R}^d)$ norm coincide for $C^\infty_c(\Omega)$ functions. The third is in general different: If you let $d = 1$ and $\Omega = \m …

The best constant is just the multiplicative inverse of the smallest positive eigenvalue of the Laplacian on the sphere. On $\mathbb{S}^N$ this the smallest eigenvalue is $N$, so $C(N)$ in that case e …

I don't know the literature, but it maybe helpful to understand exactly what this notion of differentiability gains for us. First, unlike the Sobolev notions, this does not handle functions which be …

Since the OP asked for a discussion of features, I provide one by way of an explanation of Christian Remling's counterexample: Holder's inequality states that $$ | \int fg | \leq \| f\|_p \|g\|_q $$ i …

A bit of a pet peeve of mine: the negative Sobolev spaces are spaces of distributions. However, your question (phrased as asking $f = 0$ a.e.) presupposes that elements of $H^{-1}$ can be represented …

Since you tagged reference-request: In the PDE/harmonic analysis literature this is a consequence of the Calderon-Zygmund Inequality, it is one of the main tools for studying elliptic regularity theor …

It is at least true for $k = 2\ell$ and $p \in (1,2]$. The restriction on $k$ is just to make the computation simpler, and shouldn't be too hard to extend to $k$ being odd. The restriction to $p\in (1 …

The key computation is the commutator $$ \Lambda^s (xf) - x \Lambda^s f. $$ You can check this "classically" in the case $s = 4$ to find $$ (1 - \Delta)^2 (xf) = x (1-\Delta)^2 f - 4 (1-\Delta) f'$$ w …

Taking the Fourier transform and using $L^2$ orthogonality you are equivalently trying to estimate $$ \sum_{i + j +k = 0} \hat{u}_i \hat{u}_j \hat{u}_{k} |i+j|^{n+1}|k|^{n} $$ Now from the equality …

If $u \in H^2(\mathbb{R}^N)$, then its Fourier transform satisfies $\hat{u} \in L^2$ and $(1 + |\xi|^2) \hat{u} \in L^2$. By Holder inequality you have $$ \|\hat{u}\|_{q} \leq \| (1 + |\xi|^2)^{-1} …

See Jonsson, A.; Wallin, Hans, A Whitney extension theorem in (L^p) and Besov spaces, Ann. Inst. Fourier 28, No. 1, 139-192 (1978). ZBL0369.46031. Proposition 7.1 in Section 7.3 is exactly what you ar …

As noted by Christian Remling, the Fourier transform of $|x|^{-1}$ is $|\xi|^{-2}$, so the relevant integral you wish to bound is $$\iiint |\eta - \xi|^{-2} \bar{\hat{u}}(\xi) |\eta|^{2\alpha} \hat{u} …

The assumption that $\Omega$ is bounded is in fact required. (So your attempt is the correct proof, once you fix the omission in the statement.) Counterexample: let $\Omega$ be the upper half plane. L …

$$\int \partial_1 (\langle x\rangle^{\delta-1} x_1 u^2) \mathrm{d}x = 0 $$ So $$ \int [ (\delta - 1) \langle {x}\rangle^{\delta - 3} |x_1|^2 + \langle{x}\rangle^{\delta - 1}] u^2 ~\mathrm{d}x = -\int …