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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17
votes
0answers
436 views

How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
8
votes
0answers
87 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
5
votes
0answers
87 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
votes
0answers
108 views

Operator norm of a convolution

Consider the operator on $L^2(\Bbb R)$, $f\rightarrow f*g$, where $g\geq 0$ is some $L^1$ function. Show the operator is a bounded linear operator with operator norm equal to $||g||_1$. Showing ...
5
votes
0answers
132 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 ...
4
votes
0answers
65 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 ...
4
votes
0answers
95 views

$f(y-x)$ integrable implies $f=0$ a.e.

If $f(y-x)$ is in $L^p(\mathbb R^d\times\mathbb R^d)$, then I seem to conclude that $f=0$ a.e. (which seems wrong). My reasoning is that by Fubini and the integral's shift invariance (assume $p=1$ for ...
4
votes
0answers
118 views

When does $|f*g|_{p}=|f|_{1}|g|_{p}$?

From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4 Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
4
votes
0answers
107 views

A question about functions in $L^p(E)$

I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
3
votes
0answers
40 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 $X=\mathbb{N}$. ...
3
votes
0answers
32 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
3
votes
0answers
70 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
3
votes
0answers
38 views

Relationship between complex and real Lp spaces.

The theory of integration of real functions is (as far as I know) usually extended to the complex case as follows: Let $X$ be a set. Given a function $f:X\to\mathbb C$, define ...
3
votes
0answers
75 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
3
votes
0answers
164 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
2
votes
0answers
21 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
2
votes
0answers
34 views

Sequence which does not admit weakly convergent subsequences

Let $f_h:[0,1]\to\mathbb{R}$ be defined as $f_h(x)=h$ if $0\le x\le 1$ and $f_h(x)=0$ otherwise. Prove that it does not exist a subsequence $(f_{h_k})_k$ weakly convergent in $L^1(0,1)$. Attempt of ...
2
votes
0answers
25 views

Proof verification-density of smooth compactly supported functions

I am trying to show that $C_{c}^{\infty}(\mathbb{R})$ (smooth compactly supported functions) is dense in $C_{c}(\mathbb{R})$ (in the $L^{p}$ sense). Can anyone check if my proof is correct? Let $f ...
2
votes
0answers
63 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
2
votes
0answers
25 views

References on Weak Convervenge

I am looking for a good reference on weak convergence in L^p spaces, can anyone recommend anything? Thanks a lot in advance!
2
votes
0answers
48 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
2
votes
0answers
38 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
2
votes
0answers
48 views

“Dual Lorentz characterization of L^q”

I am trying to do Exercise A.4 of Terence Tao's book on nonlinear dispersive equations (available here: http://www.math.ucla.edu/~tao/preprints/chapter.pdf, the exercise is on p.342 in Appendix A). ...
2
votes
0answers
19 views

Laplace Transform of $L^{2}$ Function.

I know the Fourier transform is an isometry of $L^2$ functions. I've read that the Laplace Transform of an $L^2$ function is $L^2$ but cannot prove it nor can I find a proof. Does anyone know of a ...
2
votes
0answers
45 views

Is this possible to prove that $L^p[E]$ is complete without using the concept of “rapidly Cauchy” sequences?

Is this possible to prove that $L^p[E]$ is complete without using the concept of "rapidly Cauchy" sequences ? In Royden's book, this how it is done. I was curious if there is an alternative approach ...
2
votes
0answers
42 views

Inequality in H-curl function space

Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := ...
2
votes
0answers
47 views

How to determine the limit?

How to solve following: Let $p\geq 1$ and $f\in L^p[0,1]$. For $\alpha\in\mathbb{R}$ determine $\lim_{x\rightarrow 0} x^\alpha\int_0^x f(t) dt$. This problem probably has several cases, but I'm ...
2
votes
0answers
184 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 ...
2
votes
0answers
44 views

A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
2
votes
0answers
82 views

An almost orthogonality principle for $L^p$

If two functions are far from being orthogonal, their difference cannot be too large in $L^2$. A precise statement (easily verified with the Pythagorean theorem) is as follows: let ...
1
vote
0answers
43 views

Prove that $f\in L^p[0,1]$ for all $p\in[1,2)$:

Given that $f:[0,1]\to[0,\infty)$ in $L^1$ such that $\int_E f$ $dm\leq\sqrt{m(E)}$ for every $E\subseteq[0,1]$ measurable, prove that $f\in L^p[0,1]$ for all $p\in[1,2)$. This is a qualifying ...
1
vote
0answers
28 views

Differentiate a log of $L^p$ norm, don't understand this result

I'm reading this paper. In it, the authors show this lemma: And then they prove this lemma My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log ...
1
vote
0answers
47 views

If $a_n \to a$ in $L^2$ and $F:\mathbb{R} \to \mathbb{R}$ is continuous, does $\int F(a_{n_j}) \to \int F(a)$?

