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4

For (2), the answers to Does there exist a continuous function of compact support with Fourier transform outside L^1? may be helpful. For (3), the answer is yes, although one always needs to rely on some theoretical background. My preferred argument is to note that every bounded linear map $C_0({\bf R}) \to L^1(X,\mu)$ ($X$ and $\mu$ arbitrary) is weakly ...

3

In the literature you can sometimes see them called Fourier-Lebesgue spaces, with notation $\mathcal{F} L^p(\mathbb{R}^n)$, consisting of the set of all tempered distributions whose norm (as you wrote) is finite. See, e.g., http://arxiv.org/abs/0804.1730 and http://arxiv.org/abs/0801.1444

1

By the compactness of the torus, there is a uniform radius of convergence $r>0$ working for every point. You can extend $f$ to complex variables and use Cauchy's formula to find $|\partial^k f|\le Ck!(r/2)^{-k}$, with the constant $C$ only depending on $\sup_{|\Im z|\le r} |f|$. Then you get the same sort of bound for $|\xi|^k \hat f(\xi)$. Sum over $k$ ...

2

Suppose that $u_0=0$, otherwise $L^p$ decay estimates are well-known. Let $\{T(t)\}_{t\geq0}$ denote the heat semigroup, i.e., $T(t)$ for $t>0$ is convolution with $(4\pi t)^{-d/2}\,e^{-|x|^2/(4t)}$. Notice that, for $p\geq q$, $$\|T(t)\|_{L^q\to L^p} = \left\|\frac1{(4\pi t)^{d/2}}\,e^{-\,\frac{|x|^2}{4t}}\right\|_{L^r} = ... 1 O.V Besov, V.P Il'in, S.M Nikolskii. Integral Representations of Functions and Embedding Theorems. Most of results are stated for arbitrary domains G\subset \mathbb R^n. 2 Hormander: The Analysis of Linear Partial Differential Operators II, 1983, page 13 ff. These spaces are B_{k,p}(\mathbb R^n)\cap \mathcal E'(X), where X is open in \mathbb R^n. 3 One has, for f,\,g\in \dot{H}^{-1/4}(\mathbb R),$$ \begin{aligned} \left|\int_{-\infty}^\infty\int_{-\infty}^\infty |x-y|^{-1/2}f(x) \,\overline{g(y)}\,dx dy\,\right| &= C_0 \left|\left((-\Delta)^{-1/4}f,g\right) \right| \\ & \leq C_0\|(-\Delta)^{-1/4}f\|_{\dot{H}^{1/4}}\|g\|_{\dot{H}^{-1/4}} = C_0\|f\|_{\dot{H}^{-1/4}} ...

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