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$\newcommand{\D}{\mathbb{D}}$ Let $H^\infty(\D)$ be the space of bounded analytic functions in the unit disc $\D$. For a function $f(z) = \sum_{n=0}^\infty a_nz^n$ we can define its truncation as $$T_ …

The answer to your question is yes, and it is a pretty well-understood topic. First of all $X$ is more or less irrelevant for the bounds in 4) so let us take $X = Y$, say, for convenience. Second, w …

I think the answer is no, there is no such infinite-dimensional subspace. Say, we have such a subspace $X$. I will assume that $X$ lies in the Sobolev space $W^{1,2}([0, 1])$, because otherwise it is …

$\newcommand{\a}{\alpha}$ $\newcommand{\b}{\beta}$ $\newcommand{\g}{\gamma}$ We want to prove that $ (f(ax+by)g(ax+by))^\g \ge a(f(x)g(x))^\g + b (f(y)g(y))^\g$ for every $x, y$ and $a+b = 1$. Since …

It seems to me that the answer is no for all $1 < p < 2$: consider $x_n = \frac{e^{2\pi i nx}}{n^{r}}, n = 1, 2, \ldots$ for $r = \frac{1}{p}$. It is easy to see that the sequence $\{ |x_n|\}$ is not …

No, linear combinations of functions of the form $|x-y|$ are dense in $C([0, 1])$ (because you can approximate any continuous function by the piecewise-linear), so (if $f\in L^1(0,1)$, otherwise the i …

Let me try to give an intuitive explanation for why it is conceivable that inequality like this can hold for Lipschitz functions (with bounded Lipschitz norm) and yet fail for $f$ with $f'\in L^2$. Th …

I will show that it is possible. Moreover, the functions $f_k^m$ I will construct will be independent of the numbers $c_l$ (I'm not sure if this is needed for you, but clearly this extra condition can …

(Un?)fortunately, there are no such functions. The idea is simple, yet powerful: if the double integral $\int\int d\xi d\eta$ is finite, then for almost all $\eta$ the $d\xi$ integral is finite as wel …

I might be misunderstanding the definitions, but isn't it kinda trivial to do with the Fourier transform? On the Fourier side $Q_t$ is a multiplication by $e^{-t |x|^2}$, and $L^2(A)$ for any $A\subse …

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\phi}{\varphi}$ I will answer it for all $0 < q < \infty$, $1\le n\le p$ and all non-negative measures $\mu$. The desired inequality holds if and only if $\m …

The answer is yes if your domain has compact embedding $H^1(\Omega)\to L^2(\Omega)$ (that is, for any regular enough domain), or you consider $H^1_0(\Omega)$, but no in general. Counterexample is as f …

The answer is yes. I will prove it using the assumption only for $\xi > 0$ (almost the same proof works if we only assume it for $\xi < 0$). For $z\in U = \{z: Im (z) \ge 0, |z| \ge 1\}$, consider th …

Umm, $A$ is path-connected and it is fairly simple -- just change the coordinates one by one. Say we want to go from $u\in A$ to $v\in A$. On $[0, \frac{1}{2}]$ change linearly $u_1$ to $v_1$, then on …

If $\sigma_1(B) = 0$ then $B = 0$. Now pick your favorite $2\times 2$ matrix $A$ with $\sigma_1(A) = 1$, $\sigma_2(A) = 0$ and get a counterexample ($A\otimes A \neq 0$, but you want $\sigma_1(A\otime …