User user17240 - MathOverflow

Smoothness of the convolution of a singular measure with itself

2012-06-27T03:25:57Z

Let $\gamma:I\subset\mathbb{R}\rightarrow\mathbb{R}^2, s\mapsto \gamma(s)$, denote the arclength parametrization of a smooth, convex curve $\Gamma:=\gamma(I)\subset\mathbb{R}^2$. Equip $\Gamma$ with arclength measure $\sigma$.

Here is my question: what extra conditions on $\Gamma$ will ensure that the triple convolution $\sigma\ast\sigma\ast\sigma$ is a smooth function of $x\in\mathbb{R}^2$ inside its support (i.e. the set $\Gamma+\Gamma+\Gamma\subset\mathbb{R}^2$), up to the boundary? I am also interested in partial regularity results, and counterexamples.

Here is a possible approach: for any $x\in\mathbb{R}^2$ and $\epsilon>0$, $$\int_{|y-x|<\epsilon} \sigma\ast\sigma\ast\sigma(y)dy=|\{(s,s',s'')\in I^3: |\gamma(s)+\gamma(s')+\gamma(s'')-x|<\epsilon\}|.$$

Understanding how these sets look like will probably help. I am familiar with results in this spirit for double convolutions (e.g. Fefferman 1970), but not for triple ones.

References would be much appreciated.

Thank you.

$L^1$ norm of the Fourier transform of a truncated Gaussian

2011-10-13T21:28:23Z

I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes:

Consider the Gaussian $G(x):=e^{-x^2}$ on the real line, and localize it to the region $|x|\sim 2^k$ by multiplying it by an appropriate smooth cut-off. More precisely, take $\phi\in C_0^\infty(\mathbb{R})$ supported in the region $${x\in\mathbb{R}: \frac{1}{2}<|x|\leq2}$$ such that $0\leq\phi\leq 1,$ and let $\phi_k(x):=\phi(2^{-k}x)$. Consider: $$G_k(x):=\phi_k(x)G(x).$$ It is straightforward to check that $\|G_k\|_{L^1}\lesssim 2^ke^{-4^k}$, which tends (very quickly) to $0$ as $k\rightarrow\infty$. Also, using Young's convolution inequality one can easily show that $\|\widehat{G_k}\|_{L^1}\leq \|\phi\|_1\|G\|_1$, but this gives no decay in terms of $k$.

My question is: does $\|\widehat{G_k}\|_{L^1}$ decay as $k\rightarrow\infty$? If so, how fast? Can you prove sharp bounds (in $k$)?

Thank you.

Geometric meaning of derivatives of the curvature

2011-05-27T01:41:21Z

Let $\Gamma$ be a sufficiently smooth, closed, convex curve in the plane parametrized by arclength $s$. Further assume that $\kappa>0$ and that the second derivative of $\kappa$ with respect to $s$ (denoted $\kappa''$) is such that \begin{equation} (\kappa''\cdot \kappa^{-3})(p)< C \end{equation} for every point $p\in\Gamma$ at which $\kappa$ attains a global minimum (for some absolute constant $C<\infty$).

What can be said about the curve $\Gamma$? In particular, is the quantity $\kappa''\cdot \kappa^{-3}$ of geometric significance and has it been considered before? Can it for instance be expressed in terms of the (centro-)affine curvature of $\Gamma$? References would be most appreciated!

Motivation comes from trying to reduce several local problems involving the restriction of the Fourier transform to $\Gamma$ to the corresponding (more tractable) problem for an appropriate osculating conic.

Thank you.

Phase perturbations in oscillatory integrals

2011-05-11T21:02:07Z

I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in Stein's terminology).

I have the following concrete problem, which originated from studying the restriction problem for the Fourier transform on the 2-sphere. Let $f\in C_0^\infty (\mathbb{R}^2)$ be supported in $B(0,R_1)$, and consider the phase $\phi_\epsilon (y)=\frac{|y|^2}{2}+\epsilon\frac{1}{8}|y|^4.$ Look at the integrals

$$I_\epsilon(x,t)=\int_{\mathbb{R}^2}e^{-ix\cdot y}e^{-it \phi_\epsilon(y)}f(y)dy$$ and

$$J_\epsilon(x,t)=\int_{\mathbb{R}^2}e^{-ix\cdot y}e^{-it (\phi_\epsilon(y)+\epsilon^2 |y|^6)}f(y)dy.$$

Let $R_2>0$ be sufficiently large. Is it true that, for $|(x,t)|\geq R_2$,
$$|I_\epsilon(x,t)|^4-|J_\epsilon(x,t)|^4=C\epsilon^2 |(x,t)|^{-4} \;\;\textrm{as }\;\; \epsilon\rightarrow 0^+,$$ where the constant $C$ is allowed to depend on $R_1$ and $R_2$ (and possibly on some appropriate norm of $f$ and its derivatives), but on nothing else?

