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 classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
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$ L^p ( X ) \cap L^{\infty}( X) $ is a Banach space with respect only to the $p$-norm $\| \cdot \|_p$, $p<\infty$?
The space $πΏ^π(π) \cap πΏ^\infty(π)$, $p<\infty$, with the norm $||π||_{πΏ^π \cap πΏ^\infty}=||π||_π+||π||_\infty$ is a Banach space. I imagine that if we remove the norm $||π||_\infty$ ...
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Eliminating Neumann boundary condition for elliptic PDE
In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:
I am now wondering if this also works with Neumann boundary ...
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Is there a smooth function, which is in $L^1$, but not in$L^2$? [closed]
I am studying measure theory. While going over $L^p$-spaces I asked myself, whether there is $f\in C^\infty(\mathbb{R})$ s.t. $f\in L^1(\mathbb{R})\setminus L^2(\mathbb{R})$? I assume there could be ...
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Spectrum of the laplacian outside of a compact
Let us consider $A$ a translation invariant lower semi-bounded operator on $L^2(\mathbb{R}^n)$ with domain $D(A)$ and with empty discrete spectrum (I exclude bound states). I have the following ...
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Linear Analysis β Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$
Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$?
I know how to prove for $p=2$ ...
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Equivalence of Fourier Transform on $\ell_2(\mathbb{Z}_+)$ and $L_2(\mathbb(R)_+)$ via equivalence of $H_p( \mathbb{D})$ and $H_p(\mathbb{C}_+)$?
Throughout I'll use the fact that the Hardy space $H_2$ is the set of $L_2$ functions on the boundary with vanishing Fourier coefficients.
We know that the Fourier Transform is an isometric ...
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Weighted $L^2$ space on Torus.
I'm studying weighted $L^2$ spaces in the circle $[0,2\pi]$
Definition 1
A weight is a function $w\colon [0,2\pi]\to \mathbb{R}^+$ (non negative)
Definition 2 The weighted $L_w^2([0,2\pi])$ is defined ...
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What justifies the use of global coordinates when computing the $L^p(\mathbb{T}^n)$ norm?
Consider the $n$ dimensional torus $\mathbb{T}^n$. The $L^p$ spaces over $\mathbb{T}^n$ is defined as consisting of an equivalence class of functions satisfying:
$$\int_{\mathbb{T}^n}|f|^p < \infty....
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Weighted inequality on torus
In the Torus (circle). Let $[0,2\pi]\to\mathbb ]0,\infty[\colon \theta\mapsto w(\theta)$ a weight function, i.e. nonnegative and integrable on $[0,2\pi]$. If $\mathbb{Z}\to\mathbb{R}\colon k\mapsto m(...
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What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?
I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.''
Here's an attempt at a ...
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Is there a Hilbert space of HenstockβKurzweil square-integrable integrable functions?
As is well-known, the space of square-integrable functions (say, on $[0,\,1]$) where the integral is a Riemann integral is not complete. If one completes it, one obtains the $L^{2}([0,\,1])$ Hilbert ...
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Limit depending on parameter and $L^1$ function
What is the $\lim_{n\to\infty} n^a\int_0^1 \frac{f(x)dx}{1+n^2x^2}$ depending on $a\in\mathbb{R}$, if $f\in L^1(0,1)$?
By Banach-Steinhaus theorem I deduced that the limit is zero for $a\leq 0$, but I ...
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Auxiliar inequality for Rellich-Kondrachov theorem
To prove the Rellich-Kondrachov Theorem it is used the following statement
If $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then $||\...
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$L^{\infty}$ (uniform) decay of Dirichlet heat equation $u_t=\Delta u$
Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^N$. Consider the following initial-boundary value problem for the heat equation:
\begin{equation}
\begin{cases}
u_t=\Delta u\quad\quad\quad\;...
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Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
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