# Tagged Questions

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### The trivility of Besov space for large parameter

For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define $$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$ and ...
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### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definition We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...
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< ... 0answers 150 views ### Is there an asymptotic bound for this oscillatory integral? I have an oscillatory integral: $$\int u(x,y) e^{i\lambda f(x,y)} dx$$ with$f(x,y)\in \mathbb{C}^{\infty}$a complex-valued function in a neighborhood of$(0,0)$satisfying: $$\text{Im} f \geq ... 1answer 186 views ### Spectrum of this ODE I noticed something interesting studying this Sturm-Liouville Problem:$$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ... 0answers 75 views ### Variational Principle for a System of Differential Equations I am studying a differential operator of the form $$L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ... 0answers 85 views ### elliptic regularity when right hand side in weak L^p I am interested in the following question (whose answer i assume is well known) but just not by me. Suppose u,f are smooth functions defined on B_1 and \Delta u = f in B_1 with u=0 on ... 0answers 54 views ### Holder continuity of Poisson equation with divergence free drift I am interested in the following PDE. Suppose u_m is a smooth solution of a elliptic equation of the form$$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$with u_m=0 on ... 2answers 208 views ### Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof) Let m \geq 2 and let m' be its conjugate. Let w_j for j=1, ..., k be a basis of H_1 \cap L^{m'}. The task is to show that there is a u(t) \in \text{span}(w_1, ..., w_k)=:A such that ... 2answers 352 views ### 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) = ... 1answer 196 views ### Can I approximate Schwartz functions which integrate to zero by C_0^\infty functions which integrate to zero? Let X be the closed subspace of Schwartz space \mathcal{S}(\mathbb{R}^N) defined by \begin{equation*} X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}. \end{equation*} My ... 1answer 242 views ### smooth Luzin theorem For measurable functions f(x), g(x) on [0,1] define the distance \rho(f,g) as a Lebesgue measure of the set \{x:f(x)\ne g(x)\}. Then Luzin's famous theorem states that C[0,1] is dense with ... 2answers 285 views ### Abstract ODE; PDE; uniqueness of solution I have a somewhat vague question regarding an abstract ODE in a Banach space. Suppose A:D(A) \subset X \rightarrow X is some linear operator (let's assume it's closed) and maybe add some other ... 2answers 612 views ### What can be said about the Fourier transforms of characteristic functions? What can be said about the Fourier transform of the characteristic function 1_A, where A\subset \mathbb{R}^n is of finite Lebesgue measure? In particular, What properties are common to ... 1answer 203 views ### Theorem with an example [closed] i have this theorem in the paper they gives an example: but here H_1 is not satisfied ! How to correct it please? 2answers 151 views ### Existence for ODE in Banach space (accretive operators and Crandall-Liggett) There is a theory of mild solutions u \in C^0(0,T;X) where X is a Banach space for equations of the form$$\frac{du}{dt} + Au = f$$where A is an accretive nonlinear operator under some ... 1answer 160 views ### A differentiable version of the Michael selection theorem Assume that X and Y are Banach spaces and T:X\to Y is a bounded surjective linear map. Is there a Gateaux differentiable function g:Y\to X such that T\circ g=Id_{Y}? 1answer 229 views ### A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable Assume that X is a Banach space. Is there a continuous map f:X\to X such that f is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ... 0answers 215 views ### Inverse Transpose of Jacobian Matrix Let f:\mathbb{R}^n\mapsto\mathbb{R}^n be a bijective function. Fixed a\in\mathbb{R}^n. For any x closes to a, using Taylor's series we can approximate f(x) by f(x)\approx ... 1answer 133 views ### Is the speed of a curve in \ell^\infty zero a.e. if the derivative of each component is zero a.e.? Let A be an \mathcal{H}^1-measurable subset of \mathbb{R} and \gamma \colon A \subseteq \mathbb{R} \to \ell^\infty be a Lipschitz mapping with the Lipschitz constant L . Also, assume ... 0answers 95 views ### A two dimensional integral equation I have the following integral equation: \phi(x, y) = \frac{a}{x-y} \int_y^x \phi(s, y) ds + \frac{b}{x-y} \int_y^x \phi(x, s) ds where a > 1 and b> 1 are constants, and x \geq y. The ... 0answers 152 views ### matrix Khintchine inequality The usual Khintchine inequality says that if \{\epsilon_n\}_{n = 1}^N are i.i.d. random variables with \mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2} for each n then \begin{equation*} \left( ... 2answers 410 views ### What is the translation in Fourier transform for a function to have exp. decay at x\to -\infty It is known that smooth functions with exponential decay at \pm\infty are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ... 2answers 2k views ### Dynamical properties of injective continuous functions on \mathbb{R}^d Let \varphi:\mathbb{R}^d\to\mathbb{R}^d be an injective continuous function. Denote by \varphi_n the n-th iterate of \varphi, i.e. \varphi_n(x)=\varphi_{n-1}(\varphi(x)) for all ... 3answers 254 views ### What are the basis functions for a product space? Let X=L^1\left([0,1]^3\right), for numerical purpose, what are the possible basis function for X? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ... 2answers 195 views ### specific improper integral involving erf I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help:$$ \int_{1}^{\infty} ... 