Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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Nonnegative eigenvector form the point of view of variational characterization

Let $\Omega$ be a smooth domain in $\mathbb{R}^N$ $(N\geq 3)$. We denote by $G(x, y)$ the Green function for the boundary value problem $$ -\Delta_x G(x, y)=c_n \delta(x-y) \quad \text { in } \Omega, ...
Davidi Cone's user avatar
4 votes
2 answers
462 views

How to get this inequality in Santambrogio's book about optimal transport?

Let $\hat{\varrho}, \tilde{\varrho}$ be probability density functions on $\mathbb R^d$ where $\tilde{\varrho} \in L^{\infty} (\mathbb R^d)$. For $\varepsilon \in [0, 1]$, we define $\varrho_{\...
Akira's user avatar
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3 votes
1 answer
122 views

Oscillation functions and similar constructs

For given $f$ from reals to reals, the associated oscillation function is defined as follows: $$\textstyle osc_f(x):= \lim_{n\rightarrow \infty} [\sup_{y \in B(x, \frac{1}{2^n}) } f(y)-\inf_{z \in B(x,...
Sam Sanders's user avatar
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1 answer
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Possible research directions in analysis? [closed]

I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
TaD's user avatar
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Are analytic solutions for the Navier-Stokes equations sufficient?

Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space. However, I am wondering, whether it is possible to consider just analytic ...
tobias's user avatar
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8 votes
1 answer
485 views

A fractional weighted Poincaré inequality

Does there exists a constant $C>0$ such that $$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$ for all $u\in C^{\infty}_0((-1,1))$?
Ali's user avatar
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8 votes
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223 views

The approximate mean value theorem / Rolle's theorem in pure constructive mathematics

In the replies of this very similar question, there is a fascinating answer that is beautiful in its simplicity. In particular, it seems to use perhaps the most minimal assumptions one can possibly ...
SpectreDNZ's user avatar
1 vote
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59 views

A possible upper bound for a function that satisfies a singular integral inequality

I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality: $$ |v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left( |...
Takieddine Zeghida's user avatar
3 votes
3 answers
267 views

Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure

Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure. Let $n \geq 2$ be an integer, and $E \subset \mathbb R^n$ be a set of finite $\mathcal H^{n-2}$ measure. Suppose $f: \mathbb R^...
Nate River's user avatar
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1 vote
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Dependence of Sobolev embedding theorem constant on smoothness

Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that $$ \|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
user515999's user avatar
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"Calculus" of metric entropy: How does metric entropy behave with respect to binary operators?

This is a general question, more of a reference request. tl;dr: Is there a "calculus" for computing metric entropy bounds? Given a function space $\mathcal{F}$, we may define its metric ...
kim341's user avatar
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1 vote
1 answer
102 views

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 ...
Isaac's user avatar
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Does there exist a continuous choice of maximizing balls for the Hardy Littlewood maximal function?

Note: Here $\overline B_r (x)$ denotes the closed ball of radius $r$ around $x$. Let $f \in L^1 (\mathbb R^d)$. We define the averages $A(x, r)$ for $x \in \mathbb R^d$ and $r > 0$ by $$A(x, r) := \...
Nate River's user avatar
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2 votes
1 answer
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Equivalent characterization of weak derivative in Bochner space

Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff $$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
Mandelbrot's user avatar
2 votes
0 answers
115 views

On the integer of the form p^a q^b closest to a given integer N

If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
Azoth's user avatar
  • 41
1 vote
1 answer
335 views

Topological degree of differentiable map using line integrals?

Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$ I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
António Borges Santos's user avatar
1 vote
1 answer
61 views

Boundedness of maximisers of parametric strictly concave functions

Let $L:[0,1]\times \mathbb R^m\times \mathbb R^n\to \mathbb R$ be defined by $$L(\lambda, x,y):=\sum_{1\le i\le m}\alpha_i x_i + \sum_{1\le j\le n}\beta_j y_j -\sum_{1\le i\le m, 1\le j\le n} p_{i,j}\...
Fawen90's user avatar
  • 975
7 votes
2 answers
955 views

Method of characteristics for higher order PDEs in more than two variables

I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
Puzzled's user avatar
  • 8,842
2 votes
0 answers
112 views

Product of marginals absolutely continuous with respect to a Borel probability measure

Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
KhashF's user avatar
  • 2,588
9 votes
2 answers
538 views

Does this integral condition characterize $L^\infty$?

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ with smooth boundary. For any $f \in L^1 (\Omega)$, is it true that $f \in L^\infty (\Omega)$ if and only if the following condition ...
Nate River's user avatar
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5 votes
0 answers
172 views

Elliptic regularity and Sobolev spaces

Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e. $$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$ where $a$ are ...
G. Blaickner's user avatar
  • 1,137
1 vote
0 answers
97 views

Periodicity in one Fourier variable

Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument. We can thus write $f$ as $$ f(x) = \sum_{n \in \...
António Borges Santos's user avatar
4 votes
0 answers
102 views

Lipschitz extension of a flow can still be a flow?