As a consequence one would have that $\|J_\epsilon\|_{L^4_{x,t}(\mathbb{R}^3)}^4=\|I_\epsilon\|_{L^4_{x,t}(\mathbb{R}^3)}^4+O(\epsilon^2)$ (which is what I am really interested in). Thank you!

A tricky integral

2010-11-24T04:55:07Z

Let $\alpha>0$ and $\beta\in\mathbb{R}$. I am looking for an explicit formula for the integral

$$\int_{-\infty}^{\infty} (1+x^2)^{-1/2}e^{-\alpha x^2}e^{-i \beta x}dx.$$

I tried several changes of variables, and contour integration doesn't seem to work.

Motivation comes from the following closely related kernel $$K(s,t)=e^{-\frac{(s-t)^2}{4}}\int_{-\infty}^{\infty}(1+x^2)^{-1/2}e^{-\frac{(s-t)^2}{4} x^2}e^{-i (s^2-t^2) x}dx,$$ which provides an example of a compact integral operator on $L^2(\mathbb{R})$ that is not Hilbert-Schmidt. I would like to check the details.

Thank you!

Range of the Fourier transform on L^1

2009-12-07T07:22:49Z

It is well known that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^d)$ into, but not onto, $\overline{C_0^0}(\mathbb{R}^d)$, where the closure is taken in the $L^\infty$ norm. This is a consequence of the open mapping theorem, for instance.

My question is: what's an explicit example of a function in $\overline{C_0^0}(\mathbb{R}^d)$ which is not in the image of $L^1(\mathbb{R}^d)$ under the Fourier transform?

I would also like to know whether there is a useful characterization of $\mathcal{F}(L^1(\mathbb{R}^d))$.

Remark: it is easy to see that the Banach space $\overline{C_0^0}(\mathbb{R}^d)$ consists of all continuous functions $f$ on $\mathbb{R}^d$ such that $f(\xi)\rightarrow 0$ as $|\xi|\rightarrow\infty$.

Thank you!

Decomposition of Hölder continuous functions

2009-12-03T07:23:12Z

Let $\alpha\in(0,1)$ and $\eta\in\Lambda_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is possible to decompose $$\eta=f+g$$ in such a way that $f\in C^n(\mathbb{R})$ and $||f||_{C^n}=O(R^C)$ , and $g\in L^\infty(\mathbb{R})$ with $\|g\|_{L^\infty}=O(R^{-1})$ .

Here $C$ is a universal constant. On the other hand, the real parameter $R$ can be chosen as large as we want (at the expense of increasing $\|f\|_{C^n}$).

Thank you!

Answer by user17240 for What is the first interesting theorem in (insert subject here)?

2009-12-04T07:06:45Z

Harmonic Analysis: Plancherel's theorem

Comment by user17240

2011-10-27T00:06:53Z

Well, "decay to 0" is equivalent to some kind of bound. My second sentence can be rephrased as "how will this imply that the $L^1$ norm of $\widehat{G_k}$ tends to $0$ as $k\rightarrow\infty$?". $L^1$ and $L^\infty$ norms are in general not comparable...

Comment by user17240

2011-10-26T01:04:51Z

I agree that the $L^\infty$ norm of $\widehat{G_k}$ is small (it is bounded by $\|G_k\|_{L^1}$). But how will this imply bounds on the $L^1$ norm of $\widehat{G_k}$? I don't think dominated convergence will do it...

Comment by user17240

2011-10-13T22:27:45Z

@Yemon: Yes, I should have mentioned that. Thanks!

Comment by user17240

2010-11-24T18:35:09Z

@Willie: edited, thanks!

Comment by user17240

2009-12-04T08:31:29Z

By the way, I meant to get C independent of f but not necessarily n. I should have been more clear about that.

Comment by user17240

2009-12-04T08:30:37Z

@Julián: Thank you for your answer. I understand the bound on the remainder since $\eta\ast\phi_R\rightarrow\eta$ uniformly as $R\rightarrow\infty$, but how do you get the bound on '$\|f\|_{C^n}$'?