# Questions tagged [real-analysis]

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

3,095
questions

**-1**

votes

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11 views

### When is weighted average decreasing?

Suppose we have sequences $\{w_n(x)\}_{n\in\mathbb{N}}$ and $\{f_n(x)\}_{n\in\mathbb{N}}$ that are both strictly decreasing in $x$. Further, we have $w_n(x)> 0,f_n(x)\geq 1\quad \forall n\in\mathbb{...

**2**

votes

**1**answer

76 views

### Duality form of $L^q$ norm, without assumption that $\int fg$ defined?

The following theorem is found, for example, in the Real Analysis books by Folland, by Yeh, and (in a slightly different form) by Royden.
Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space.
Let ...

**5**

votes

**0**answers

64 views

### Points where singular sum is small

We consider $x_1,..,x_N$ points in the plane $\mathbb{R}^2.$
We define the sum
$$F(x):=\frac{1}{N^2}\sum_{i=1}^N \sum_{j \neq i} \vert x_i-x_j \vert^{-2}.$$
I am looking for a statement of the ...

**4**

votes

**1**answer

110 views

### Elliptic estimates for self-adjoint operators

Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$
Let $T$ be a densely defined and closed operator ...

**8**

votes

**2**answers

203 views

### Recognizing Lipschitz functions up to change of target metric

Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).
Let $f:K \rightarrow \mathbb{R}^m$ be a continuous function.
Is ...

**3**

votes

**1**answer

121 views

### Isoperimetric inequality for analytic functions on an annulus

Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...

**2**

votes

**1**answer

104 views

### Proof of a discrete isoperimetric inequality

The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions:
$$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{...

**-1**

votes

**0**answers

19 views

### norm inflation for cubic NLS

I'm trying to understand the norm inflation result for the cubic NLS with initial data in negative Sobolev spaces, specifically, Theorem 1.1 in this paper. I'm have the trouble in verifying one ...

**-1**

votes

**1**answer

58 views

### Minimal covering sets of continuous endomorphisms

For any topological space $(X,\tau)$, let $\text{End}(X)$ denote the set of continuous functions $f:X\to X$. We say that ${\cal C}\subseteq \text{End}(X)$ covers $\text{End}(X)$ if for every $f\in \...

**2**

votes

**1**answer

69 views

### Hardy-Littlewood-Sobolev for “componentwise product” of Riesz kernels

Let $d\in\mathbb N$ and $0<\alpha<d$. Define the Riesz kernel $K_\alpha(x):=|x|^{\alpha-d}$, and the associated convolution operator
$$K_\alpha f(x):=\int\frac{f(y)}{|x-y|^{d-\alpha}}~dy.$$
The ...

**1**

vote

**0**answers

36 views

### A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...

**3**

votes

**0**answers

74 views

### Building strong topologies on the space of continuous functions

Let $\{K_k\}_{k=1}^{\infty}$ be a compact exhaustion on $\mathbb{R}^n$ (ie $\bigcup_{k=1}^{\infty} K_k = \mathbb{R}^n$). Equipe each $X_k:=\{f \in C(\mathbb{R}^n):\, \operatorname{supp}(f)\subseteq ...

**20**

votes

**1**answer

336 views

### Is every function $f: \mathbb R \to \mathbb R$ differentiable at at least one point when restricted to some everywhere dense subset of $\mathbb R$?

I was doing some fairly simple research a few hours ago and I almost asked a similar question with the word continuous instead of differentiable in the title, but then I found this question asked by ...

**1**

vote

**0**answers

26 views

### Independence of variables in curvilinear coordinate systems

Let $U$ be a connected open subset of $\Bbb{R}^n$, and let $(\xi_1,\dots,\xi_n)$ be a curvilinear smooth ($C^\infty$) coordinate system on $U$. Suppose $1\leq k<n$. A smooth function $f:U\...

**6**

votes

**1**answer

94 views

### Equivalence of antiderivative in L1 sense and in the usual sense

We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that:
$$
\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0
$$
where $\Vert \cdot \Vert_1$ is the $L_1$ ...

**3**

votes

**0**answers

62 views

### Approximation of Lipschitz functions

Let $X$ be an open set in $\mathbb{R}^N$ and call $C^{1,1}(X)$ the set of functions $X\rightarrow \mathbb{R}$ that are $C^1$ with Lipschitz first derivatives.
I have a $C^{1,1}$ function $f$ and a ...

**0**

votes

**0**answers

254 views

+50

### A function fitting method: can we recover binary step functions?

A function fitting method
Data $(p_i,a_i) \in \mathbb{T}^m\times\{-1,1\}$, $i = 1,2,...n$.
Let $C_{\lambda}(f) = \sum\limits_{i=1}^{n}(f(p_i)-a_i)^2 + \|f\|_{L^2}^2 + \lambda\|\nabla^kf\|_{L^2}^2$ ...

**1**

vote

**0**answers

54 views

### Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$

I'm currently trying to work with the non-local operator given by
$$
(-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy,
$$
where $f :\mathbb ...

**7**

votes

**1**answer

135 views

### When is the cut-locus normal coordinate collared

Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold, $p \in M$ be fixed and let $C_p$ be the cut-locus of $p$.
Other than when $M$ is non-positively curved (in which $C_p=
\emptyset$ by ...

**0**

votes

**1**answer

46 views

### Estimate difference $f(x)\,H(f(x)-M) - g(x)\, H(g(x)-M)$, with $H$ the Heaviside function and $M>0$ fixed, in terms of the difference $|f - g|$? [closed]

Let $f,g: \mathbb R \to \mathbb R$ such that $f,g \in L^\infty(\mathbb R)$.
Fix $M>0$ and let $H$ denote the Heaviside function. How can we estimate the difference
$$f(x)\,H(f(x)-M) - g(x)\, H(g(...

