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13

The answer is yes. Fix $x,y$, and write $e(\alpha) := e^{2\pi i \alpha}$. Using a Littlewood-Paley partition of unity and the triangle inequality, we may bound $$ |f(x,y)| \leq \sum_N a_N$$ where $N$ ranges over powers of two, $$ a_N := \left|\sum_{n \in {\bf Z} \backslash 0} \psi( \frac{n}{N}) \frac{1}{n} e(x n + yn^2)\right|, $$ and $\psi$ is a suitable ...


5

Prof. Tao's answer is excellent. I also found two research papers answering the question so I list them below as complementary reference: G.I.Arkhipov and K.I.Oskolkov, On a special trigonometric series and its applications, 1989 Math. USSR Sb. 62 145 Link to the article: http://iopscience.iop.org/0025-5734/62/1/A10 See Theorem 1. E.S.Stein and S.Wainger, ...


5

The answer is "yes". We have $$f(x)=\int_{-\omega}^\omega g(t)e^{2\pi itx}dt,$$ so $$|f'(x)|^2\leq\left(2\pi\int_{-\omega}^\omega |t||g(t)|dt\right)^2\leq\frac{8\pi^2\omega^3}{3}\| g\|_2,$$ by Cauchy-Bounyakovski-Schwarz inequality. It remains to notice that $\| g\|_2=\| f\|_2$ according to Parseval . Equality when $g(t)=t$.


3

I think the legend is that an official introduction to $BMO$ was given in the paper "Of Functions of Bounded Mean Oscillation" by F. John and L. Nirenberg , but the initial mention was in the paper by F. John "Rotation and Strain". F. John was looking at mappings and rotations. For example, the class of mappings $f(x)$ satisfying the following: ...


3

The place to look is Section 17 of Thanigasalam's earlier paper, http://matwbn.icm.edu.pl/ksiazki/aa/aa46/aa4611.pdf. In particular, it is not the case that $G(10)\leq 105$ implies directly $H(10)\leq 107$, but rather that the same argument can be used for both. That is, if one can control the minor arcs using $s_2$ summands additively Hua's lemma style, ...


3

I think all your questions are answered by the following calculation (assume $m\geq 1$ and $\Re(s)>1$): $$ \sum_{c=1}^\infty\frac{r_m(c)}{c^{2s}} = \sum_{c=1}^\infty\frac{1}{c^{2s}}\sum_{\substack{\text{$d$ mod $c$}\\{(d,c)=1}}}e\left(m\frac{d}{c}\right) = \sum_{c=1}^\infty\frac{1}{c^{2s}}\sum_{\text{$d$ mod ...


2

Here is an example where $T(f,g)\notin l^2$ while $m\in L^2$ and $f=g\in l^2$: Let $a(n)=|n|^{-1+s}$ with $\frac38<s<\frac12$ (so that $a\in l^2$), $m(\xi,\eta)=\hat{a}(\xi)\hat{a}(\eta)$. Take also $f=g=a$. One has $\hat{a}(\xi)\propto |\xi|^{-s}$ near $0$. Then $\hat{a}(\xi)\propto |\xi|^{-2s}$, so that $(a * a)(n)\propto n^{-1+2s}$ for large $n$. ...


2

assuming $c>0$, this limit is dominated by the upper end of the integration interval, so $f(\omega)=a/2$ independent of $\omega$; as a check, try $\gamma=3$, when a closed form expression exists, $$f(\omega) = \lim_\limits{R \to \infty} \frac{a}{R^2}\left( 1 - (1+bR)e^{-cR}\right) \int_{r=0}^{R}{re^{i \omega r^{-3}}} \mathrm{d}r$$ $$=\lim_\limits{R \to ...


2

You ask for a Fourier reconstruction mechanism that avoids the Gibbs phenomenon. One strategy in this direction was developed by David Gottlieb and collaborators, in a series of papers entitled "On the Gibbs phenomenon: Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function" (part I, part II, part III, part IV, part ...


1

The real Hardy spaces are equivalent to $L^p(\mathbb{R}^n)$ space when $p>1$, and they are much easier to to use than $L^p(\mathbb{R}^n)$ when $p\leq 1$. Since $H^p(\mathbb{R}^n)$ has a maximal function and singular integral generalisation, $H^p(\mathbb{R}^n)$ gives us an extension of maximal function/singular integral results to $p\leq 1$, when they were ...


1

Let $\frac12<s<1$, let $u\in H^s(\mathbb R)$, then also $u\in C_0$, and let $f$ be Lipschitz on $[-||u||_\infty \ ,+||u||_\infty]$ with $|f(x)-f(y)|\leq M|x-y|$. Then $$\int\int \frac{|f(u(t))-f(u(t'))|^2}{|t-t'|^{1+2s}}dt\ dt'\leq M^2\int\int\frac{|u(t)-u(t')|^2}{|t-t'|^{1+2s}}$$With $f(0)=0$ you also have $|f(u)|\leq M|u|$ so that $f(u)\in L^2$. ...


1

Let $\mathbb{H}$ denote the field of quaternions and $\mathbb{S}$ denote the unit sphere of imaginary quaternions. We have a Riemann mapping theorem for axially symmetric sets inside the field of quaternions $\mathbb{H}$. The set $\Omega$ is axially symmetric if the set $\{ x+Jy~|~J\in\mathbb{S} \},~J\in\mathbb{S}$ are contained in $\Omega$ (we have rotation ...


1

The following papers may help: "Besov Spaces on Domains in $\mathbb{R}^d$" by R. Devore and R. Sharpley This paper is for Besov spaces $B^{\alpha}_{q}(L_p(\Omega)),~p,q,\alpha \in (0,\infty )$ on domains $\Omega\subset \mathbb{R}^d$ http://www.ams.org/journals/tran/1993-335-02/S0002-9947-1993-1152321-6/S0002-9947-1993-1152321-6.pdf "Elliptic and ...



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