All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
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Young's convolution inequality for weighted norms
Young's convolution inequality states that, for $1/p+1/q=1/r+1$ ($1\leq p,\, q, r\leq \infty$), we have $$\lVert f * g \rVert_r \leq \lVert f\rVert_p \lVert g\rVert_q.$$
It is implicit here that the ...
- 75
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Sobolev embedding in the space of continuous functions [duplicate]
Let $I = \mathbb{R}$ and let $W^{1,2}(I,\mathbb{R})$ be the Sobolev space of function from $I$ to $\mathbb{R}$ (one time weakly differentiable and contained in $L^{2}$) and $C^{0}(I,\mathbb{R})$ be ...
- 11
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380
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Is this operator invertible?
Let $T(t)$ be a strongly continuous semi-group on a Banach space $X$, and let $A(\cdot)\in C(0,\tau; \mathcal{L}(X))$ for some $\tau>0$. The operator $G:C(0,\tau;X)\to C(0,\tau;X)$ maps every $h\in ...
- 395
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337
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Bochner measurable; continuous operator
It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $...
- 536
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76
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Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 765
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133
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A question about the maximal function
Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
- 155
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An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 825
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A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$",
but it seems much more nontrivial than I expected... so, I post an ...
- 3,477
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143
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$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
- 101
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2k
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Product of Dirac delta function
The following equation may be meaningful, but how can we make it well-defined
$$\delta(x-a)\cdot\delta(x-b)=0$$
Question: How do we defined this equation? Or more broadly define product between ...
- 13
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210
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On a property of complex exponentials
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
- 4,143
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Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$
Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$.
To prove that, as $\epsilon \to 0$, the weak solution ...
user139844
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135
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Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$
Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that
$$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$
and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
- 123
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198
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Convergence of the regularized gradient of a Lipschitz function
Let $\varphi:\mathbb R^d\to\mathbb R_+$ be given as
$$
\varphi(x) := \begin{cases}
c\exp\big(1/(|x|^2-1)\big) & \mbox{if } |x|\le 1 \\
0 & \mbox{otherwise},
\end{cases}
$$
...
user128095
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366
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Can $X^*$ characterize a smooth norm on $X$?
I conjecture following statement in the Banach space $X$ with norm $\|.\|$.
Let $B \subseteq X$ be the unit closed ball in $X$ then,
$\|.\|$ is smooth, in sense $\|.\|$ is differentiable at any ...
- 369
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210
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Eigenvalues of the double layer potential
Consider the double layer potential $K: L^2(S^2)\to L^2(S^2)$
$$(Kf)(x)=\int_{S^2}f(y)\frac{\partial}{\partial v_y}E(x,y)dS_y,$$
where $E(x,y)=||x-y||^{-1}$ and $\frac{\partial}{\partial v_y}$ means ...
- 171
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2
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923
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Spectrum of Mathieu equation
I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
user37929
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276
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Question on Morse inequalities
I want to understand why: From K.C Chang's book "Infinite Dimensional
Morse Theory and Multiple Solution Problems":
if i have
then $(4.1)$ is formal : it means that
EDIT1: $(4.1)$ tel us that $\...
- 277
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859
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Continuous and dense embeddings and the density of sets in Hilbert space
Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose $B$...
- 113
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3k
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In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
- 11
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3k
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Is point to set distance continuous?
Assume $\mathbf{d}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}_0^+$ is a metric such that the function $\psi(x)=\mathbf{d}(x,y)$ for any $y\in\mathbb{R}^n$ is continuous in the Euclidean ...
- 27
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91
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Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
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121
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An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,143
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162
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Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,104
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249
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Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points
Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds.
My question: Does a 'reasonable' sufficient ...
- 671
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Unable to understand an application of Minkowski's inequality
Consider the following exerpt from the paper "Non-linear Quantum Processes" by Segal:
with the norm $\|F\|=\left(\int\|F(x)\|^p \, d x\right)^{1 / p}$, then the operator $T_1^{\prime}: F \...
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120
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Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
- 227
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Does convolution with heat kernel converge to pointwise evaluation?