0answers 90 views ### growth bound for solution of an ordinary integro-differential equation I am considering the following ordinary integro-differential equation $$A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2$$ where $$A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2 + ... 1answer 283 views ### Pseudo-differential operators with compactly supported symbols If the symbol p(x,\xi) of a pseudodifferential operator P has compact x-support, then for any Schwartz function f, Pf has compact x-support. Is the reverse true? Namely that if some PDO ... 0answers 153 views ### A contradiction to do with continuity? (involves chain rule) Suppose for each t, S(t) \subset \mathbb{R}^n is a domain (hypersurface). We have a diffeomorphism D^0_t:S(0) \to S(t) for each t such that it solves the ODE$$\frac{d}{dt}D^0_t(\cdot) = ... 0answers 199 views ### decreasing rearrangements: why the asymmetry of measure-preserving maps? Ryff proved in 1970 that the decreasing rearrangement$f^*$of a, say, continuous function$f:[0,1]\to\mathbb{R}$admits a measure preserving map$\phi$such that$f=f^*\circ\phi$. In general it is ... 1answer 292 views ### On the convergence of the the function series$\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$Let$f$be a smooth real function defined around origin. If we informally differentiate from the series$\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$term by term we get ... 0answers 82 views ### Conditions on a measure to satisfy certain relation on moments. Suppose we have a measure$\mu$on$\mathbb R_+$such that$\forall s>-1t^s\in L^1(\mathrm d\mu(t))$. I'd like to impose some conditions on$\mu$so the function $$f:p\to \frac{\int_0^\infty ... 2answers 261 views ### Bruhat-Schwartz functions and derivatives in p-adic numbers First of all, I am not an expert in neither classical, nor p-adic functional analysis, but anyway, I stumbled over the following lately: Let \varphi:\mathbb{Q}_p\rightarrow\mathbb{C}. ... 0answers 204 views ### Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let M be a compact, orientable n-dimensional ... 0answers 92 views ### Fundamental solutions for degenerate elliptic equations Hello, I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elliptic equations L = ... 1answer 189 views ### Variation on Fatou's lemma for Sobolev norms Recall that Fatou's Lemma says that for every sequence f_n of non-negative measurable functions$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$If I am not ... 1answer 295 views ### Could we interpolate the compactness of compact operators? Classical theorems of Marcinkiewicz and Riesz and their extensions to general Banach spaces by CalderÃ³n, Lions, Peetre, et al. allow us to interpolate the continuity of two operators, viz., the ... 2answers 274 views ### A sufficient condition for a probability measure to have compact support Consider a probability measure \mu on, let's say, \mathbb R. Is there a necessary and sufficient condition so that \mu has compact support Supp(\mu) ? I agree this question is too vague, ... 2answers 426 views ### For what spaces is the Hardy-Littlewood maximal operator of strong type (p,p) if and only if p > p_0 > 1? (This is essentially a continuation of my previous question, here.) Let (X,d,\mu) be a metric measure space, i.e. \mu is a Borel measure on the metric space (X,d). Further assume (though you ... 3answers 1k views ### Does the derivative of log have a Dirac delta term? Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics": \frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x), see http://adsabs.harvard.edu/abs/1947pqm..book.....D ... 1answer 150 views ### weak*closure of {f:||f||=1} in dual. What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology. So what is the weak* closure of this set. Thanks. 2answers 329 views ### A graduate course on Sturm Liouville theory? Dear Math community, ... 1answer 152 views ### Convolution in \ell_p when 0<p<1 Background It is known that given real sequences a = (a_n)_{n \in \mathbb Z} \in \ell_p and b = (b_n)_{n \in \mathbb Z} \in \ell_q, their convolution defined as$$ a * b (n) = \sum_{k \in \mathbb ... 1answer 167 views ### Weierstrass Approximation Theorem with an additional condition Good afternoon: Let$f$be a continuous function defined on an closed interval$[a, b]\subset\mathbb R$. By Weierstrass Approximation Theorem, for any$\epsilon>0$, there is a polynomial$p$such ... 1answer 287 views ### Riemann-Lebesgue lemma for measures Riemann Lebesgue Lemma states that Fourier transform of an$L^1$function,$\hat{f}(\lambda)$is continuous and goes to zero as$|\lambda|\to \infty$. If$\mu$is a finite nonatomic measure then is it ... 1answer 259 views ### For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)? Let$(X,d,\mu)$be a metric measure space, i.e.$\mu$is a Borel measure on the metric space$(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ... 1answer 232 views ### Poincaré lemma in infinite dimensions Hi everyone, Is the PoincarÃ© lemma true in infinite dimensions? Here's a precise statement: Let$X$be a Banach (or maybe Hilbert) vector space,$U$a simply connected open set in$X$. Is it true ... 2answers 353 views ### Proving that a complicated function is eventually concave I have a function$f:\mathbb{R}^+ \to \mathbb{R}^+$that I want to prove is eventually concave - i.e. that there exists$\gamma _0 > 0$such that for every$\gamma>\gamma_0$,$f(\gamma)$is ... 0answers 145 views ### Continuity of the Shadow of a Nondecreasing Function So I'm working a lot with monotone nondecreasing functions$f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means$x < y \Rightarrow f(x) \leq ...
I am looking for as general a class as possible of real functions defined on $\mathbb{R}^+$ that are guaranteed to have a finite number of zeroes - no, polynomials are not enough :). Specifically, ...