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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2
votes
1answer
52 views

$X$ be Banach , $T:X \to \mathcal l ^{\infty}$ be linear , $(Tx)_n$ the $n$-th term of $T(x)$;$f_n(x)=(Tx)_n$ ; if each $f_n$ is bdd then so is $T$?

Let $X$ be a Banach space , $T:X \to \mathcal l ^{\infty}$ be a linear transformation , for each $x\in X$ and each $n \in \mathbb N$ , $(Tx)_n$ be the $n$-th term of $T(x)$ and for each $n \in \...
1
vote
1answer
16 views

$f(t)=t^{2r-2p-1} (e^{2t}-e^{-2t})^{-1/2}\in L^{1}(0, \infty)$?

Let $r>3/4,$ and $p>1/2.$ My Question: Can we expect $\int_0^{\infty} t^{2r-2p-1} (e^{2t}-e^{-2t})^{-1/2} dt < \infty$? I am trying to analyze the above integral. Any suggestions/...
1
vote
0answers
25 views

If $f \in L^{\infty}(\Bbb R^d) \cap C_u(\Omega)$, then $\rho_{\epsilon} * f \to f$ uniformly over $\Omega$

Consider the following statement: If $(\rho_{\epsilon})_{\epsilon > 0}$ is an approximation identity and $f \in L^{\infty}(\Bbb R^d) \cap C_u(\Omega)$, then $\rho_{\epsilon} * f \to f$ ...
1
vote
2answers
67 views

Characterization of weak convergence in Lp

Weak convergence in $X=L^p(0,1)$ for $1<p<\infty$ can be characterized as following: $f_n\rightharpoonup f$ if and only if $f_n$ is bounded in $X$ and $\int_{(0,t)}\;f_n\;\rightarrow \int_{(0,t)}...
2
votes
1answer
27 views

If $f^p\in L^1([0,1])$ it's bounded a.e.

We know that being Lebesgue integrable does not imply boundedness of the function (e.g. $g(x)=\frac{1}{\sqrt x}$). However function in $L^p$ spaces are functions with some decay conditions. Suppose ...
3
votes
1answer
52 views

Linear operator on a dense subset of $L_p$ which is unbounded when extended to $L_p$

Let the linear operator $L:C^{\infty}_0([-1,1]) \rightarrow C(\{0\})$ be defined as $f(0)$ (evaluating a function in $C^{\infty}_0([-1,1])$ at $0.$ I would like to show that extending this to ...
0
votes
2answers
23 views

If $f_n + g_n \to h$ in $L^2(\Omega)$ and $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$?

On a bounded domain $\Omega$, if $f_n + g_n \to h$ in $L^2(\Omega)$ and each $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$? I feel like this should be true since each sequence is non-negative, so ...
0
votes
3answers
39 views

Let $f\in L^p$. Can we say $\|f\|_{L^{p}} \leq \epsilon$ on $|x|\geq R$ for large $R$?

Let $f\in L^{p}(\mathbb R), (1\leq p <\infty)$ and $\epsilon>0.$ My Question: Can we expect to find $R>1$ (may be large) so that $\|f\|_{L^{p}(B_R)} \leq \epsilon$ on $B_{R}=\{x:|x|...
1
vote
1answer
37 views

Continuity on $L_p$ spaces

Consider a nonlinear and continuos function $f:\mathbb{R} \rightarrow \mathbb{R}$ and we define the functional \begin{equation} F(u) = \int_{[0,1]^2} f(u(x,y)) dxdy \end{equation} where $u$ is an ...
1
vote
1answer
23 views

Approximations of $L^p$ functions, convolutions, mollifiers, etc. (resource needed)

What is a good resource in which I can read about mollifiers, basic theorems regarding convolutions, smooth approximations of $L^p$ functions and the like? (the presence of exercises would be great, ...
3
votes
1answer
42 views

Proof that $\|S_N\|_p < \infty $ is equivalent to $\|S_N f - f\|_p \to 0$ as $N \to \infty$

I am having difficulties with the proof of proposition 1.9 in the book "Classical and multilinear harmonic analysis, Vol. 1" by C. Muscalu and W. Schlag. The following statements are equivalent ...
1
vote
1answer
26 views

Showing $u'=v$ a.e. given $u_k \to u$ and $u'_k \to v$ in $L_2(\mathbb{R})$.

