All Questions
3,628 questions with no upvoted or accepted answers
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The category of discontinuous Banach spaces
A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
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108
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A bilinear estimate in Lp space
Let $\varphi(D)$ be a Fourier multiplier with symbol $\varphi(\xi) = \xi/(1+|\xi|^2)$. It's easy to prove that
\begin{equation}
\|\varphi(D)u^2\|_{H^s(R)}\lesssim \|u\|^2_{H^s(R)} \quad (*)
\end{...
- 425
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125
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Convergence of solutions of the volterra integral equation with convergent kernels
Consider the following Volterra integral equation
$$
g(t) = \int_0^t K_n(t,s)w_n(s) ds
$$
where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
- 111
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163
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Reference : Special case of Banach-valued function integration by parts
Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in ...
- 171
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93
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Schoenberg correspondence on $L^p$
Schoenberg correspondence states that $\psi: \mathbb R^d\longrightarrow \mathbb C$ is conditionally positive definite and hermitian if and only if $e^{t\psi}$ is positive definite for each $t>0$. ...
- 630
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115
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Free Endpoint of Minimization Problem
Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > 0$...
- 516
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Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$
Consider the equation
$$u' + Au = f$$
$$u|_{\partial \Omega} = 0$$
$$u(0) = u_0$$
where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...
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525
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Separability of the space $C(C[0, 1], \mathbb{R})$
Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that ...
- 199
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Defining density of a random function using Radon-Nikodym Theorem
Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...
- 213
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decomposition of tempered distributions by entire analytic functions
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...
- 1,051
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91
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Representing a Pullback as an Infinite Matrix
Let $M$ and $N$ be manifolds and let $T: M \to N$ be a bijective map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) be the space of all functions from $M$ (resp. $N$) to $\...
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Properties of a function from its pullback
Edit: I have now removed the duplication previously referred to. Thank you.
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{...
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Request for reference about bound on zeroes of the Laguerre polynomials
Consider the sequence of polynomials given as, $p^{a}_k (x) = (1 - a \frac{d}{dx})^k x^n $ for some parameter $a>0$ and $k$ being a positive integer. For any positive integer $d$ it seems to be ...
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Difference Quotients Evans
There is a theorem in Evans partial differential equation book as follows:
if $u \in W^{1,p}(U)$ then for each compact $V$ in $U$ we have that:
$ |D^hu|_{L^p(V)} \leq C |Du|_{L^p(U)} $
for all $ |h|...
- 4,143
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Rate of convergence in narrow convergence
Does anyone help me in the following question?
I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...
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139
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Can we define log-convex operators?
Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq [...
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205
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Inversion of Fourier transform of a multivariate gamma distribution in polar form?
Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on $\...
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96
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Reference needed for Hilbert-Schmidt result regarding basis of $V \subset H$
I am seeking a reference that says:
If $V \subset H \subset V^*$ is a Gelfand triple with all spaces Hilbert spaces and if $V \subset H$ is a compact embedding, then there is a basis of $V$ which ...
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136
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A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
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Is a finite depth-index irreducible subfactor, intermediate of a depth ≤ 3 one?
Let $(N \subset M)$ be a finite depth-index irreducible subfactor.
Main question: Is $(N \subset M)$ the intermediate of a finite index depth $\le 3$ irreducible subfactor?
(In others words, is ...
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182
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Laplacian mapping on various function spaces
I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< \infty$...
- 539
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218
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Compact embedding
Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact.
Is it true that the embedding $H^1_0(\Omega) \rightarrow L^2_K(\Omega)$...
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101
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On existence of right units with control of their norms
Does, there exist a Banach algebra with a family of right units with norms converging to 1, but without right unit of norm 1?
- 1,697
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153
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On sequence of functions $(h_n)$ satisfying $\Vert\sum_{n=1}^\infty f * h_n\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert$ for all $f\in L_1(G)$
Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property
$$
\left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert
...
- 1,697
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$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?
Let $\mathbb T$ be a circle group.
In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, $L^...
- 1,051
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123
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$L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?
Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact (...
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Clarkson's inequalities for Banach space valued functions
In standard analysis, Clarkson's inequalities expresses the norms of the sum and difference of two functions in $L^p$ in terms of the norms of the individual functions. In particular, one may use the ...