**1**

vote

**1**answer

55 views

### Integral average near a point of dispersion

Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that ...

**-1**

votes

**0**answers

31 views

### Hessian matrix of the function defined with implicit function theorem [closed]

Let $x=(x_1,...,x_n) \in \mathbb{R}^n, y\in \mathbb{R}$ and let $F(x,y)=F(x_1,...,x_n,y) \in C^2(\mathbb{R}^{n+1})$.
Suppose we have all the hypothesis for the existence of the function $f(x)=y$ ...

**0**

votes

**1**answer

64 views

### Distance function and its approximation

An easy and quick question:
Consider a function $u\in C(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$.
Define a function $Q$ that measures the distance of a point $(x,y) \in\mathbb{...

**1**

vote

**1**answer

110 views

### Injectivity of an integral operator

Consider the operator
$$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$
with $k\in L^2((0,1)\times(0,1)).$
I want to know under what assumption the kernel is reduced to zero. i....

**4**

votes

**0**answers

116 views

### A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic

I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...

**-1**

votes

**1**answer

72 views

### Sum of $\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}$ [closed]

I want to calculute or estimate of order $O(n^{2-\varepsilon})$, where $\varepsilon>0$, of the following sum for $0<\alpha<1$
$$\sum_{1\leq k\leq k'\leq n}\frac{k^{\alpha}}{k'^{\alpha}}.$$

**2**

votes

**0**answers

47 views

### Measure of the convex hull of a ball and a point

I need to prove the following statement:
Let $B_s(z)$ be a ball centered at $z$ of radius $s$ s.t. $0\not\in B_s(z)$. Moreover let $K_s(z)$ the convex hull of $\{0\}\cup B_s(z)$.
Then
$$ \...

**3**

votes

**0**answers

91 views

### Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates.
In one of his papers, Lin proves the following result:
Let's consider a ...

**3**

votes

**0**answers

67 views

### Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...

**4**

votes

**1**answer

87 views

### Every convex set is of locally finite perimeter

I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter.
$E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ s.t. the Gauss ...

**0**

votes

**0**answers

86 views

### Equivalence of weak/strong uniform Glivenko-Cantelli classes

Let $(S, \sum, P)$ be a probability space and $H$ a class of real valued measurable functions on $S$. Then we say that $H$ is a weak uniform Glivenko-Cantelli class if for any $\epsilon > 0$
$$ \...

**0**

votes

**3**answers

182 views

### Does the generalised directional derivative satisfy any version of the chain rule?

Is there a chain rule of any kind for the generalised directional derivative (of the Clarke type)? There is certainly a chain rule for the generalised gradient.
The generalised directional derivative ...

**0**

votes

**1**answer

111 views

### Finding the conjugate of a function

I know that the Fenchel conjugate of a function is
$$f^*(x^*) = \sup_x\{\langle x, x^*\rangle - f(x)\}.$$
However, how do I find the Fenchel conjugate of the function
$$f(x) = \frac{1}{p}\sum\limits_{...

**2**

votes

**1**answer

57 views

### Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...

**2**

votes

**0**answers

38 views

### Second derivative estimates for a subsolution of linear elliptic equation

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...

**-2**

votes

**1**answer

46 views

### Does having the derivative in the limit suffice to solve the function at the limit? [closed]

Suppose that I have a function $f(x, \epsilon)$ and I know that
$$
\lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x).
$$
Now let $g(x)$ be the function whose derivative appears above. How can I ...

**0**

votes

**0**answers

65 views

### Existence of the inverse Fourier transform, Carr Madan

I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant.
So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...

**2**

votes

**0**answers

64 views

### First Dirichlet eigenvalue below second Neumann eigenvalue?

Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...

**4**

votes

**0**answers

196 views

### An inequality in harmonic analysis with the BMO flavour

I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes).
Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in
...

**2**

votes

**0**answers

202 views

### Cardinal numbers and the Bolzano-Weierstrass theorem

Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...

**1**

vote

**0**answers

58 views

### Intuition from Hopf lemma (boundary point lemma )

Consider the classical boundary point lemma:
Let $L$ be an elliptic operator.
Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...

**1**

vote

**1**answer

91 views

### 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 $S_{r,s}^{\theta}$: $\...

**3**

votes

**1**answer

171 views

### Aleksandrov maximum principle for semi-convex function

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...

**0**

votes

**0**answers

83 views

### Are periodic real-analytic functions dense in the Frechet space $C^\infty(S^1)$?

The question is as above:
Are periodic real-analytic functions dense in the Frechet space $C^\infty(S^1)$?
Here I gave $C^\infty(S^1)$ the metric topology in which the convergence is the uniform ...

**1**

vote

**0**answers

32 views

### Example of periodic semidifferentiable function without absolutely convergent Fourier series

Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but
with a non-absolutely convergent Fourier ...

**4**

votes

**0**answers

79 views

### Asymptotic of a functional as $x\rightarrow \infty$

Consider the following functional :
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},
$$
where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...

**0**

votes

**0**answers

109 views

### Making area/volume calculations that use SIA rigorous

There are some intriguing "proofs" using Smooth Infinitesimal Analysis of theorems concerning areas and volumes. Some examples:
A proof that $\sin'(0) = 1$.
A proof that the surface area of a cone is ...

**2**

votes

**2**answers

92 views

### Convergence of fraction of expectation values

Let $X_1,...,X_n$ be iid normal random variables.
I am looking for a strategy to establish the following limit for fraction of expectation values
$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i ...

**1**

vote

**0**answers

35 views

### Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...

**2**

votes

**1**answer

49 views

### Estimates on divergence-type operator for the matrix

Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form
$$
{\rm div}(Av)=f
$$
where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...