# Tagged Questions

For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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### Measure Theory and $L^{p}$ spaces

I have the two following very simple questions regarding measure theory that I want to show: If $f \in L^{p}(X, \mathcal{M}, \mu)$ for $1 \leq p < \infty$, then $f < \infty$ $\mu$-almost ...
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### Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
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### Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on $\mathbb{R}^k$...
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### Inclusion of $L^r(\mu)$ in $L^q(\mu)+L^p(\mu)$

Let $(X,\mathcal{M},\mu)$ a positive measure space, then $L^r(\mu)\subset L^p(\mu)+L^q(\mu)$. How can I prove this? I know the standard inclusions for finite measure spaces, and spaces without ...
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### Cyclic vector in $\ell^2(\mathbb{Z})$ space

Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$. Define the right-translation operator by R:\{c(n)\}\mapsto \{c(...
It is known that for every $k\in\mathbb{N}$ there is $N_k\in\mathbb{N}$ such that every $N_k$-dimensional subspace of $\ell_p$, $1<p<\infty$, contains uniformly complemented copies of $\ell_2^k$ ...