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198
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Measurability of a map that takes a functional to its composition with a linear operator
Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated.
Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with the ...
user12713
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Comparison principle using truncation for porous medium equation
For a porous medium equation (eg. $u_t - \Delta \Phi(u) = f$), is it possible to obtain a comparison principle for very weak solutions (eg. if $u_0 \geq 0$ and $f \geq 0$ then $u \geq 0$ a.e.) using ...
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170
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A question about Smulian lemma
Smulian lemma says Let $(X, ||.||)$ be a Banach space and$(X^*, ||.||^*)$ and let $x\in S_X=\{x\in X:||x||=1\}$ then
(i) $||.||$ is Frechet diffrentiable at $x$ iff $\lim\limits_{n\to\infty}||f_n-...
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Is this function space a "classical" Sobolev space?
I apologise if this is indeed classical but my functional analysis is quite rusty...
My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...
- 4,123
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Determining the exact form of a projection in a Hilbert space
Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$
where $\mathcal{L}^2[0,T]$ is the set of Lebesgue square-...
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218
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Is the closed ball of a normed space closed in any Hausdorff locally convex topology, weaker than the norm topology?
Assume that we have a normed space $X$ and a subspace $Y$ of $X^{*}$ such that $Y^{\perp}=\{0\}$. They form a non-degenerate dual pare.
Moreover, $\|y\|=\sup_{x\in B_{X}}|\langle x,y\rangle|$, where $...
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Differentiable Path of Operators and their Inverses
Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
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convergence of supergradient
Let $\{g_n\}$ be a sequence of concave functions defined on $\mathbb{R}$ and set
$$\lambda_n(x)=\lim_{\Delta x\to 0+}\frac{g_n(x+\Delta x)-g_n(x)}{\Delta x}$$
Assume there exists a concave function ...
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217
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convergence of concave envelope
Let $\{f_n\}$ be a sequence of uniformly upper bounded functions defined on $\mathbb{R}$ s.t. for every $x\in\mathbb{R}$
$$f_n(x)\to f(x),~ n\to\infty$$
Define $g_n$ and $g$ as the concave envelope ...
- 1,835
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380
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If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?
I have stumbled upon the following problem during my research:
Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
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Looking for CDFs that I can integrate a particular transformation of
I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate
$$ \int_{-\infty}^t \lambda(a(x+b))dG(x), $$$a>0,b\in\Re$. As far as I can tell, there exist no functions that ...
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Existence of orthogonal projections generating Von Neumann algebras
Let $V$ and $V'$ be Abelian Von Neumann algebras of projections on some Hilbert space $H$, and let $V_{1}$ and $V_{2}$ be minimal sub-algebras of $V$ and $V'$ generated by projections $P_{1} \in V$ ...
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Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
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134
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on high order Laplacian
Roughly speaking, we have good understanding of the solution to heat equation $u_t-\Delta u=0$, on bounded or unbounded domain. For example, we know the decay rate, we know it generates analytic ...
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127
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Mixed Tsirelson Norm
A couple of days ago I posted this question on Mathematics Stack Exchange. Surprisingly, so far, I haven't received any answers or comments about it (besides my own possible answer). Maybe I can get ...
- 125
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331
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Relationship between weak Lp and strong Lq topologies for q<p
Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
- 203
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74
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Closed for the motion of an interacting particle system
I am dealing with interacting particle systems approximately in the sense of http://www.math.vu.nl/~rmeester/onderwijs/Interacting_Particle_Systems/liggett.pdf p. 5 except I am reading a book by the ...
- 277
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198
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Passing to the limit in a PDE (subsequence problems)
For $w \in L^2(0,T;H^1)$, consider the PDE
$$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$
where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and $f$...
- 155
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191
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A "coarse" version of position operator and its properties
I am wondering if someone has ever studied the following operator in the context of quantum mechanics:
$$Q : L^2(\mathbb{R}) \to L^2(\mathbb{R})$$
$$(Q f)(x) := s(x) f(x),$$
where $s(x)$ is the ...
- 1,435
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84
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Parabolic partial differential equation, initial conditions
Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
- 11
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164
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How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
- 1,051
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246
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Fractional Derivatives Of Sums
I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...
- 96
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103
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Regularity of weak solutions for a quasilinear problem
Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in C^{1,\beta}_0(\...
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