Suppose $(u_k)$ is a sequence of differentiable functions in $L_2(\mathbb{R})$ satisfying (1) There is a $u \in L_2(\mathbb{R})$ so that $\| u_k - u\|_2 \to 0$. (2) There is a $v \in L_2(\mathbb{R})$...
2
votes
1answer
32 views

Prove that $\rho_n \star f \to f$ in $L^p(R^N)$.

Let $\rho \in L^1(R^N)$ with $\int_{}^{} \rho=1$ .Set $\rho_n(x)=n^N\rho(nx)$. Let $f\in L^p(R^N)$ with $1\leq p<\infty$. Prove that $\rho_n \star f \to f$ in $L^p(R^N)$. My try: Since $f \in L^...
1
vote
1answer
35 views

Quotients of $L_1$

I know the rather standard fact in Banach space theory that every separable Banach space is a quotient of $\ell_1$. Is it true that every (possibly non-separable) Banach space is a quotient of some $...
1
vote
1answer
29 views

Subspaces of quotients of $L^p$ spaces

Is the collection of subspaces of quotients of $L^p$ spaces considered to be a large class of Banach spaces?
0
votes
1answer
27 views

Proving that $l_r$ is dense everywhere in $l_p$ $1\leq r \leq p$

$$l_p=\{(x_i)^{\infty}_{i=1}|\sum_{i=1}^{\infty}|x_i|^p<\infty\}$$ The answer is given, but this proof makes no sense to me. If somebody could explain the logic, idea here, I would be very ...
2
votes
0answers
33 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 $V_{n-2}(r)$. ...
0
votes
2answers
44 views

Duality of $L^p$ spaces

Let $p,q\in(1,\infty)$ be such that $1/p+1/q=1$ and let $(\Omega, \mathcal A,\mu)$ be a $\sigma$-finite measure space. Claim: The map $$\phi:L^q(\Omega)\to \left(L^p(\Omega) \right)^*,\quad \phi(g)...
1
vote
1answer
48 views

Boundedness and norm of a sequence operator

Let $s = \{s_{n}\}_{n=1}^{\infty}$ be a fixed and bounded sequence of real numbers, i.e. $s \in (\ell^{\infty},\|\cdot\|_{\infty})$. Consider the operator $T_{s} : \ell^{2} \to \ell^{2}$ defined by ...
0
votes
1answer
39 views

$L^2$ and $L^\infty$ normed inequality for PDE solution: Which one is more informative and why?

I have the following inequalities $$max_{t \in [0,T]} \lVert u_1(t, \cdot)-u_2(t, \cdot) \rVert_{L^2(\mathbb{R})} \leq C \lVert g_1(x) - g_2(x) \rVert_{L^2(\mathbb{R})}$$ and $$max_{t \in [0,T]} \...
3
votes
1answer
59 views

Rudin's RCA Q3.4

I'm trying to solve the following question from Rudin's Real & Complex Analysis. (Chapter 3, question 4) : Suppose $f$ is a complex measurable function on $X$, $\mu$ is a positive measure on $...
1
vote
0answers
25 views

About a sequence of functions that converges locally but not globally

Good morning. During my thesis, I have come to the following problem: suppose $(M, g)$ is a closed Riemannian manifold of dimensione greater than $2$. You have a function $\varphi \in C^0(M)$ s.t. $\...
3
votes
1answer
43 views

Nonlinear elliptic PDE - passing to the limit

In the notes I am trying to follow one can find the following argument (part of a longer proof on existence of a weak solution to a certain type of nonlinear elliptic pde): Let $V = H^1_0(\Omega)$ ...
0
votes
1answer
42 views

Does $\lim_{p \to \infty}\biggl(\int_0^T \Bigl(\int_\Omega |f|^{\frac{3p}{1+2p}}\Bigr)^{\!\frac{1+2p}{3}}\biggr)^{\!\frac 1p} < \infty$ exist?

