All Questions
Tagged with lp-spaces lebesgue-integral
509
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How do we know the dual pairing between Lp spaces is well defined? [closed]
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
- 365
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$L_p$ norm estimate of a sum
Let $R>0$, and $(B_k)_{k \in \Bbb{N}}$ a collection of disjoint balls of radius R and let $f$ be a measurable function on $\mathbb{R}^n$ of the form
$$f = \sum_{k=1}^{\infty} a_k X_{B_k}$$
for ...
- 22k
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Given $f \in L^p_{\text{loc}}(\Omega) \setminus L^\infty(\Omega)$, does it follow that $A \cap S(f,K) \neq \emptyset$ for all $K > 0$?
Context. Throughout this post I will be dealing with the Lebesgue measure over $\mathbb R^n$. Moreover, I denote the measure of a measurable set $E \subset \mathbb R^n$ by $|E|$ and $\Omega \subset \...
- 1,141
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Does convolution preserve local integrability in the sense that if $f \in L^p_{\text{loc}}$ and $g \in L^1$, then $f \ast g \in L^p_{\text{loc}}$?
Consider the Euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is a fixed integer, equipped with the Lebesgue measure. Let $L^p_{\text{loc}}(\mathbb R^n)$($1 \leqslant p < \infty$) denote the ...
- 1,141
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Equivalent characterisation of the space $L^p_{\operatorname{loc}}(\Omega)$, where $\Omega$ is a non-empty open subset of $\mathbb R^n$?
Consider the euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is an arbitrary integer, equipped with the usual Lebesgue measure. Moreover, let $\Omega \subset \mathbb R^n$ denote an arbitrary ...
- 1,141
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Applying the Lebesgue Dominated Convergence Theorem without explicitly building a sequence of functions.
Let $\Omega \subset \mathbb R^n$ denote an arbitrary non-empty open set.
The classical formulation of the Lebesgue Dominated Convergence Theorem (LDCT, for short) is as follows.
LDCT. Let $(f_n)_{n \...
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How pathological are the $L^{p}$ functions?
The motivation for my question is simple: for Riemann integrable functions, we had, at worst, almost everywhere continuity. It's not that bad behavior wise, but it is integration wise. So we come up ...
- 161
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Boundedness of an Integral Operator on $L^p(\mu)$
I realize this question has been asked way too many times, for instance, it's this exact problem which I will put below for convivence:
Let $(X,\Omega, \mu)$ be a $\sigma$-finite measure space with $...
- 1,399
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With finite $\mu$, suppose $f_n$ converges in measure to $f$, and for all $n$, $||f_n||_2\leq 1$. Prove $||f_n-f||_1\rightarrow 0$
This is actually a part b), where part a) was to show $f\in L^2$. This was simple: as $f_n$ converges in measure to $f$, there exists a sub-sequence which converges $\mu$-a.e., thus Fatou's gives: $\...
- 150
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Does the weak limit of a sequence in $L^2([0,1])$ vanish on the limit set of vanishing sets?
Suppose $h_n$ is a sequence of non-negative functions in $L^2([0,1])$ converging weakly to $h$ (i.e., for every $g\in L^2([0,1])$ it holds $\int g\cdot h_n \,d\lambda \to \int g\cdot h\, d\lambda$).
...
- 365
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$fg \in L^1$ for every $f \in L^2$ implies $g \in L^2$
Suppose that $f g$ is in $L^1([a,b])$ for every $f$ in $L^2([a,b])$, I understand that this implies that $g$ is in $L^2([a,b])$. Is this assertion true and, if so, how do I prove it?
This question was ...
- 119
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Convergence on $L^1$ space
I come across this problem while reading hints for Exercise 3.15 (Brezis):
Let $\Omega=(0,1)$ and sequence $f_n$ defined by $f_n(x)=ne^{-nx}$. Prove that:
$$\displaystyle \int_{\Omega} \varphi f_n\...
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119
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Define multiplication on $L^1(\mathbb{R})$
I want to define multiplication on $L^1(\mathbb{R})$, of course, convolution is an allowed multiplication which makes $(L^1(\mathbb{R}),*)$ is a Banach algebra, I want to know if there are other ...
- 11
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$p$-integrability of a certain Bessel function
I read the following statement in a book:
$$ \frac{J_{\frac{n-2}{2}}(2\pi r)}{r^{\frac{n-2}{2}}}\text{ is in }L^p\text{ if and only if }p>\frac{2n}{n-1}\text{ ;} $$where $n\geq3$, $r$ is non-...
- 371
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If $u\in L^\infty$, does the inequality $\int_\mathbb R u(x) |v(x)|^2 dx\le \|u\|_{L^\infty} \int_\mathbb R |v(x)|^2 dx$ hold true?
Let $u, v:\mathbb R\to\mathbb R^3$. Suppose that $u\in L^\infty(\mathbb R, \mathbb R^3)$ and $v\in L^2(\mathbb R, \mathbb R^3)$.
Consider the integral
$$\int_\mathbb R u(x) |v(x)|^2 dx.$$
The question ...
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