Let $G(t, x) := \frac{1}{\sqrt{4 \pi t}} \exp\left( -\frac{x^2}{4 t }\right)$ for all $(t, x) \in (0, T) \times \mathbb{R}$ be the fundamental solution to the heat equation $\partial_tu = \partial_{...
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131
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A generalized form of the approximation to identity?
This question is an extension of the one I posted on ME: https://math.stackexchange.com/questions/4701500/if-alpha-nx-int-lvert-x-y-rvert-leq-1-n-lvert-x-y-rvert2-d-muy
It might be elementary for here,...
- 3,477
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160
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On an integral equation
Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...
- 4,143
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102
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Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?
I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$.
I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...
- 3,477
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Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?
Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
- 399
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588
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Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
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143
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A question on the self-adjointness of an operator
Given a Hilbert space (separable) $\mathcal{H}$ with an orthonormal basis $\{e_i\}_{i=1}^{\infty}$, define an operator $T$ with domain $\mathcal{D}(T)$ equal to the span of $\{e_i\}$ by $Te_i:=\...
user471371
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412
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Support of a measure
Let $T:X\to X$ be a continuous function on a compact manifold $X$ and let $\text{Leb}$ be the Lebesgue measure normalized so that $\text{Leb}(X)= 1$. We denote by $\mathcal{M}(X)$ the space of all ...
- 1,043
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225
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How to prove the reverse Hölder inequality for Laplace equations?
Let $ u\in H^1(2B) $ be a weak solution of $ \Delta u=0 $ in $ 2B $, where $ B=B(0,1) $ is a ball with center $ 0 $ and radius $ 1 $. Then there exists some $ p>2 $ such that
\begin{eqnarray}
\left(...
- 1,219
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195
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Eigenvalues of operator
In the question here
the author asks for the eigenvalues of an operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
Here I would like to ask if one can extend ...
- 192
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287
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Examples of $C^{k,1}$ functions which are not $C^{k+1}$?
I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which:
$$
\|...
- 5,405
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106
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'Partial boundedness' of continuously parametrised power series
Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space.
Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by
$...
- 463
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148
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Understanding a family of Sobolev-type inequalities
I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following:
Denote the following inequality as $S_{r,s}^{\theta}$: $\...
- 11
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185
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Interpolation of $L^p$ spaces
Let $\Omega_x$ and $\Omega_y$ be sets of finite Lebesgue measure.
We can then look at the space $X_1:=L^2(\Omega_x \times \Omega_y).$
This space is contained in the larger space
$$X_0:=L^2(\...
user69109
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1
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83
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Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set
I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
- 401
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1
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245
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Evaluation of a double definite integral with a singularity
How to compute the
$$\int_{0}^{1} \int_{0}^{1} \frac{(\log(1+x^2)-\log(1+y^2))^2 }{|x-y|^{2}}dx dy.$$
Is it possible to compute the integral analytically upto some terms. I believe it should involve ...
- 407
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154
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BV function with absolutely continuous divergence
Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and ...
- 839
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2
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138
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Unboundedness of the Sobolev norm of a sequence : Does it follow from Sobolev embedding?
Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...
- 698
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124
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Do functions exist and are they dense? Or does it depend on the basis?
Consider an orthonormal basis $(\varphi_n)_{n \in \mathbb N}$ of $L^2(\mathbb R).$
We consider the functionals $\Phi_n$ given by $$ C^b(\mathbb R) \ni f \mapsto \left\langle \varphi_n, f \varphi_{n+1}...
- 25
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1
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737
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$L^2$ function in Schwartz space?
Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...
- 177
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2
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434
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Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix [closed]
I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$
is it sufficient to prove the norm of ...
- 159
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642
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Interchange of integration order (of a not absolutely convergent integral with sinus)
Can we interchange the integral order of this integral to start integration on $x$ ? (Taking $g$ and $f$ two functions of rapid decrease which are $o(x^2)$ near zero)
$$A=\int_{0}^\infty \int_0^{\...
- 1,199
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2
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333
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What gives a "Parseval like" equation mixing cosine and sine Fourier transforms ?
Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by:
$$\mathcal{F}^c [f (x)]=\int_0^{\infty} f(t) \cos(xt) dt $$
$$\mathcal{F}^s [f (x)]=\...
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