Let $f \in L^2(0,T;L^2(\Omega))$ on a bounded set $\Omega$. Does the following limit exist? $$\lim_{p \to \infty}\biggl(\int_0^T \Bigl(\int_\Omega |f|^{\frac{3p}{1+2p}}\Bigr)^{\!\frac{1+2p}{3}}\biggr)...
1
vote
0answers
31 views

Extending $L^{p}$ Duality to $\sigma$-finite Spaces

Let $1 \leq p < \infty$, $(X,\mathcal{M},\mu)$ be a sigma-finite measure space. Let $L$ be a continuous linear form on $L^{p}(X,\mathcal{M},\mu)$. Then, show that $\exists g \in L^{p'}$ such that: ...
0
votes
1answer
38 views

hint on exercise about weak $L^p$ space

I'm working on a problem from Grafakos, Classical Fourier Analysis. Let $(X, \mu)$ be a measure space and let $E$ be a subset of $X$ with $\mu(E) < \infty$. Assume that $f$ is in $L^{p,\infty}(...
2
votes
0answers
28 views

Lebesgue Space/Bochner Space interpolation Theorem

I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain, $$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{p_2}(I;L^{p_2}(\Omega))\hookrightarrow L^...
2
votes
1answer
22 views

Alternative characterisation of weak derivatives

Let $\Omega \subseteq \mathbb{R}^n$. The textbook I am reading defines the space $W^{k,2}(\Omega)$ as follows: An element $u$ is in $W^{k,2}(\Omega)$ if there exists a sequence $(u_m)$ in $C^{\...
1
vote
2answers
41 views

Counterexample for $L^p$ Inclusion

The general question is : disprove that $L^p(\mathbb R)\subset L^q(\mathbb R)$ and $L^q(\mathbb R)\subset L^p(\mathbb R)$ for $q<p\leq\infty$ I managed to find counterexamples for the finite cases ...
4
votes
1answer
70 views

$L^p \subset L^q$ for $p\neq q$.

Let $1\leq p \leq q \leq \infty$. It's well known that on a finite measure space $(X,\mathcal{M}, \mu)$, we have the inclusion $L^q(X,\mathcal{M}, \mu) \subset L^p(X,\mathcal{M}, \mu)$. Questions ...
3
votes
0answers
97 views
+500

Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - \...
0
votes
2answers
32 views

Mixed up with hierarchy of $L_p$ spaces

Consider the interval $[0,1]$ and define $$ X_1:= \left[0, \frac{1}{2}\right], ~~~X_2 := \left[\frac{1}{2}, \frac{3}{4}\right], ~~ X_3 := \left[\frac{3}{4}, \frac{7}{8}\right], ...$$ Define a ...
0
votes
2answers
62 views

Monotone increasing sequence in Lp convergent a.e.

I´m having trouble with the proof of the following theorem in measure theory: Consider the sequence $(f_n)_{\,n \in \Bbb N}\subset L^p(X)$ with $\;0\le f_n(x)\le f_{n+1}(x) \;\;\;\forall x\in X\;\...
3
votes
1answer
68 views

Does the convergence of integrals against $H^1_0$ functions imply boundedness in $L^2(\Omega)$?

In order to show something, I would like to have this strange side result, which seems obvious yet surprisingly I cannot find a way to show it "rigoriously". Suppose we have a sequence $(u_n) \subset ...
0
votes
1answer
20 views

$L^p(\Bbb R^n)$ spaces decomposition

Let us fix $1\le p<r<q$; fix an $l>0$ and take an $f\in L^r(\Bbb R^n)$; then we split the function $$ f=\underbrace{f\chi_{\{|f|>l\}}}_{=:f_1}+\underbrace{f\chi_{\{|f|\le l\}}}_{=:f_2} $$ ...
1
vote
1answer
41 views

Problem with $L^p(\Bbb R^n)$ spaces (Marcinkiewicz Theorem)

Marcinkiewicz Theorem says that, if $1\le p<q$ given a bounded operator $T$ from $L^p(\Bbb R^n)$ to $L^{p,w}(\Bbb R^n)$ (the last one is the $L^p$ weak space) AND EVEN from $L^q(\Bbb R^n)$ to $L^{q,...
3
votes
0answers
37 views

$L^p$-bounding inequality

Do we have that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$. ...
5
votes
2answers
65 views

Weak convergence in different $L^p$ spaces

Consider $p \ge \alpha \ge 1.$ If a sequence converges weakly in $L^p,$ say $u_n \rightharpoonup u$, is it true that: $$u_n^{\alpha} \rightharpoonup u^{\alpha} \text{ in $L^{p/ \alpha}$}$$ This ...
0
votes
1answer
26 views

