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.

14
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291 views

If $V\subset L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$, then $V$ is finite dimensional

If $V$ is a linear subspace of $L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$ for all $f\in V$, then $V$ is finite dimensional. The proof is an explicit calculation: Since $L^\infty[0,1] \subset L^...
12
votes
0answers
1k views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
11
votes
0answers
329 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^...
10
votes
0answers
803 views

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 $n$ ...
8
votes
0answers
131 views

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 ...
7
votes
0answers
216 views

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 ...
7
votes
0answers
143 views

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 $L^{\infty}$
7
votes
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641 views

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 $\Omega \times \Omega$ ...
7
votes
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111 views

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 ...
6
votes
0answers
92 views

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 $(L_1,*)$ ...
6
votes
0answers
188 views

Relating primal and dual characterization of an (interpolation) norm on $\ell_1+\ell_2$

For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$: $$ \lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\ell_2,\ a'+a''=a\...
6
votes
0answers
172 views

Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\...
6
votes
0answers
576 views

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 $V_{n-2}(r)$. ...
6
votes
0answers
218 views

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 ...
6
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130 views

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 ...
6
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347 views

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 ...
6
votes
0answers
363 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
5
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0answers
42 views

Determining if $f\in L^{p}(\mathbb R)$ from a bound on the measure of the level sets $\{|f|>\lambda\}$ for all $\lambda>0.$

$\textbf{The Problem:}$ Let $f$ be a measurable function on $\mathbb R$ with respect to the Lebesgue measure $m$. $\textbf{a)}$ Suppose that $$m(\{\vert f\vert>\lambda\})\leq(1+\lambda)^{-1}$...
5
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0answers
39 views

blocks of a normalized basis dominated by lp

I would like to know whether the following conjecture is true, possibly with additional assumptions such as unconditionality. Conjecture 1. Suppose $(x_i)_{i=1}^\infty$ is a normalized basis for a ...
5
votes
0answers
92 views

Are the canonical representatives of the Hilbert space $L^2$ basis-dependent?

The space $\mathcal{L}^p(\mathbb{R}^n)$ of functions $f$ such that $\int |f(x)|^p\, d^nx$ converges is only a seminormed rather than a normed vector space, because any function $f$ whose support has ...
5
votes
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193 views

Schauder basis $L^p(\mathbb{R})$

Let $\{e_{n}(x)\}_{n=0}^{\infty}$ be orthonormal basis of Hilbert space $L^2(\mathbb{R})$. If $\{e_{n}(x)\}_{n=0}^{\infty} \subset L^p(\mathbb{R})$ for some $p\geq 1$, is the $\{e_{n}(x)\}_{n=0}^{\...
5
votes
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195 views

Is an essentially bounded continuous function bounded?

I've just started working with $L^p$ spaces, and I've learned that a function $u:\mathbb{R}^n \rightarrow \mathbb{R}$ is essentially bounded if there exists a constant M such that $\{x \in \mathbb{R}^...
5
votes
0answers
70 views

Complemented $\ell_q$ subspaces of $(\oplus_n\ell_p^n)_\infty$

Fix $1\leq p<\infty$, and denote \begin{equation*}(\oplus_n\ell_p^n)_\infty=\left\{\left((a_i^{(n)})_{i=1}^n\right)_{n=1}^\infty:\left(\|(a_i^{(n)})_{i=1}^n\|_p\right)_{n=1}^\infty\in\ell_\infty\...
5
votes
0answers
543 views

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(...
5
votes
0answers
203 views

Limit of $L_p$ norm as $ p \rightarrow 0$

I have reviewed Ayman Houreih's proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean. While I have found the outline of the proof very ...
5
votes
0answers
2k views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
5
votes
0answers
554 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
5
votes
0answers
5k views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
4
votes
0answers
57 views

Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown ...
4
votes
0answers
75 views

Limits of a multiple integral function

Problem Let $f(x)\in L^{1}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$ and $S_t=\left\{x\in\mathbb{R}^N: |x_1|\le t\right\}$ with $t>0$. Let $\phi(t)$ the integral function $$\phi(t)=\int_{\mathbb{...
4
votes
0answers
47 views