Consider a map $\Phi: [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$, and assume that there exists a set $U \subset \mathbb{R}^d$ such that $\Phi\rvert_{[0,T] \times U}$ is $L$-Lipschitz. It is well ...
tommy1996q's user avatar
7 votes
0 answers
146 views

The space of analytic associative operations

This question is a follow-up to this old one of mine. Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
Noah Schweber's user avatar
2 votes
0 answers
165 views

finite dimensionality of a subspace of a Banach space

Let $H$ be the space of measurable functions on $(0,1)$ such that $$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$ Let $C>0$ be a constant. Suppose that $W \...
Ali's user avatar
  • 4,067
6 votes
1 answer
307 views

A combinatorial identity involving increasing functions from $\{1, \dots, n\}$ to itself

This is related to the post An order statistics problem with some interesting geometry. The following identity arose in the context of the problem. Fix an integer $N \geq 2$. Let $\mathcal S_N^+$ ...
Nate River's user avatar
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22 votes
5 answers
2k views

PDEs and algebraic varieties

Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
Puzzled's user avatar
  • 8,842
5 votes
1 answer
422 views

Does coefficient-wise limit preserve real-rootedness?

Let $P_n$, $n=1,2,\ldots$ be polynomials with real roots only (and real coefficients), and $P_n$ converge to a non-zero polynomial $Q$ coefficient-wise. Does it follow that $Q$ has real roots only? ...
Fedor Petrov's user avatar
2 votes
0 answers
110 views

Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator $$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$ There are two ...
B.Hueber's user avatar
  • 987
0 votes
0 answers
34 views

It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?

Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
Ilovemath's user avatar
  • 585
7 votes
0 answers
238 views

$C^0$-limit of volume-preserving maps on $\mathbb R^n$

Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
Tian LAN's user avatar
  • 135
11 votes
2 answers
927 views

Twice continuously differentiable implied by existence of limit

I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that $$ \frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x) $$ for all $x\in X$ when ...
Sonam Idowu's user avatar
2 votes
0 answers
40 views

Blow up for certain classes of distributions

Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
Ali's user avatar
  • 4,067
3 votes
1 answer
92 views

Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?

We endow the space $\mathcal P_2^a (\mathbb R^d)$ of absolutely continuous probability measures with finite second moment with the Wasserstein distance $W_2$. Let $\mathcal H (\mu)$ be the relative ...
Akira's user avatar
  • 815
2 votes
2 answers
383 views

In what sense does the Hermite expansion of a bounded smooth function converge?

Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function. If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as \...
Isaac's user avatar
  • 2,727
22 votes
1 answer
4k views

A challenging (for me) limit calculation

How to calculate the following limit $$ \lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?} $$ ${}{}$
C. WANG's user avatar
  • 539
2 votes
1 answer
148 views

A compact embedding claim

Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms $$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$ Let $H_2$ be the weighted Sobolev space with the ...
Ali's user avatar
  • 4,067
1 vote
0 answers
101 views

Proving that a quantity is positive (Gaussian density and Gaussian CFD)

$\def\R{\mathbb R}$$\def\aha{{1/2}}$$\def\maha{{1/2}}$ Hi everyone, I am interested in the following problem: Let consider the heat equation problem: $$\forall (t,x) \in \mathbb{R}_+\times\mathbb{R}, ~...
NancyBoy's user avatar
  • 175
2 votes
2 answers
235 views

Is a simple closed curve always a free boundary arc?

Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points? For a simple closed curve $\...
Pacific saury's user avatar
1 vote
3 answers
175 views

Evaluating a sinusoidal series

Define the sequence of functions $$f_n(x)=\sum_{m=n}^\infty(-1)^m\frac{x^{2m}}{(2m+1)!} {m \choose n} $$ Is there a closed form expression for arbitrary $n$? It is clear that the result should assume ...
DinosaurEgg's user avatar
1 vote
1 answer
79 views

The number of roots of pseudo-exponential polynomials

Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
ABB's user avatar
  • 3,898
3 votes
1 answer
126 views

Weaker version of the lemma of K.L. Chung

Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds: $$...
giorgi nguyen's user avatar
2 votes
1 answer
239 views

Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?

As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
SBK's user avatar
  • 1,141
3 votes
2 answers
198 views

Domain of spectral fractional Laplacian

Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
B.Hueber's user avatar
  • 987
2 votes
0 answers
58 views

A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
IMOS's user avatar
  • 121
5 votes
1 answer
235 views

Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?

Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by $$M f_{\varepsilon} (x) = \sup_{r \...
Nate River's user avatar
  • 4,822
2 votes
0 answers
164 views

When do these ODE have positive solutions?

Consider the ODE \begin{equation} x'' + q(t) x = 0 \end{equation} in the unit interval $(0,1)$, with a potential function $q(t) = 4\pi^2 - \frac{Ct}{(1 - t)^2}$ depending on a positive constant $C >...
Leo Moos's user avatar
  • 4,968
0 votes
0 answers
42 views

Bounds on integral of length of intersection of segments with an arbitrary small set

Consider a cube $C$ with edges of length 1 in $\mathbb{R}^n$, centered in the origin, and consider a set $E$ of measure $\varepsilon$ in the cube. No regularity besides measurability. Now, take a ...
tommy1996q's user avatar
3 votes
2 answers
635 views

Are $L^p$ norms absolutely continuous?

Let $1 < K \leq \infty$, and suppose $f \in L^p (X)$ for all $1 \leq p \leq K$, for $X$ some $\sigma$-finite measure space with no atoms. Question: Is the function $p \to \|f\|_{L^p}$ absolutely ...
Nate River's user avatar
  • 4,822
-5 votes
1 answer
244 views

Calculus based on pdf [closed]

Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
Marius S.L.'s user avatar

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