Tagged Questions

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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Sobolev Space $W_0^{1,p}(I)$ and the boundary of $I$

Given $I \subset\mathbb{R}$ an open interval, the Sobolev Space $W_0^{1,p}(I)$ is defined as $W_0^{1,p}(I)=\overline{C^1_c(I)}^{W^{1,p}(I)}$ (The closure of $C^1_c(I)$ on the space $W^{1,p}(I)$) . ...
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What is the Stein Interpolation Theorem for any strip?

Stein Interpolation Theorem: Assume $T_z$ is an operator depending analytically on $z$ in the strip $0\leq Re (z) \leq 1.$ Suppose $T_z$ is bounded from $L^{p_0}$ to $L^{r_0}$ when $Re (z)= 0,$ and ...
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Sobolev embedding fails for $p=n$

As everyone knows, the Sobolev embedding fails fails for $n\ge 2$ if we assume $p=n$. The standard example is the function $u(x)=\log \log \bigl(1+\tfrac{1}{x}\bigr)$. This function is obviously ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $||\cdot||_{1,\infty}$) ...
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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 ...
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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 ...
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$L^{\infty}$ 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 ...
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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: ...
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$\forall g\in L^2(\Omega)$ 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Continuity on $L_p$ spaces

Consider a nonlinear and continuos function $f:\mathbb{R} \rightarrow \mathbb{R}$ and we define the functional $$F(u) = \int_{[0,1]^2} f(u(x,y)) dxdy$$ where $u$ is an ...
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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, ...
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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 ...
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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 ...
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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?
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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 ...
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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 $V_{n-2}(r)$. ...
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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 ...
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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. ...
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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 $V = H^1_0(\Omega)$ ...
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Let $f \in L^2(0,T;L^2(\Omega))$ on a bounded set $\Omega$. Does the following limit exist? $$\lim_{p \to \infty}\biggl(\int_0^T \Bigl(\int_\Omega ... 0answers 31 views Extending L^{p} Duality to \sigma-finite Spaces Let 1 \leq p < \infty, (X,\mathcal{M},\mu) be a sigma-finite measure space. Let L be a continuous linear form on L^{p}(X,\mathcal{M},\mu). Then, show that \exists g \in L^{p'} such that: ... 1answer 32 views hint on exercise about weak L^p space I'm working on a problem from Grafakos, Classical Fourier Analysis. Let (X, \mu) be a measure space and let E be a subset of X with \mu(E) < \infty. Assume that f is in ... 0answers 23 views Lebesgue Space/Bochner Space interpolation Theorem I need the embedding, for I\subset\mathbb{R} is a bounded intervall and \Omega\subset\mathbb{R}^n is a bounded domain,$$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{p_2}(I;L^{p_2}(\Omega))\hookrightarrow ...
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 ...
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 ...