Search Results

If that property is satisfied, then "hypoelliptic analyticity" holds, which means that $\mathcal L f$ analytic implies $f$ analytic. For constant coefficient operators that property is equivalent to e …

In one dimension, if $\mu$ has a density $f$ with respect to the Lebesgue measure, $$ (\tilde L f)(p)=\int_0^{+\infty} e^{-p^\alpha y}f(y^{\frac{1}{\alpha}})\frac{1}{\alpha} y^{\frac{1}{\alpha}-1} dy= …

Your equation falls in the category of regular singular differential equation. Writing your equation as $$ x z'=a z+g(x,z), \tag{$\ast$}$$ the singularity is called regular because the exponent of th …

Let me start by altering a bit your notations: we consider a differential operator $P$ defined by $$ P=\sum_{1\le j\le n}p_j(x) D^j, \quad D=-i\frac{d}{dx}. $$ The formal adjoint is (there is a typo i …

Let us look at the local problem: taking $X$ a non-zero smooth vector field in a neighborhood of 0 in $\mathbb R^3$, you may choose local coordinates such that $X=\partial_z$. If $π_1, π_2$ are smooth …

You may be able to read the Sato-Kawai-Kashiwara lecture notes if your algebraic geometry background is sufficient for this non-trivial task. On the other hand, the book by J. Sjöstrand "Singularité …

There is a classical counterexample due to Plis of an elliptic differential operator with Hölder continuous coefficients without Cauchy uniqueness. This was refined with a counterexample in divergence …

I strongly recommend the following book Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators Lloyd N. Trefethen & Mark Embree The first chapter is partly devoted to Toeplitz m …

There are general results on $C^\infty$ functions of several variables which are invariant under the action of a group, many of them due to Georges Glaeser. One of most beautiful is the following: tak …

With $q=1/p$, let me write your equation as $$ \vert D\vert u+ q u= h_0, \quad u(\pm 1)=0. $$ Multiplying the equation by $u$, we get $$\Vert{u}\Vert_{H^{1/2}_0}^2\le \Vert{u}\Vert_{H^{1/2}_0}^2+\unde …

The Hilbert transform of $\mu$ is the inverse Fourier transform of the product $$ -i\hat\mu(\xi){\pi \text{sign}\xi}, $$ using the definition $\hat u(\xi)=\int e^{-2i\pi x\cdot \xi} u(x) dx$ so that t …

There is a serious difficulty with the notion of products for distributions ; as a matter of fact Laurent Schwartz, one of the creator of Distribution Theory, wrote an article expressing the impossibi …

Since your manifold is compact, $H^s_{loc}$ regularity will be equivalent to $H^s$ regularity. To check the local regularity, you can use cutoff functions and the coordinate charts.

Too long for a comment. Let us consider for $X\in \mathbb S^{n-1}$, $ \langle X,(e^{i \alpha k})_{1\le k\le n}\rangle_{\mathbb C^n}. $ The question at hand is $$ \max_{X\in \mathbb S^{n-1}}\vert\langl …

Formally you get $ u(t)=e^{it p(D)} u_0 $ and this means that $$ u(t,x)=\int e^{i2π x\cdot \xi}e^{it p(\xi)}\widehat{u_0}(\xi) d\xi. $$ Obviously, you have to require something to give a meaning to …