From convergence in $L^2$ to convergence of the squares in $L^1$

Given a function $f\in L^2(X,\mu)$ for some measure space $(X,\mu)$, we of course have that $|f|^2$ is a positive function in $L^1(X,\mu)$. I now came across the following. Suppose $\int_X |f_n(x)-g_n(...
1
vote
1answer
55 views

Poincaré's Inequality on Sobolev Spaces in One Dimension

The following is a version of Poincaré's inequality: Let $I$ be a bounded interval, then there exists a constant $C$ dependent on $I$ such that $$\|u\|_{W^{1,p}(I)} \leq C\|u'\|_{L^p(I)} \ \ \ \ \...
1
vote
2answers
72 views

Jensen's inequality; what's the need for the probability measure?

Jensen's inequality states that if: $\mu$ is a probability measure on $\Omega$, $f$ is integrable function (on $\Omega$) and $\phi$ is convex on the range of f then: $\phi \left( \int_{\Omega}g(x)\,d\...
6
votes
5answers
262 views

Why define norm in $L_p$ in that way?

Who first defined the norm in $L_p$ space as $$\left(\int{\lvert f(x) \rvert^p}\right)^{1/p}$$ Is there any reference for this? Is it just an simple extension from $L_2$? $L_p$ space has some ...
3
votes
1answer
48 views

Prove (possibly using the Uniform Boundedness Principle) that g in L^q

The question is Prove that, if $(X, M, \mu)$ is a $\sigma$-finite measure space, $1\leq p\leq\infty$, and $\frac{1}{p} + \frac{1}{q} = 1$ then: If $f: X\rightarrow\mathbb{C}$ is a ...
4
votes
1answer
48 views

Unbounded operator, when is it dense?

Let $E = L^p(0, 1)$ with $1 \le p < \infty$. Consider the unbounded operator $A: D(A) \subset E \to E$ defined by$$D(A) = \{u \in W^{1, p}(0, 1),\text{ }u(0) = 0\} \text{ and }Au = u'.$$My two ...
3
votes
0answers
30 views

Characterization of the function $u(x) = (1 + |\log x|)^{-1}$ [closed]

Let $u(x) = (1 + |\log x|)^{-1}$. Are the following all true? $u \in W^{1, 1}(0, 1)$ $u(0) = 0$ ${{u(x)}\over x} \in L^1(0, 1)$
1
vote
1answer
44 views

In which sense is the Fourier transform the continuous analogue of the discrete Fourier transform? A mathematical point of view.

I try to understand in which sense the Fourier transform is the continuous analogue of the discrete Fourier transform. I know, there are many books and many questions on this site concerning this ...
2
votes
2answers
76 views

Is the product $f_ng_n$ of weak-star convergence sequences $(f_n)$ and $(g_n)$ in $L^\infty$ also weak-star convergent?

Suppose $X$ is a finite measure space, and $f_n$ is uniformly bounded and converges to $f$ in the weak-star topology of $L^\infty(X)$. This means $\int f_n\phi \to \int f\phi$ for all $\phi\in L^1(X)$....
1
vote
0answers
24 views

Notation: meaning of $L^{p}(\mathbb{R}^{N},\mathcal{A})$

Let's say $\mathcal{A}$ is some subset of $\mathcal{M}_{N}(\mathbb{R})$ (square matrices of order $N$ with real coefficients). What does $L^{p}(\mathbb{R}^{N},\mathcal{A})$ mean (where $p\in\mathbb{N}...
1
vote
2answers
37 views

Showing that the following set is closed in $L^p$

Question: Let $1\le p < \infty$ and $1\le q \le \infty$. Prove that the following set is closed in $L^p$. $$ \{f \in L^p \cap L^q : |f|_q \le 1 \} $$ My try: Let $f_n$ be a sequence in the above ...
4
votes
2answers
57 views

Is $D(A)$ necessarily dense in $E$? Is $G(A)$ necessarily closed in $E \times E$?

Let $E = L^p(0, 1)$ with $1 \le p < \infty$. Consider the unbounded operator $A: D(A) \subset E \to E$ defined by$$D(A) = \{u \in W^{1, p}(0, 1),\text{ }u(0) = 0\} \text{ and }Au = u'.$$I have two ...