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 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 problem is a $L^p$ space.

728 questions
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Dual of $\ell^p$ Direct sum

I am asked to show that the $\ell^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $\ell^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
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Is the normalized derivative of a holomorphic function Sobolev?

Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic. More precisely, I assume that $f$ is holomorphic on the interior $\text{int}(B)$, and ...
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Question on a step of the proof of Theorem 1.25 of Introduction to Fourier Analysis on Euclidean Spaces

Theorem 1.25: Suppose $\phi \in L^1(\mathbb{R^n})$ and $\int_{\mathbb{R^n}} \phi =1$ . Also, let $\phi_{\epsilon}(x)=\frac{\phi\left(\frac{x}{\epsilon}\right)}{\epsilon^n}$.Moreover , suppose ...
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In which $L^p$ metric is $\pi = 3.5$?

In which $L^p$ metric is $\pi = 3.5$? I am interested because it's well known that $\pi$ can range from $3.14...$ to $4$ in $L^{\infty}$
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Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
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Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
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Show that any non-trivial ideal of $(L_1,*)$ is dense

This is related to this other question, I mean, the linked question comes to my mind trying to solve the following exercise: Show that any non-trivial ideal of $(L_1,*)$ is dense. Here $(L_1,*)$ ...
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Volume of n-dimensional ball in L1 norm with change of variables

For a homework problem, I need to find a recursive equation that relates the volume of an $n$-dimensional ball $V_n(r)$ of radius $r$ to that of an $(n-2)$-dimensional ball, expressed by $V_{n-2}(r)$. ...
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Equivalent descriptions of Sobolev spaces on compact manifolds

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces. The first one, valid only for compact manifolds (because it needs to globalize ...
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Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure.

Let $(X,M, μ)$ be a measure space and $0 < p < q ≤ ∞$. Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure. My work: I proved the ...
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Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
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$\mu(X)=\infty$, and $f\in L^p (X)$ for some $1<p<\infty \Longrightarrow f\in L^1 (X)$

What are some interesting examples of a measure space $(X,\mu)$ such that $\mu(X)=\infty$ and $f\in L^p (X)$ for some $1<p<\infty \Longrightarrow f\in L^1 (X)$ All the examples I have found ...
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Morawetz's two $L^p$ inequalities

I'm reading Morawetz's Two $L^p$ Inequalities, in which two global inequalities for $p>1$ are presented. Here $\lVert \cdot \rVert$ is an $L^p$ norm. \frac{\lVert X\rVert^p + \lVert Y \rVert^p}{...
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If $xf \in L^2(\Bbb R)$, is $f \in L^1(\Bbb R)$?

I tried using Holder's inequality with $|f| = |f| \frac{(1+|x|)^{\frac{1}{2}}}{(1+|x|)^{\frac{1}{2}}}$, or variant but I can't seem to make it work. Any help is appreciated.
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Bounded variation functions with $\|f\|:=\|f\|_{L^1}+\|f'\|_{L^1}$: Banach space (after identifying any a.e. equal $f,g$)?

Let $[a,b]$ be a real bounded interval and let $\mathfrak{P}([a,b])$ be the set of all partitions of $[a,b]$, i.e. $\mathfrak{P}([a,b])=\{\{t_0,\dotsc,t_n\}:a=t_0<t_1<\dotso<t_n=b\}$. Define: ...
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Approximation theory in Lp spaces (Reference Needed)

I am looking for some reference on approximation theory in Lp spaces. I have found a number of papers like: paper1 , paper2 etc. I was wondering if there is a book or a monograph that will contain ...
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$L^p$ norm converges to $L^\infty$ norm

The question is: Let $(X,\mathcal M,\mu)$ be an arbitrary measure space. Let $f$ be a function in $L^r$ for some $0<r<\infty$. Show that $||f||_p$ converges to $||f||_\infty$ as $p\to \infty$. ...
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Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
Confusion on $\vert\vert \varphi\vert\vert_{p}$ where $p \in [1,\infty[$
Let $\varphi \in C^{\infty}(\mathbb R^n)$ and for $\epsilon > 0$ define $\varphi_{\epsilon}(x):=\epsilon^{-n}\varphi(x/\epsilon)$ such that $\varphi_{\epsilon} \in C^{\infty}(\mathbb R^n)$ with ...