Questions tagged [lp-spaces]
For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
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Is weighted $L^p$ space isomorphically $L^p$ spaces?
Let $w:\mathbb{R}^n\to ]0,\infty[$ continuous (a weight)
Is the weighted $L^p$ spaces $L_w^p(\mathbb{R}^n)$ isomorphically to $L^p$ space?
My attempt: Let $L_p(\mathbb{R}^n)\to L_w^p(\mathbb{R}^n)\...
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Given $f \in L^p_{\text{loc}}(\Omega) \setminus L^\infty(\Omega)$, does it follow that $A \cap S(f,K) \neq \emptyset$ for all $K > 0$?
Context. Throughout this post I will be dealing with the Lebesgue measure over $\mathbb R^n$. Moreover, I denote the measure of a measurable set $E \subset \mathbb R^n$ by $|E|$ and $\Omega \subset \...
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Understanding the proof of $L^{\infty}$ is complete.
I got lost when reading the proof of $L^{\infty}$ is complete. The book proceed the proof as follows:
We show that each absolutely convergent series in $L^{\infty}(X,\mathscr{A},\mu)$ is convergent. ...
- 2,493
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"Elementary" $L^2$ inequality for combination of compactly supported functions
I am reading the paper Ondelettes et poids du Muckenhoupt by Lemarié and, at some point, he needs to prove a certain operator is bounded in $L^2(\mathbb R)$ to apply weighted inequalities theory. The $...
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Convergent subsequences in $L^p(\mathbb R^n)$
This is a particular fact that I ended up proving in the process of attempting one of my recent homeworks, but I don't think I've seen this particular fact online even though it feels like a fairly ...
- 1,503
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Mikhlin, Marcinkiewicz theorem on weighted space $L^p$ spaces.
This theorem is in D. Guidetti, “Vector valued Fourier multipliers and applications,” Bruno Pini Mathematical Analysis Seminar, Seminars 2010 (2010). It is a variant (I think, a easier variant) of ...
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Given $f \in L^p_{\text{loc}}$ and $\phi \in S$ (Schwartz class), do we have that $f \ast \phi \in C^\infty$? What if $f,\phi$ have compact support?
Context. Consider the usual Lebesgue measure on $\mathbb R^n$ and let $L^p_{\operatorname{loc}}$ denote the space of measurable functions that are locally $p$-integrable on $\mathbb R^n$, with $1 \...
- 1,141
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Prove that the derivative of the mollification approaches the strong $L^p$ derivative
Let $g \in C_c(\mathbb{R})$ be such that $\int_\mathbb{R} g dx = 1$. Suppose that $f \in L^p(\mathbb{R}$ has a strong $L^p$ derivative, i.e. there is some $h$ such that $\displaystyle\lim_{y \to 0} \...
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Show $\|f\|_q\leq c\|f\|_r+\frac{1}{c^a}\|f\|_p$
Let $1\leq p<q<r<\infty$ and $f\in L ^p(\Omega)\cap L^r(\Omega)$. Then for $a=\frac{\frac{1}{p}-\frac{1}{q}}{\frac{1}{q}-\frac{1}{r}}$ and $c>0$ the inequaltiy
$$\|f\|_q\leq c\|f\|_r+\frac{...
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Smooth approximation of $L^p$ functions with zero average
Let $U \subseteq \mathbb R^n$ be a bounded open set and suppose that $f : U \rightarrow \mathbb R$ is in the Banach space $L^p(U)$ for some $1 \leq p < \infty$, that is,
$$
\int_U |f|^p dx < ...
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Does convolution preserve local integrability in the sense that if $f \in L^p_{\text{loc}}$ and $g \in L^1$, then $f \ast g \in L^p_{\text{loc}}$?
Consider the Euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is a fixed integer, equipped with the Lebesgue measure. Let $L^p_{\text{loc}}(\mathbb R^n)$($1 \leqslant p < \infty$) denote the ...
- 1,141
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Vector space generated by translates
I'm going through a proof in the book Introduction to Banach Spaces: Analysis and Probability and I don't understand some things (I believe that the context is enough, because this is just the ...
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Equivalent characterisation of the space $L^p_{\operatorname{loc}}(\Omega)$, where $\Omega$ is a non-empty open subset of $\mathbb R^n$?
Consider the euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is an arbitrary integer, equipped with the usual Lebesgue measure. Moreover, let $\Omega \subset \mathbb R^n$ denote an arbitrary ...
- 1,141
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Measurability of $\|f(\cdot, x_{2})\|_{L^\infty(X_{1})}$ (proof of Minkowski's inequality)
I am trying to prove Minkowski's inequality for integrals in the case $p = \infty$.
Suppose $(X_{1}, \mu_{1})$ and $(X_{2}, \mu_{2})$ are two $\sigma$-finite measure spaces and $f(x_{1}, x_{2})$ is a ...
- 1,187
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Confusion about Riesz representation theorem in $L^p$
Riesz-Representation Theorem (RR): let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space. Then for every $T\in H'$, there exists a unique $v_T\in H$ s.t.
$$T(u)=\langle u,v_T \rangle\quad \forall u\...
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