Suppose $a_n \to a$ in $L^2(\Omega)$. Let $F:\mathbb{R} \to \mathbb{R}$ be continuous with $F(0) = 0$. We have that $F(b) \in L^1(\Omega)$ if $b \in L^2(\Omega)$ and $|F'(x)| \leq C_1 + C_2|x|$. I ...
1
vote
0answers
37 views

Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
1
vote
0answers
46 views

Condition for a product of two function sequences in $L^1$ to be in $L^1$

We have: 1. $\{f_n\} \subset L^1(E, \Sigma, \mu)$ 2. $g_n \subset L^\infty(E, \Sigma, \mu)$ 3. $\|f_n-f\|_{L^1}\to0$ 4. $g_n \to g \text{ }\mu\text{-a.e.}$ 5. $\{g_n\}$ is uniformly bounded. ...
1
vote
0answers
32 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
1
vote
0answers
56 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
1
vote
0answers
22 views

The relationship of L^1(U) and C(U)

Let $U$ be a open set of $\mathbb{R}^n$, C(U) is all continuous functions on U, for example C(0,1), when $U=(0,1)$. And $L^1(U)$ is lp-space where $p=1$. It was said that $L^1(U)$ is the completion ...
1
vote
0answers
55 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
1
vote
0answers
26 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
1
vote
0answers
21 views

Criteria for a parameter-dependent integral to be squared integrable

I'm currently stumbling over a criteria of integrability. Let's suppose we have a real-valued function $h$ defined on the positive real axis which is integrable. We then define the function $K ...
1
vote
0answers
45 views

Norm of multiplication operator

I have that $(X,\Omega,\mu)$ is a sigma finite space, and I have that $g$ is a measurable function. Assume that $fg\in L^p$ for all $1\leq p\leq \infty$. I want to show that $g\in L^\infty$. My idea ...
1
vote
0answers
42 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
1
vote
0answers
43 views

$L\log L$ and $L^p$ embedding

My question is a simple one: I am aware of the embedding $L^p(\Omega)\in L\log L(\Omega)$ for finite measure spaces, with constant $\frac{cp}{p-1}$. Does this embedding hold on for instance, the whole ...
1
vote
0answers
76 views

Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ a.e. on D.

I want to prove this result: Let $D$ be an open subset of $\mathbb{R}^n$, $p \in[1,\infty)$ and $f$ be in $L^p(D)$. Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ ...
1
vote
0answers
22 views

Show $\lim_{k \rightarrow \infty} (f_{n_k}, g) = (f,g)$ for a subsequence $f_{n_k}$ of $f_n$ and $g$ in $L^2([0,1])$

I would like to show that for any sequence $f_n(x) \in L^2([0,1])$ with $||f_n||_2=1$ for all $n$, there exists an $f\in L^2([0,1])$ and a subsequence $(f_{n_k})$ such that for every $g\in L^2([0,1])$ ...
1
vote
0answers
98 views

ess-sup property

Let $(X, \mathcal{A}, \mu)$ be a measured space, and let $f \in L^p(\mu)$. I would like to show that $$ |f| \leq \text{ess-sup}(|f|) \text{ a.e. on } X, $$ where $$ \text{ess-sup}(|f|) := \inf \{ ...
1
vote
0answers
53 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
1
vote
0answers
94 views

Orthonormal basis for this $L_2$ space of probability measures?

I couldn't find this on math.se or by searching the internet. Thanks for any help! OK, so I have a set $\Omega$. For now, let's think of $\Omega$ as finite, consisting of $n$ elements. Now consider ...
1
vote
0answers
62 views

Transpose of the Hilbert-Schmidt operator

Let $X = L^2(\Omega)$, $\Omega \subset \mathbb{R}^N$ be an open set (or a $\sigma$-finite measure space), $B \in L^2( \Omega \times \Omega)$. Then the Hilbert-Schmidt operator $T \in \mathcal L(X)$ ...