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Too long for a comment. For $d=1$ should be true. Let me simplify a bit (hopefully without loss of generality). If $\Omega=(-1,1), f\in C(-1,1)$. Then what you ask implies that \begin{equation} f(x) \ …

Suppose that $u $ is a smooth function real valued function defined on an open neighborhood of the unit disc $ \mathbb{D} $, which satisfies a second order elliptic partial differential equation; \beg …

Yes I think this is true. Let $\mathcal{G} := \{ \times_{i=1}^n[k_i,k_i+1] : k_i\in \mathbb{Z} \} $ be the grid of cubes of side length $1$ and vertices in $ \mathbb{Z}^n$. For $Q\in \mathcal{G} $ le …

No I do not think this is always possible. Take for example a subsequence of $\mathbb{N}$, call it $\{ m_k \}$ and consider the random variables \begin{equation} X_i = \chi_{(0,1/2)},\quad m_k \leq i …

The following Tauberian theorem is true (see Theorem I.11.1 of ''Tauberian theory: A century of developments''). Let $ a_n $ a sequence of real numbers. If $f(x) = \sum_{n=1}^\infty a_n x^n $ converge …

I think the paper "A singular function with a non-zero finite derivative on a dense set with Hausdorff dimension one" answers exactly this question.

The opposite inequality cannot be true. If that were true, then consider a positive function $g$ with the property such that for all $s\in \mathbb{T}$ it holds that $g(s,x) \leq C g(s,y)$ whenever $|x …

Partial answer: About linear independence, it is true that if $f$ is non constant then the dilations $f_n(x)=f(nx), n\in \mathbb{N}$ are linearly independent. In fact suppose for a finite sum we hav …

I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $ \lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-upcro …

Suppose you have a power series with coefficients $a_n$ $$ f(z):= \sum_{k=1}^\infty a_k z^k .$$ Then the coefficients of $f^2$ are exactly $c_n$. Also if we denote by $\odot$ the Hadamard multiplicati …

You can probably prove that $$ \Big( \int_\mathbb{A_r} |f(z)|^2 \frac{dxdy}{\pi(1-r^2)} \Big)^{1/2} \leq \int_{ \mathbb{T}} |f(e^{i\theta})| \frac{d\theta}{2\pi}+\int_{ \mathbb{T_r}} |f(re^{i\theta}) …

So first of all, what is claimed in Love's paper is slightly different, It says that for a sufficient large choice of the parameter $c>0$ the function $$ g(x): = \sum_{n=0}^\infty c^{-n/p}\cos(c^n \pi …

By Frostman's lemma, if $E$ is a compact set of positive $\alpha$-Hausdorff content, then there exists a probability Borel measure $\mu$ supported in $E$ such that $\mu(I) \leq c |I|^\alpha $ for ever …

It should be true for $p>1$. For a function $\varphi$ in the Schwartz class it holds that \begin{equation} D_1D_2 \varphi(x) = -R_1 R_2 \Delta \varphi(x), \end{equation} where $R_1, R_2$ are the Riesz …

I think similar questions always translate in the $L^p$ boundedness of a Fourier multiplier. In this case you want a Fourier multiplier which "exchanges the operator $D_1D_2D_3$ with the operator $D_1 …