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
2answers
42 views

Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
0
votes
0answers
14 views

Iterated convolutions w.r.t. different variables of a function

I do not understand a claim from a paper: Let $b:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a bounded function and let $$b^{n} (t,x) = b(t,x) \ast \psi_n(t) \ast\phi_n(x), $$ where $\...
3
votes
1answer
33 views

Show that $L^1$ function is $0$ a. e.

Suppose $f \in L_1(\mathcal{R})$ satisfies for every measurable $A \subset \mathcal{R}$ $$ |\int_A f| \leq [m(A)]^{(1+\epsilon)} $$ for some $\epsilon >0$. Prove $f=0$ a.e. This is a problem on ...
4
votes
1answer
60 views

$p$-norm inequality

Let $p \ge 2$ and $q$ such that$${1\over p} + {1\over q} = 1.$$Is it true that there exists a constant $c$ such that for all $x$, $y$ such that $\|x\|_q \le 1$ and $\|y\|_q \ge 1$$$\left\|x - {y\over{\...
1
vote
1answer
41 views

What does it mean “a Lebesgue point of $f$”?

Follan - Real Analysis p.243 Theorem 8.15 As far as I know, the term, Lebesgue point $x$, is defined for $L^1$ functions such that $\lim_{r\to 0} \frac{1}{m(B(x,r))} \int_{B(x,r)} |f - f(x)| dm =0$. ...
1
vote
1answer
37 views

Questions on proof that $\Vert \cdot \Vert_p$ is a norm when dealing with $L^p$ spaces

Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so ...
4
votes
2answers
45 views

Is $f(x)\exp(-x^2)$ summable if $f$ is square summable?

Suppose that $f \in L^2(\mathbb R)$; i.e. $$ \int_{- \infty}^\infty \vert f(x) \vert^2 dx < \infty. $$ Can we from this infer that $$ \int_{- \infty}^\infty \vert f(x)\vert e^{-x^2} \, dx < \...
0
votes
1answer
38 views

Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
5
votes
1answer
112 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 $\Omega \times \Omega$ ...
0
votes
1answer
26 views

subspace of $L^2$

Let $(X,B(X),\mu)$ a measurable space, for a positive finite measure $\mu$, we consider $H=L^2(X,d\mu)$, Let $A$ a closed subspace of $H$, we know that $A$ is a hilbert space, can we say that it exist ...
2
votes
2answers
53 views

If $f \in L^2[-\pi, \pi]$, then $f \in L^1[-\pi, \pi]$ and$\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$

I'm learning about Fourier analysis and need help with the following problem (which is part of a subchapter on $L^p$ spaces): Using the Cauchy-Schwarz inequality show that if $f \in L^2[-\pi, \pi]...
0
votes
1answer
41 views

The Lebesgue Fundamental Theorem of Calculus

Let $f,g:\mathbb{R}\to\mathbb{R}$ be two Lebesgue integrable functions. If we have $$f(b)-f(a)=\int_a^bg(x)dx$$ for almost all $a,b\in \mathbb{R}$. How can we modify $f$ on a set of measure zero to ...
1
vote
1answer
49 views

If $||g_j-g||_{1,\infty}\to0$, then $||g_j||_{1,\infty}\to||g||_{1,\infty}$

I have some problems with my notes: my teacher wrote that if a sequence $\{g_j\}_j\subseteq L^{1,\infty}(\Bbb R^n)$ (which is the weak $L^1$ space, endowed with the quasinorm $||\cdot||_{1,\infty}$) ...
2
votes
0answers
41 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
0
votes
1answer
19 views

Compactness Sobolev embedding for even functions on $\mathbb{R}$.

It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial ...
0
votes
1answer
33 views

$L^{\infty}$ convergence for random variable

I am slightly confused with this borderline case regarding $L^p$ convergence. In some probability books, they clearly state that $p<\infty$ whereas the online sources do not impose this ...
1
vote
1answer
59 views

If an $H^1$ function vanishes on a set of positive measure, its $L^2$ norm is controlled by the gradient

I am trying to solve question 15 from Evans' PDE book, chapter 5. You have a set of positive measure, subset of the unit ball $B$, such that $u$ is equal to zero on that set. Then, one can show that: $...
2
votes
0answers
81 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

Let $\Omega\subset \mathbb R^d$ ($d=2,3$) is a bounded Lipschitz domain. Question: Is it true that for each function $g(x)\in L^2(\Omega)$ one can find a sequence $\{g_n\}_1^\infty$ of $H_0^1(\...
2
votes
2answers
50 views

Showing that a subspace of $L^p$ is closed

I would like to prove that a particular subspace of $L^p([1,\infty[)$ (for some $p \in [1,\infty[$) is closed, but I'm not sure how to do it properly. For any sequence $(x_n) \in \ell^p$, let ...
2
votes
1answer
51 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
24 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
64 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 ...
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
36 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
22 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
38 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
34 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
43 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
38 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
57 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
40 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
34 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
23 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
21 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
38 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
67 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
43 views

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
31 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 ...