Weak convergence in $H^1(\mathbb R^3)$ implies convergence of integrals

Suppose that $f_n \rightharpoonup f$ in $H^1(\mathbb R^3)$ (weak convergence). Then $$\int_{\mathbb R^3} \frac{\lvert f_n(x) \rvert^2}{\lvert x \rvert} dx \stackrel{n\to \infty}{\longrightarrow} \int_{...
4
votes
0answers
84 views

How to prove that $ ||T_{\lambda}f||_{L^{q}(\mathbb{R}^{n})}\leq C_{p, q}\lambda^{n(1/p-1/q)}||f||_{L^{p}(\mathbb{R}^{n})},\quad \lambda>0. $

Suppose $ h\in C_{0}^{\infty}(\mathbb{R}) $ supported in $ (0, \infty) $ and for $ \lambda >0 $ $$ T_{\lambda}f(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}} e^{ix\cdot\xi}h\left(\frac{|\xi|}{\lambda}\right)...
4
votes
0answers
57 views

Is $\sum_{m\in \mathbb Z} f(x-m)f(x-n) \in L^2(a,b)$ if $f\in L^2(\mathbb R)$?

Let $f\in L^2(\mathbb R)$ and $0<a<b< \infty,$ put $A= \{x\in \mathbb R: a<|x|<b\}.$ Fix $n\in \mathbb Z.$ Define $$ F_n(x)=\sum_{m\in \mathbb Z} f(x-n-m)f(x-m)$$ We note that $F_n$ is ...
4
votes
0answers
195 views

Equivalent norms in polynomial space

Let $\mathbb{P}_k(\mathbb{R}^d)$ be the space of multivariate polynomials of degree $\le k$ defined on $\mathbb{R}^d$. Let $Q=[0,1]^d$ and let $B(0,2)$ be the ball centred at the original with radius ...
4
votes
0answers
180 views

Embedding of some function spaces

Consider the strictly monotone continuous function $d:\mathbb{R^+}\to\mathbb{R^+}$, and denote by $\mathcal{D}$ the space of all measurable functions $f:[0,1]\to\mathbb{R}$ such that: $$\int_0^1 d(|f(...
4
votes
0answers
174 views

Show that if $L^p$ is not a subset of $L^q$ for $q \gt p$ then $L^p$ contains indicators of sets of arbitrarily small, positive measure

Show that if $L^p$ is not a subset of $L^q$ for $q \gt p$ then $L^p$ contains indicator functions of sets of arbitrarily small, yet positive measure. Since $L^p$ is not a subset of $L^q$, there ...
4
votes
0answers
135 views

Injectivity of a multiplication operator implies that the corresponding function is supported almost everywhere

Suppose that we have given any measure space $(\Omega, \Sigma, \mu)$ (such that $L^2(\mu)$ is not trivial) and consider the multiplication operator $M_g : L^2(\mu) \rightarrow L^2(\mu)$ given by $M_g (...
4
votes
0answers
65 views

$\mu(X)=\infty$, 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 ...
4
votes
0answers
126 views

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}{...
4
votes
0answers
64 views

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.
4
votes
0answers
155 views

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: ...
4
votes
0answers
141 views

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 ...
4
votes
0answers
1k views

$L^p$ 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 $p\to \infty$. ...
4
votes
0answers
135 views

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 ...
4
votes
0answers
459 views

Weak convergence and trace operator

Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to L^p(\...
4
votes
0answers
526 views

Properties of $L^{\infty}$

I'm trying to get a better grasp on the idea of $L^{\infty}$. What are the implications if we are given that $f \in L^{\infty}$? Also, how do we write $\|f\|_{\infty}$ in terms of the inf of a set of ...
4
votes
0answers
72 views

Why $\|f-g\|=0$ if and only if $f=g$?

I'm learning Fourier Transformation lately, and in the Course Reader page 23, it defines $\|f\|=\left(\int_0^1 \left|f(t)\right|^2 dt\right)^{1/2}$. And then $\|f-g\|=0$ if and only if $f=g$. My ...
4
votes
0answers
115 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
3
votes
0answers
38 views

Does multiplication by a test function stay in a Sobolev space?

Let $u \in D^{1,\vec{p}}(\Omega)$ and $\phi \in C_c^{\infty}(\Omega)$. Then do we necessarily have $u\phi \in D^{1,\vec{p}}(\Omega)$? My attempt What we need to show is that $\partial_i (u \phi) \...
3
votes
0answers
51 views

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 ...